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Theory of Computing Report

Tuesday, October 06

3SUM is false, even falser

from Emanuele Viola
I was just pointed to arxiv.org/abs/2610.06783v1 wow! Another item for my growing list of disproved conjectures in my book… told ya that P=NP! For fun I plugged the paper in chatgpt and the bounds can apparently already be improved. Here’s the model’s output: The method of Alman and Vassilevska Williams can be optimized to give […]

I was just pointed to https://arxiv.org/abs/2610.06783v1 wow! Another item for my growing list of disproved conjectures in my book… told ya that P=NP! For fun I plugged the paper in chatgpt and the bounds can apparently already be improved. Here’s the model’s output:

The method of Alman and Vassilevska Williams can be optimized to give O(n^{1.99601}) expected Las Vegas time and O(n^{1.99896}) deterministic time for 3SUM on integers of magnitude n^{O(1)}, on a word RAM with O(\log n)-bit words.

These bounds follow from optimizations of their algorithm. Their paper states a deterministic bound of O(n^{1.9992}):

https://arxiv.org/abs/2610.06783

The main improvement is to stop computing unused encodings.

1. Prune the encoding phase.

In the paper’s notation, D=4^m, the recursion has L levels, and

\displaystyle M=\binom{L}{m}9^{L-m}.

Every leaf contributing to a wanted output has at most m occurrences of the private term P_0. Retain only these leaves, forming a set T. When L=cm for a fixed constant c>10, we have |T|=O(M).

Their encodings can be computed in O(|T|) operations. After transforming k coordinates, each retained prefix has 9^{L-k} distinct completions in T, but only 7^{L-k} remaining input coordinates. Therefore the intermediate array has size at most

\displaystyle |T|(7/9)^{L-k}.

The retained prefixes are closed under taking parents, so these intermediate arrays suffice to compute every required encoding. Summing the geometric bound replaces the paper’s encoding cost by \widetilde O(N\sqrt M).

2. Use the exact binomial tail.

Set L=cm and t=\lceil\theta m\rceil. Write

\displaystyle H(x)=-x\ln x-(1-x)\ln(1-x).

Define

\displaystyle \Gamma=\frac{cH(1/c)-cH((1-\theta)/c)-\theta\ln9}{\ln4},

\displaystyle q=\frac{H(\theta)+\theta\ln9}{\ln4},

and

\displaystyle E=\frac{\frac12cH(1/c)+(c-1)\ln3}{\ln4}.

The same box algorithm now has preprocessing time

\displaystyle \widetilde O(N^2D^{-\Gamma}+ND^E)

and query time \widetilde O(D^q).

The sharper saving \Gamma follows from the paper’s exact count

\displaystyle \beta_d=\binom{L}{m-d}9^{L-m+d},

rather than its geometric estimate. Indeed, the tail beginning at t is bounded by a constant times \beta_t, and

\displaystyle \frac{\beta_t}{M}=D^{-\Gamma+o(1)}.

3. Apply the direct integer reduction.

The Kopelowitz–Pettie–Porat reduction produces set-disjointness instances with \widetilde O(n) sets, universe size D, and \widetilde O(n^2/\sqrt D) queries. Thus the total expected time becomes

\displaystyle \widetilde O\left(n^2D^{-\Gamma}+nD^E+n^2D^{q-1/2}\right).

The reduction verifies surviving candidates, so the algorithm always answers correctly; only its running time is random.

Choose c=21, \theta=14/125, and D=\Theta(n^{0.0575}). The three costs, including their polynomial exponents but suppressing logarithmic factors, are:

Boxes: n^{1.996007125}.

Encodings: n^{1.994729933}.

Queries: n^{1.996002337}.

Consequently, the logarithmic factors fit within O(n^{1.99601}).

The direct reduction is described here:

Click to access higher3sum.pdf

For the deterministic improvement, also preserve the actual middle dimension pk in the Exact Triangle reduction, where p is the modulus and k is the piece size. Taking c=22, \theta=0.116, p=\Theta(s^{0.028525}), and k=\Theta(s^{0.026435}) gives Exact Triangle time

\displaystyle s^{2.99791+o(1)}.

The paper’s deterministic 3SUM reduction then gives

\displaystyle n^{1.998955+o(1)}=O(n^{1.99896}).

The substantive change is the pruned encoding phase. The remaining gains come from sharper counting and reduction parameters.

By Manu

TR26-232 | Reed-Solomon Codes at Capacity: Algorithmic List-Decoding and Proximity Gaps | Prahladh Harsha, Mrinal Kumar, Ramprasad Saptharishi

from ECCC Papers
Understanding the limits of list-decodability of Reed-Solomon codes has been one of the most important open problems in algebraic coding theory. Recently, Brakensiek, Chen, Putterman, Zhang, and Zheng, in a remarkable breakthrough, showed that Reed-Solomon (RS) codes over fields of large characteristic are algorithmically list-decodable all the way up to capacity. Building on this result, Jeronimo subsequently extended these techniques to solve the proximity-gaps question for RS codes. In this article, we give a unified and transparent exposition of these results.
Understanding the limits of list-decodability of Reed-Solomon codes has been one of the most important open problems in algebraic coding theory. Recently, Brakensiek, Chen, Putterman, Zhang, and Zheng, in a remarkable breakthrough, showed that Reed-Solomon (RS) codes over fields of large characteristic are algorithmically list-decodable all the way up to capacity. Building on this result, Jeronimo subsequently extended these techniques to solve the proximity-gaps question for RS codes. In this article, we give a unified and transparent exposition of these results.

Too Big to Fail

from Ben Recht
The obsession with and futility of macroeconomic forecasting

Hi there, argmin readers! Today’s post is a live blog of Class 10 of my graduate seminar “Forecasting: A Critical Retrospective.” The syllabus and list of past posts are here.

All of my cybernetically inclined friends are into Friedrich Hayek, but my research and teaching keep bringing me back to John Maynard Keynes. These two gentlemen occupy the two poles of the dialectic of the American Experiment. Hayek famously introduced the notion of markets as distributed computers, where prices carry knowledge across the economy. Keynes, with his economic theory of central banking, set the stage for centralized computing of economic variables to govern those markets.

In Keynes’ paradigm-shifting 1936 work, The General Theory of Employment, Interest, and Money, he lays out an oxymoronic formula for central planning in capitalist societies.1 The basics of the Keynesian model are laid out in the appendix of Robert Evans’ paper from this week’s reading. The national economy has four key variables: the amount of investment firms make into the economy, the amount of savings firms accumulate in financial instruments or by paying down debt, the demand for money in the economy to facilitate purchases and sales, and the supply of money from the government. To change these variables, the government can enact various policies. For example, it can invest in infrastructure, raise taxes, increase the money supply, or change interest rates. Keynes argues that the government can dictate the economy’s output—and hence the general welfare of all citizens, who are players in the big macroeconomic game—by properly executing its policy apparatus.

This would set the stage for the subsequent 90 years of democratic capitalist monetary policy. The government has to make policy to ensure a well-run economy. To do this, it has to know the current state of national investment and savings. It also has to be able to forecast what these variables will be if no policy changes are enacted. This last requirement has driven the major investment in macroeconomic forecasting.

Forecasting in macroeconomics is thus primarily a tool of control. Here I mean control in the academic sense: the theory of dynamical systems with inputs and outputs and the design of subsystems to drive outputs to desired targets. Keynes casts the economy as a giant control problem, where the goal is to deftly change policy to ensure a particular state of economic output and employment. It should be no surprise that tools from control, notably the work of Rudolf Kalman on optimal filtering and control of linear systems with quadratic objectives, play a central role in macroeconomics.

Now, to filter and control, we need to make predictions. Keynes didn’t tell us how to generate those predictions. But his disciples, in what is often called “Keynesian” macroeconomic forecasting, write down structural equations of the economy, fit the parameters of these equations using varied means, and then make forecasts directly from the fitted models.

This program of prediction ran into several obstacles. First, it requires mathematical equations that predict all of the aspects of the economy needed to precisely determine optimal policies. Second, it requires a massive measurement system to pin down all the relevant factors in the model. Both were terribly daunting and required a great deal of expert judgment.

Economists want their methods to be “scientific,” since they influence decisions with major consequences. However, with so many variables, so little stationarity in economic conditions, and so much politics involved, building a truly objective and replicable system seems like a fool’s errand. Beatrice Cherrier details some of the typically arbitrary, political nature of macroeconomic sausage-making in this blog post. Evans details the amount of analytical flexibility and expert judgment forecasters necessarily employ in their predictive techniques.

Beyond these nuances of modeling and measurement, however, a fundamental problem of feedback control remains insurmountable. The models have a ton of parameters that are fit to historical data. Different policies yield different parameters. These parameters change when a policy changes. And you can’t predict what the parameters will be after a policy changes. So what on earth are we doing?

The critique in the previous paragraph was levied at macroeconomic forecasting by Robert Lucas in 1976. That macroeconomic forecasting is still an influential practice 50 years later certainly tells us something. Macroeconomic forecasters occupied positions of power and held a sense of civic duty. So they took Lucas’ critique as a challenge, not a reason to close up shop.

I’m not going to hash out the various attempts to build complex, nonparametric macroeconomic models that add ornate complexity while failing to dodge the fundamental problem. Theoretical critiques can carry only so much weight. The fact that the Great Recession was substantially caused by terrible financial policy and inadequate forecasting should have been the nail in the coffin. In 2003, Robert Lucas himself declared that macroeconomics had been a great success:

“My thesis in this lecture is that macroeconomics in this original sense has succeeded: Its central problem of depression prevention has been solved, for all practical purposes, and has in fact been solved for many decades.”

Oops.

Economists didn’t see a problem with the deep instability created by hyperfinancialization. Indeed, though Evans did his ethnographic research on macroeconomists a decade before the crash, his point rings true:

“...[E]conomic policy cannot be based on a quantitative calculus of costs and benefits and must, instead, rest on the considered judgement of a community of experts. Macroeconomic modellers may be those experts, but to expect anything more from them is to expect too much.”

So who should we trust? The Obama administration hoped economists could help get us out of the mess. But 8 years of attempted neoliberal patching of the American system ended in such broad dissatisfaction that… well, you know what happened. We’ve since had a decade of federal unrest as we try to unmoor ourselves from the expertise of economists. While I don’t believe the Biden and Trump administrations have found themselves a suitable alternative, I make no predictions about whose policy advice we’ll be deferring ot next.

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1

The quote from last week’s post was from a rebuttal Keynes wrote to critics of this book.

By Ben Recht

Designated Sets in Concurrent Simplex and Alpenglow

from Decentralized Thoughts
Recent consensus protocols like Alpenglow or Concurrent Simplex run two confirmation paths concurrently: a “slow” path with two voting rounds, each using a small quorum, and a “fast” path with a single voting round using a large quorum. The “slow” path can actually finish first because distances are not uniform in a geographically distributed system. Some parties are nearby, often on the same continent, while others are across an ocean....

By Ittai Abraham, Clément Burgelin, Antoine Murat, Joachim Neu

Recent consensus protocols like Alpenglow or Concurrent Simplex run two confirmation paths concurrently: a “slow” path with two voting rounds, each using a small quorum, and a “fast” path with a single voting round using a large quorum. The “slow” path can actually finish first because distances are not uniform in a geographically distributed system. Some parties are nearby, often on the same continent, while others are across an ocean....

By Ittai Abraham, Clément Burgelin, Antoine Murat, Joachim Neu

TR26-231 | Tarski Fixed Points in Quasi-FPT Queries | Xi Chen, Ruiquan Gao, Yuhao Li, Aviad Rubinstein, Mihalis Yannakakis

from ECCC Papers
We study the query complexity of finding a Tarski fixed point over $[n]^k$. Previous work has left a large gap between $\smash{{\Omega}(\log^2 n)}$ and $\smash{\log^{O(k)}n}$. We show that both of the previous upper and lower bounds were far from tight: for every $k\geq 3$, $$ \Omega\left((\log n)^{\frac{1}{2}\lceil \log k\rceil}\right)\le \textrm{Tarski}(n,k)\le O\left(5^k (\log n)^{\lceil \log k\rceil}\right).$$ Succinctly, up to the fixed-parameter factor of $5^k$, the complexity is settled at $(\log n)^{\Theta(\log k )}$. In particular, we obtain the first super-polynomial query lower bound for this problem.
We study the query complexity of finding a Tarski fixed point over $[n]^k$. Previous work has left a large gap between $\smash{{\Omega}(\log^2 n)}$ and $\smash{\log^{O(k)}n}$. We show that both of the previous upper and lower bounds were far from tight: for every $k\geq 3$, $$ \Omega\left((\log n)^{\frac{1}{2}\lceil \log k\rceil}\right)\le \textrm{Tarski}(n,k)\le O\left(5^k (\log n)^{\lceil \log k\rceil}\right).$$ Succinctly, up to the fixed-parameter factor of $5^k$, the complexity is settled at $(\log n)^{\Theta(\log k )}$. In particular, we obtain the first super-polynomial query lower bound for this problem.

Distributional Quantum Query Complexity

from arXiv: Computational Complexity

Authors: Shalev Ben-David, M. H. Ebtehaj

Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting $Q(f\circ g)=Θ(Q(f)Q(g))$ for all Boolean functions $f$ and $g$; a direct sum theorem asserting that computing $k$ copies of a function (or search problem) costs $Ω(k)$ times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires $Ω(k)$ times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution $μ$ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing $k$ copies of $f$ when the input comes from the product distribution $μ^k$ instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the $γ_2$ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.

Authors: Shalev Ben-David, M. H. Ebtehaj

Quantum query complexity enjoys a variety of pleasing joint computation properties: for example, a composition theorem asserting $Q(f\circ g)=Θ(Q(f)Q(g))$ for all Boolean functions $f$ and $g$; a direct sum theorem asserting that computing $k$ copies of a function (or search problem) costs $Ω(k)$ times as much as the cost of computing one copy; and a direct product theorem asserting that for Boolean functions, even succeeding at the direct sum problem with exponentially small probability still requires $Ω(k)$ times the cost of computing one copy to bounded error. However, all of these results are strictly for worst-case quantum query complexity. For example, if we have a fixed distribution $μ$ over inputs, the direct sum theorem says nothing about the quantum query complexity of computing $k$ copies of $f$ when the input comes from the product distribution $μ^k$ instead of being worst-case. (Note that while a standard Yao-type minimax theorem guarantees a hard distribution for the direct sum problem, there's no guarantee that this hard distribution is a product distribution.) A similar problem occurs for the direct product theorem and the composition theorem: none of these results respect distributions. In this work, we give distributional joint computation lower bounds for the composition, direct sum, and direct product problems. Along the way, we introduce some new tools for handling quantum query lower bounds, including (a) a new ``multiplicative'' variant of the $γ_2$ norm (which we use in place of the multiplicative adversary method for proving the direct product theorem), and (b) a new ``Shaltiel-free'' measure of quantum query complexity, which we show characterizes the composition behavior of distributional quantum query complexity and satisfies pleasing properties.

Natural proofs for quantum state preparation lower bounds

from arXiv: Computational Complexity

Authors: Christine Li, Natalie Parham

We identify a barrier that helps explain why proving stronger quantum state-preparation lower bounds has been so difficult. In particular, we establish a quantum analogue of the Razborov-Rudich natural proofs barrier for state-preparation lower bounds. We call a property of quantum states \emph{natural} if it holds for a sufficiently large fraction of Haar-random states and can be efficiently tested when given all of the state's amplitudes. Under a standard cryptographic assumption, we show that no natural property can prove superpolynomial state-preparation lower bounds even against a fixed level of the Magic Hierarchy. We show that several existing state-preparation lower-bound techniques are natural in our sense, including arguments based on approximate degree, not being a unique ground state of a local Hamiltonian, and mutual information.

Authors: Christine Li, Natalie Parham

We identify a barrier that helps explain why proving stronger quantum state-preparation lower bounds has been so difficult. In particular, we establish a quantum analogue of the Razborov-Rudich natural proofs barrier for state-preparation lower bounds. We call a property of quantum states \emph{natural} if it holds for a sufficiently large fraction of Haar-random states and can be efficiently tested when given all of the state's amplitudes. Under a standard cryptographic assumption, we show that no natural property can prove superpolynomial state-preparation lower bounds even against a fixed level of the Magic Hierarchy. We show that several existing state-preparation lower-bound techniques are natural in our sense, including arguments based on approximate degree, not being a unique ground state of a local Hamiltonian, and mutual information.

Characterizing Quantum Advantage for Generalizations of the Boolean Hidden Matching Problem

from arXiv: Computational Complexity

Authors: Mark Bun, Joao F. Doriguello, John Kallaugher, Nadezhda Voronova

We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $Ω(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \le 1$ and quantum protocols when $d \le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \ge 2$, the randomized classical communication complexity of this problem is $Θ(n^{1-1/d})$ while its quantum communication complexity lies between $Ω(n^{1-2/d})$ and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.

Authors: Mark Bun, Joao F. Doriguello, John Kallaugher, Nadezhda Voronova

We study the one-way communication complexity of the $f$-Boolean Hidden Partition problem. Here, Alice is given an $n$-bit string and Bob is given $Ω(n)$ disjoint blocks of its indices together with a string of labels. Under the promise that the evaluations of $f$ on these blocks either agree with all of the labels or disagree with all of them, Bob must determine which is the case using a single message from Alice. This problem generalizes the Boolean Hidden Matching and Hidden Hypermatching problems, which capture the special case where $f$ is the parity function. We establish classical and quantum communication bounds for $f$-Boolean Hidden Partition in terms of the sign degree $d$ of $f$, proving a conjecture of Doriguello and Montanaro (TQC 2020). Their prior work gave logarithmic communication upper bounds for classical protocols when $d \le 1$ and quantum protocols when $d \le 2$, as well as polynomial lower bounds for certain structured functions with larger sign degree. We show that for every $f$ of sign degree $d \ge 2$, the randomized classical communication complexity of this problem is $Θ(n^{1-1/d})$ while its quantum communication complexity lies between $Ω(n^{1-2/d})$ and $\bO{n^{1-1/\lceil d/2 \rceil} \log n}$. This completely characterizes the functions $f$ admitting polynomial quantum advantage as those for which $d \ge 2$, with a new infinite family of exponential separations given by the case $d = 2$.

Complexity separations for optimal matchgate-Clifford synthesis

from arXiv: Computational Complexity

Authors: Berta Casas, Diego García-Martín, Yuxuan Zhang

Clifford and matchgate circuits are canonical families of classically simulable quantum circuits. Their intersection, the matchgate-Clifford group, plays an important role in randomized fermionic protocols and in matchgate synthesis. Its adjoint action is isomorphic to the group of unit-determinant signed permutations of $2n$ Majorana modes, and we study optimal exact synthesis in this group. That is, given a target unitary and a gate set, output an $n$-qubit circuit implementing the target using the fewest operations. We show that the complexity of this problem strongly depends on the gate set. In particular, we study gate sets consisting of Majorana braids with different connectivity graphs. For path and complete graphs, we prove that the problem is classically solvable in $\mathcal{O}\left(n^2\right)$ time, and we provide explicit gate-optimal compilers. In addition, we prove that when the connectivity graph is a tree, the decision version of the optimal synthesis problem becomes NP-complete. Finally, we benchmark our optimal compiler on chains of up to $n=80$ qubits against those of \texttt{Qiskit} and \texttt{Tket}, obtaining circuits with constant factor improvements $\times2.57$ and $\times2.28$ in the total number of gates, respectively.

Authors: Berta Casas, Diego García-Martín, Yuxuan Zhang

Clifford and matchgate circuits are canonical families of classically simulable quantum circuits. Their intersection, the matchgate-Clifford group, plays an important role in randomized fermionic protocols and in matchgate synthesis. Its adjoint action is isomorphic to the group of unit-determinant signed permutations of $2n$ Majorana modes, and we study optimal exact synthesis in this group. That is, given a target unitary and a gate set, output an $n$-qubit circuit implementing the target using the fewest operations. We show that the complexity of this problem strongly depends on the gate set. In particular, we study gate sets consisting of Majorana braids with different connectivity graphs. For path and complete graphs, we prove that the problem is classically solvable in $\mathcal{O}\left(n^2\right)$ time, and we provide explicit gate-optimal compilers. In addition, we prove that when the connectivity graph is a tree, the decision version of the optimal synthesis problem becomes NP-complete. Finally, we benchmark our optimal compiler on chains of up to $n=80$ qubits against those of \texttt{Qiskit} and \texttt{Tket}, obtaining circuits with constant factor improvements $\times2.57$ and $\times2.28$ in the total number of gates, respectively.

Recognizers for Graph-Encoding Languages

from arXiv: Computational Complexity

Authors: Anssi Yli-Jyrä

We introduce recurrent incidence automata (RIAs), a new automaton model motivated by a decomposition of certain two-stack visibly pushdown computations. The decomposition separates vertex-local finite-state computations from recurrent one-stack interfaces connecting consecutive vertices. The construction is motivated by a two-stack visibly pushdown encoding of arbitrary ordered graphs whose strings admit a unique factorization into center-foldable vertex-local factors and whose auxiliary stack is empty at every factor boundary. Folding each factor into a sequence of pair symbols yields a local interface transformation. An RIA consists of a finite-state unit that computes these transformations and a recurrent layer that composes them across consecutive factors. Rather than manipulating an internal pushdown store, RIAs externalize long-range stack memory into recurrent interfaces between local computations. We show that nondeterministic RIA languages are closed under union, intersection, concatenation, Kleene-*, and reversal. Deterministic RIAs are closed under Boolean operations, although emptiness remains undecidable.

Authors: Anssi Yli-Jyrä

We introduce recurrent incidence automata (RIAs), a new automaton model motivated by a decomposition of certain two-stack visibly pushdown computations. The decomposition separates vertex-local finite-state computations from recurrent one-stack interfaces connecting consecutive vertices. The construction is motivated by a two-stack visibly pushdown encoding of arbitrary ordered graphs whose strings admit a unique factorization into center-foldable vertex-local factors and whose auxiliary stack is empty at every factor boundary. Folding each factor into a sequence of pair symbols yields a local interface transformation. An RIA consists of a finite-state unit that computes these transformations and a recurrent layer that composes them across consecutive factors. Rather than manipulating an internal pushdown store, RIAs externalize long-range stack memory into recurrent interfaces between local computations. We show that nondeterministic RIA languages are closed under union, intersection, concatenation, Kleene-*, and reversal. Deterministic RIAs are closed under Boolean operations, although emptiness remains undecidable.

Separating ClonableQMA and QCMA Relative to a Classical Oracle

from arXiv: Computational Complexity

Authors: Alper Cakan, Kai-Min Chung, Wei-Hsiang Hung, Tzu-Yi Yang

Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting. However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones. In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.

Authors: Alper Cakan, Kai-Min Chung, Wei-Hsiang Hung, Tzu-Yi Yang

Since the introduction of the complexity class QMA as a quantum-verifier analogue of NP (Kitaev, 1997), many have wondered whether quantum proofs are necessary or classical proofs suffice - that is, whether QMA = QCMA or QCMA != QMA (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC '07). This longstanding question was recently answered by works of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC '26) and Bostanci, Huang, and Vaikuntanathan (FOCS '26), which showed that quantum proofs are more powerful than classical ones in the classical-oracle setting. However, it remains unclear what exactly makes quantum proofs more powerful than classical ones. In the information-theoretic setting, a family of quantum states is not classicalizable if and only if it is unclonable. Indeed, these recent works also explicitly highlight the unclonability of their quantum proofs as a key mechanism behind their separations, and their arguments crucially rely on this property. This raises the question of whether unclonability is necessary for quantum proofs to be more powerful than classical ones. In this work, we show that even clonable quantum proofs can be more powerful than classical ones relative to a classical oracle by constructing a classical oracle O such that QCMA^O != ClonableQMA^O. This resolves the open question of Nehoran and Zhandry (ITCS '24), who established the analogous separation relative to a quantum oracle. We also show a classical-oracle separation between BQP/clonableqpoly and BQP/poly, and give applications of our results to quantum cryptography.

Subsequence Analysis Problems for Binary Parikh Matrices

from arXiv: Computational Complexity

Authors: Szilárd Zsolt Fazekas, Xinhao Huang, Robert Mercaş

The universality index corresponding to a word is the largest integer k such that every word of length k over the given alphabet occurs in it as a subsequence. Relative to languages this notion can be investigated with respect to both existential and universal quantifiers, with the former corresponding to the existence of a word in the language with universality index at least k, while the latter considers the index across all words that the language contains. In this work we study the existential (exists-universality) and universal (forall-universality) subsequence universality (as introduced in [Adamson et al., ISAAC 2023]) for binary languages consisting of all words with a fixed number of letters a, letters b, and subsequences ab. We prove that both exists- and forall-universality admit exact arithmetic characterizations for these languages. Extending the above notions, we end the paper by initiating the analysis of the probability of a fixed word occurring as subsequence of the words in such a language. To this end, we prove that, for every fixed pattern word w and an error tolerance, given as input the subsequence counts describing a language, the ratio of words in the language having the pattern w as a subsequence admits an additive approximation scheme that is polynomial-time in the binary encoding of the input counts.

Authors: Szilárd Zsolt Fazekas, Xinhao Huang, Robert Mercaş

The universality index corresponding to a word is the largest integer k such that every word of length k over the given alphabet occurs in it as a subsequence. Relative to languages this notion can be investigated with respect to both existential and universal quantifiers, with the former corresponding to the existence of a word in the language with universality index at least k, while the latter considers the index across all words that the language contains. In this work we study the existential (exists-universality) and universal (forall-universality) subsequence universality (as introduced in [Adamson et al., ISAAC 2023]) for binary languages consisting of all words with a fixed number of letters a, letters b, and subsequences ab. We prove that both exists- and forall-universality admit exact arithmetic characterizations for these languages. Extending the above notions, we end the paper by initiating the analysis of the probability of a fixed word occurring as subsequence of the words in such a language. To this end, we prove that, for every fixed pattern word w and an error tolerance, given as input the subsequence counts describing a language, the ratio of words in the language having the pattern w as a subsequence admits an additive approximation scheme that is polynomial-time in the binary encoding of the input counts.

Random-Oracle Unitary Synthesis is Impossible

from arXiv: Computational Complexity

Authors: Andrew Huang, Akshar Ramkumar, John Wright

A major open problem in quantum complexity is the question of unitary synthesis---namely, is it possible to efficiently implement any unitary, given the ability to evaluate any classical function in superposition? Inspired by a recent work by Brakerski and Yuen (CRYPTO 2026), we propose an ``average-case'' unitary synthesis problem, which asks whether it is possible to efficiently implement a Haar random unitary given access to a random Boolean function. We then demonstrate a superpolynomial query lower bound, establishing that unitary synthesis in this model is impossible. Along with our main result, we settle in the negative a question Brakerski and Yuen pose, of whether scalable pseudorandom unitaries (PRUs) can be implemented in the ROM-PRU model. In particular, we show that in the ROM-PRU model, no $Ω(N^{1+γ})$ unitary design on $n$ qubits, where $N=2^n$, can be efficiently achieved for any $γ> 0$. On the other hand, we construct an $O(N)$-design in $\text{poly}(n)$ queries in the same model, surpassing the previously best known results which constructed $O(\sqrt{N})$-designs.

Authors: Andrew Huang, Akshar Ramkumar, John Wright

A major open problem in quantum complexity is the question of unitary synthesis---namely, is it possible to efficiently implement any unitary, given the ability to evaluate any classical function in superposition? Inspired by a recent work by Brakerski and Yuen (CRYPTO 2026), we propose an ``average-case'' unitary synthesis problem, which asks whether it is possible to efficiently implement a Haar random unitary given access to a random Boolean function. We then demonstrate a superpolynomial query lower bound, establishing that unitary synthesis in this model is impossible. Along with our main result, we settle in the negative a question Brakerski and Yuen pose, of whether scalable pseudorandom unitaries (PRUs) can be implemented in the ROM-PRU model. In particular, we show that in the ROM-PRU model, no $Ω(N^{1+γ})$ unitary design on $n$ qubits, where $N=2^n$, can be efficiently achieved for any $γ> 0$. On the other hand, we construct an $O(N)$-design in $\text{poly}(n)$ queries in the same model, surpassing the previously best known results which constructed $O(\sqrt{N})$-designs.

Explicit Nonlinear Functions beyond the Fourier bound

from arXiv: Computational Complexity

Authors: Swastik Kopparty, Rishabh Kothary, Shanthanu S. Rai

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)> 0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = Θ(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $γ>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_γ(n)$, achieving $$ A(m,n) \leq (1 + γ)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{γ, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

Authors: Swastik Kopparty, Rishabh Kothary, Shanthanu S. Rai

We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb F_2^n \to \mathbb F_2^m$. Concretely, we want an $F$ and an $A = A(m,n)> 0$ as small as possible, so that for every affine map $L: \mathbb F_2^n \to \mathbb F_2^m$ (of the form $L(x) = M x + b $) we have: $$\mathrm{agree}(F, L) := |\{ x \in \mathbb F_2^n \mid F(x) = L(x) \}| \leq A.$$ Such questions have been studied by Nyberg (1991,1993), Carlet and Ding (2004,2007), Liu, Mesnager and Chen (2017), Nagy (2025), and Biryukov, Turecek, and Udovenko (2026). There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = Θ(2^{n-m} + 2^{n/2}),$$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $γ>0$, a function $F: \mathbb F_2^n \to \mathbb F_2^m$ with $m = O_γ(n)$, achieving $$ A(m,n) \leq (1 + γ)^n.$$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb F_2^n \to \mathbb F_2^m$, with $m = O_{γ, d}(n)$. Here the previously best bounds were of the form $A(m,n) = O( 2^{-\frac{n}{2^{d+1}}} \cdot 2^n )$ of Ben-Sasson and Kopparty (2010), based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb F_q$ in place of $\mathbb F_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

Unitary RQL Equals RQL

from arXiv: Computational Complexity

Authors: Quinten Tupker

Intermediate measurements let quantum computations discard information and reuse their workspace. Deferring all measurements can require storing their entire history, which need not preserve logarithmic space. Fefferman and Remscrim proved that measurements can nevertheless be eliminated with two-sided bounded error, and asked whether the same holds with one-sided error [FR21]. We prove RQLΓ = RQULΓ for the standard gate set Γ = {H, T, CNOT}, preserving polynomial time, logarithmic space, and exactly zero acceptance on no-instances. Our proof extends the density-matrix doubling method of Girish, Raz and Zhan [ GRZ21 ] to general channels, including resets and classical memory. We also prove measurement elimination for broader finite gate sets with exact inverses, including suitable gates with transcendental entries, and establish gate-set independence for a specified family of algebraic gate sets. A separate history-checking construction proves QMAL1,G = QUMAL1,G under explicit gate assumptions: measurements can also be eliminated from logarithmic-space quantum verification while preserving perfect completeness

Authors: Quinten Tupker

Intermediate measurements let quantum computations discard information and reuse their workspace. Deferring all measurements can require storing their entire history, which need not preserve logarithmic space. Fefferman and Remscrim proved that measurements can nevertheless be eliminated with two-sided bounded error, and asked whether the same holds with one-sided error [FR21]. We prove RQLΓ = RQULΓ for the standard gate set Γ = {H, T, CNOT}, preserving polynomial time, logarithmic space, and exactly zero acceptance on no-instances. Our proof extends the density-matrix doubling method of Girish, Raz and Zhan [ GRZ21 ] to general channels, including resets and classical memory. We also prove measurement elimination for broader finite gate sets with exact inverses, including suitable gates with transcendental entries, and establish gate-set independence for a specified family of algebraic gate sets. A separate history-checking construction proves QMAL1,G = QUMAL1,G under explicit gate assumptions: measurements can also be eliminated from logarithmic-space quantum verification while preserving perfect completeness

Fractal Gadgets for Neural Networks: The Complexity of the Narrow Regime

from arXiv: Computational Complexity

Authors: Olivier Bournez, Johanne Cohen, Laura Cohen, Adrian Wurm

We study the verification problem for deep narrow ReLU neural networks: given a network of bounded width computing a piecewise-affine map on [0,1], does some input satisfy a prescribed output constraint? Classical NP-hardness proofs for ReLU verification use one neuron per Boolean variable and say nothing about networks of small constant width, while width-1 networks are easy to verify. We show that verification of ReLU networks is NP-complete at width 4 for arbitrary inputs in [0,1]. When inputs are restricted to a natural discrete encoding set, NP-completeness already holds at width 3. Together with polynomial-time decidability at width 1, this leaves open only width 2 on the encoding set, and widths 2 and 3 on [0,1]. The technical core is a fractal preprocessing gadget: a width-2 ReLU subnetwork whose iterate vanishes precisely near a finite Cantor-like subset of [0,1] with 2^n points. It reduces verification of a continuous function on [0,1] to verification on 2^n discrete points without increasing the width, and is the missing ingredient for width-bounded hardness reductions. The same construction yields further results at width 3 on the encoding set: the universal problem is coNP-complete, counting zeros is #P-complete, a majority variant is PP-complete, and approximating the minimum output within a constant gap inherited from Max-3Sat is NP-hard. The NP, coNP and inapproximability results lift to all of [0,1] at width 4; lifting counting and majority, and lifting at width 3, remain open.

Authors: Olivier Bournez, Johanne Cohen, Laura Cohen, Adrian Wurm

We study the verification problem for deep narrow ReLU neural networks: given a network of bounded width computing a piecewise-affine map on [0,1], does some input satisfy a prescribed output constraint? Classical NP-hardness proofs for ReLU verification use one neuron per Boolean variable and say nothing about networks of small constant width, while width-1 networks are easy to verify. We show that verification of ReLU networks is NP-complete at width 4 for arbitrary inputs in [0,1]. When inputs are restricted to a natural discrete encoding set, NP-completeness already holds at width 3. Together with polynomial-time decidability at width 1, this leaves open only width 2 on the encoding set, and widths 2 and 3 on [0,1]. The technical core is a fractal preprocessing gadget: a width-2 ReLU subnetwork whose iterate vanishes precisely near a finite Cantor-like subset of [0,1] with 2^n points. It reduces verification of a continuous function on [0,1] to verification on 2^n discrete points without increasing the width, and is the missing ingredient for width-bounded hardness reductions. The same construction yields further results at width 3 on the encoding set: the universal problem is coNP-complete, counting zeros is #P-complete, a majority variant is PP-complete, and approximating the minimum output within a constant gap inherited from Max-3Sat is NP-hard. The NP, coNP and inapproximability results lift to all of [0,1] at width 4; lifting counting and majority, and lifting at width 3, remain open.

The Complexity of Computing Nash Equilibria in Colonel Blotto Games

from arXiv: Computational Complexity

Authors: Vasilis Pollatos, Andreas Kontogiannis

We study the complexity of Nash equilibrium computation in discrete Colonel Blotto games. For two-player general-sum Colonel Blotto with monotonic piecewise-constant battlefield payoffs, we show that computing an inverse-polynomial approximate Nash equilibrium is PPAD-hard, even with only two battlefields and a constant number of pieces per battlefield, thereby resolving an open question of [KPF+25]. Allowing more battlefields, the hardness persists when every battlefield payoff is $2$-piecewise constant with respect to either player's allocation. A general PPAD-membership theorem for multiplayer Blotto with arbitrary local reward functions then implies PPAD-completeness for both hardness results. Motivated by fixed-rank bimatrix games, we then identify a tractable frontier within two-player general-sum Colonel Blotto. If the social payoff has rank-$r$ structure, then, for every fixed $r$, an $\varepsilon$-Nash equilibrium can be computed in time polynomial in $B_1,B_2,k$, and $1/\varepsilon$. Thus constant-rank two-player Colonel Blotto remains tractable despite its exponentially large pure strategy spaces. Finally, we study multiplayer winner-takes-all Colonel Blotto with player-specific battlefield values and uncover a sharp tie-breaking frontier. If every highest bidder receives her full battlefield value, an exact pure Nash equilibrium can be computed in polynomial time for arbitrary binary-encoded budgets. In contrast, under ordinary equal splitting among highest bidders, we prove that computing an inverse-polynomial-accuracy Nash equilibrium is PPAD-hard even with only five players. This resolves an open question posed in concurrent work by Bichler and Ghosh [BG26], who established PPAD-hardness in the same player-specific equal-splitting setting when the number of players grows with the instance and asked whether hardness persists for a constant number of players.

Authors: Vasilis Pollatos, Andreas Kontogiannis

We study the complexity of Nash equilibrium computation in discrete Colonel Blotto games. For two-player general-sum Colonel Blotto with monotonic piecewise-constant battlefield payoffs, we show that computing an inverse-polynomial approximate Nash equilibrium is PPAD-hard, even with only two battlefields and a constant number of pieces per battlefield, thereby resolving an open question of [KPF+25]. Allowing more battlefields, the hardness persists when every battlefield payoff is $2$-piecewise constant with respect to either player's allocation. A general PPAD-membership theorem for multiplayer Blotto with arbitrary local reward functions then implies PPAD-completeness for both hardness results. Motivated by fixed-rank bimatrix games, we then identify a tractable frontier within two-player general-sum Colonel Blotto. If the social payoff has rank-$r$ structure, then, for every fixed $r$, an $\varepsilon$-Nash equilibrium can be computed in time polynomial in $B_1,B_2,k$, and $1/\varepsilon$. Thus constant-rank two-player Colonel Blotto remains tractable despite its exponentially large pure strategy spaces. Finally, we study multiplayer winner-takes-all Colonel Blotto with player-specific battlefield values and uncover a sharp tie-breaking frontier. If every highest bidder receives her full battlefield value, an exact pure Nash equilibrium can be computed in polynomial time for arbitrary binary-encoded budgets. In contrast, under ordinary equal splitting among highest bidders, we prove that computing an inverse-polynomial-accuracy Nash equilibrium is PPAD-hard even with only five players. This resolves an open question posed in concurrent work by Bichler and Ghosh [BG26], who established PPAD-hardness in the same player-specific equal-splitting setting when the number of players grows with the instance and asked whether hardness persists for a constant number of players.

Dimension Amplification for Tarski Fixed-Point Query Lower Bounds

from arXiv: Computational Complexity

Authors: Boyu Liu, Zihe Wang

We prove that finding a fixed point of a monotone map on the nine-dimensional grid $[N]^9$ requires $Ω((\log N)^3)$ deterministic queries, even when the fixed point is unique and each query returns the entire function value. The proof gives a construction that raises the dimension from $d$ to $4d+1$, preserves uniqueness, and adds a logarithmic factor to the lower bound. Iteration gives $Ω((\log N)^{r+2})$ queries in dimension $(7\cdot4^r-1)/3$, for every fixed nonnegative integer $r$, with an implicit constant that may depend on $r$. Consequently, no finite logarithmic exponent bounds the query complexity in all fixed dimensions.

Authors: Boyu Liu, Zihe Wang

We prove that finding a fixed point of a monotone map on the nine-dimensional grid $[N]^9$ requires $Ω((\log N)^3)$ deterministic queries, even when the fixed point is unique and each query returns the entire function value. The proof gives a construction that raises the dimension from $d$ to $4d+1$, preserves uniqueness, and adds a logarithmic factor to the lower bound. Iteration gives $Ω((\log N)^{r+2})$ queries in dimension $(7\cdot4^r-1)/3$, for every fixed nonnegative integer $r$, with an implicit constant that may depend on $r$. Consequently, no finite logarithmic exponent bounds the query complexity in all fixed dimensions.

Optimal and Verifiable Quantum Advantages in Communication Complexity

from arXiv: Computational Complexity

Authors: Ryan Anselm, Michelle Ding, Dar Gilboa, Sabee Grewal

We establish optimal quantum-classical separations in communication complexity for search problems. We introduce a total search problem called Pelagic Fourier Fishing and show that it admits an $n$-qubit quantum one-way protocol, whereas every randomized two-way protocol requires $Ω(2^n)$ bits of communication. We then introduce a variant of this problem whose solutions can be verified in polynomial time. This variant also admits an $n$-qubit quantum one-way protocol, while every randomized one-way protocol requires $Ω(2^n)$ bits of communication. We also construct a family of efficiently verifiable total search problems achieving an $n$ versus $Ω_d(n^d)$ separation between quantum one-way and randomized one-way communication for every fixed $d \ge 2$. In the quantum protocol, Alice prepares her message using a single unitary from the $d$th level of the Clifford hierarchy, and Bob performs a Clifford measurement. This separation is asymptotically optimal under this restriction on Alice's message. For $d = 2$, Alice's message is a stabilizer state and Bob's measurement is Clifford, so the protocol uses no magic, yet achieves an optimal quadratic quantum advantage. Finally, we discuss how these separations can be adapted to near-term quantum advantage experiments in which the demonstrated advantage is both unconditional and efficiently verifiable.

Authors: Ryan Anselm, Michelle Ding, Dar Gilboa, Sabee Grewal

We establish optimal quantum-classical separations in communication complexity for search problems. We introduce a total search problem called Pelagic Fourier Fishing and show that it admits an $n$-qubit quantum one-way protocol, whereas every randomized two-way protocol requires $Ω(2^n)$ bits of communication. We then introduce a variant of this problem whose solutions can be verified in polynomial time. This variant also admits an $n$-qubit quantum one-way protocol, while every randomized one-way protocol requires $Ω(2^n)$ bits of communication. We also construct a family of efficiently verifiable total search problems achieving an $n$ versus $Ω_d(n^d)$ separation between quantum one-way and randomized one-way communication for every fixed $d \ge 2$. In the quantum protocol, Alice prepares her message using a single unitary from the $d$th level of the Clifford hierarchy, and Bob performs a Clifford measurement. This separation is asymptotically optimal under this restriction on Alice's message. For $d = 2$, Alice's message is a stabilizer state and Bob's measurement is Clifford, so the protocol uses no magic, yet achieves an optimal quadratic quantum advantage. Finally, we discuss how these separations can be adapted to near-term quantum advantage experiments in which the demonstrated advantage is both unconditional and efficiently verifiable.

Strong Refutation for Random Quantum 3-SAT at Constant Density

from arXiv: Computational Complexity

Authors: Siu On Chan, Jeff Xu

We give a classical polynomial-time algorithm that strongly refutes random quantum $3$-SAT at sufficiently large constant constraint density, thereby disproving the quantum analogue of Feige's random $3$-SAT hypothesis. This stands in sharp contrast to classical random $3$-SAT, for which polynomial-time strong refutation is known only at constraint density $Δ\gtrsim n^{1/2}$. Although quantum $3$-SAT shares the pairwise-independence barrier of its classical counterpart, our SDP-based refutation overcomes this barrier by exploiting the noncommutativity of quantum constraints.

Authors: Siu On Chan, Jeff Xu

We give a classical polynomial-time algorithm that strongly refutes random quantum $3$-SAT at sufficiently large constant constraint density, thereby disproving the quantum analogue of Feige's random $3$-SAT hypothesis. This stands in sharp contrast to classical random $3$-SAT, for which polynomial-time strong refutation is known only at constraint density $Δ\gtrsim n^{1/2}$. Although quantum $3$-SAT shares the pairwise-independence barrier of its classical counterpart, our SDP-based refutation overcomes this barrier by exploiting the noncommutativity of quantum constraints.

A Dichotomy for Planar Graph Homomorphisms with Nonnegative Weights

from arXiv: Computational Complexity

Authors: Chenghua Liu, Boning Meng

We prove a complete complexity dichotomy for planar graph homomorphism counting with any fixed symmetric nonnegative matrix of arbitrary finite order, giving an explicit criterion for tractability. We also characterize exactly which fixed positive vertex weights preserve tractability, with both classifications extending from algebraic weights to fixed real weights in a prescribed exact representation. Our proof hinges on an entropy-based continuation argument: maximal logarithmic support identifies distance kernels as maximum-entropy completions, extending their positive definiteness throughout the parameter interval. This enables distance geometry to recover hidden product coordinates even when planar gadgets cannot distinguish colors; counting-hardness arguments then force the factors to be zero-field Boolean Ising interactions. The classification also yields complete tractability criteria for clock models, coupled Ising systems, and planar contractions of stoquastic imaginary-time kernels. All results have been formally verified in Lean 4.

Authors: Chenghua Liu, Boning Meng

We prove a complete complexity dichotomy for planar graph homomorphism counting with any fixed symmetric nonnegative matrix of arbitrary finite order, giving an explicit criterion for tractability. We also characterize exactly which fixed positive vertex weights preserve tractability, with both classifications extending from algebraic weights to fixed real weights in a prescribed exact representation. Our proof hinges on an entropy-based continuation argument: maximal logarithmic support identifies distance kernels as maximum-entropy completions, extending their positive definiteness throughout the parameter interval. This enables distance geometry to recover hidden product coordinates even when planar gadgets cannot distinguish colors; counting-hardness arguments then force the factors to be zero-field Boolean Ising interactions. The classification also yields complete tractability criteria for clock models, coupled Ising systems, and planar contractions of stoquastic imaginary-time kernels. All results have been formally verified in Lean 4.

On the Computational Complexity of Problems: Formalizing Sensitivity to Uncertainty and Parametric Complexity Classes

from arXiv: Computational Complexity

Authors: Yannis Tzitzikas

Is the classical question $P \stackrel{?}{=} NP$ the right one for understanding the complexity of real-world computation? Traditional worst-case analysis characterizes complexity solely as a function of input size $n$. Yet tasks whose cost is exponential in general often run in polynomial or even linear time once specific structural constraints or partial inputs are known. For instance, the Partition Problem is solvable in linear time, a constant-time core step following linear-time verification, when its inputs satisfy a simple structural property, although no polynomial-time algorithm is known for it in general; Integer Factorization, by contrast, resists structural knowledge: no checkable property of the input is known to improve on the sub-exponential number-field-sieve bound. Classical worst-case analysis cannot explain this difference, because it collapses a whole spectrum of difficulty into a single pessimistic bound. This paper formally develops a comprehensive parametric and uncertainty-aware framework for computational complexity. We introduce the notion of \emph{Sensitivity to Uncertainty} (SU) and formalize the cost variation $Δ_P(n, K)$ under a given input property $K$; we define the uncertainty-vanishing metric $Ψ(n, K)$ and prove a fundamental \emph{Uncertainty Reduction Theorem} bridging cost variation and input uncertainty; and we introduce the parametric classes $P_U[K]$ and $NP_U[K]$ as a more practical, instance-level foundation, focused on certifiable structure, rather than distributional typicality, that resolves the gap between theoretical complexity and real-world tractability. We further show how the same uncertainty analysis supports practice: it enables deterministic SLA via Design-by-Contract informs tool-use and reasoning splits in agentic AI, and provides an explanation of when quantum computers achieve exponential speedup.

Authors: Yannis Tzitzikas

Is the classical question $P \stackrel{?}{=} NP$ the right one for understanding the complexity of real-world computation? Traditional worst-case analysis characterizes complexity solely as a function of input size $n$. Yet tasks whose cost is exponential in general often run in polynomial or even linear time once specific structural constraints or partial inputs are known. For instance, the Partition Problem is solvable in linear time, a constant-time core step following linear-time verification, when its inputs satisfy a simple structural property, although no polynomial-time algorithm is known for it in general; Integer Factorization, by contrast, resists structural knowledge: no checkable property of the input is known to improve on the sub-exponential number-field-sieve bound. Classical worst-case analysis cannot explain this difference, because it collapses a whole spectrum of difficulty into a single pessimistic bound. This paper formally develops a comprehensive parametric and uncertainty-aware framework for computational complexity. We introduce the notion of \emph{Sensitivity to Uncertainty} (SU) and formalize the cost variation $Δ_P(n, K)$ under a given input property $K$; we define the uncertainty-vanishing metric $Ψ(n, K)$ and prove a fundamental \emph{Uncertainty Reduction Theorem} bridging cost variation and input uncertainty; and we introduce the parametric classes $P_U[K]$ and $NP_U[K]$ as a more practical, instance-level foundation, focused on certifiable structure, rather than distributional typicality, that resolves the gap between theoretical complexity and real-world tractability. We further show how the same uncertainty analysis supports practice: it enables deterministic SLA via Design-by-Contract informs tool-use and reasoning splits in agentic AI, and provides an explanation of when quantum computers achieve exponential speedup.

Separations with Immunity Relative to a Random Oracle in Computational Complexity

from arXiv: Computational Complexity

Authors: Gabriel Istrate

Rossman, Servedio, and Tan proved that the polynomial hierarchy is infinite relative to a random oracle. This paper strengthens this celebrated result by considering \emph{strong separations}, witnessed by immune languages. The main result of the paper shows that with respect to a random oracle the levels of the polynomial hierarchy separate with immunity. Specifically, with probability one over a random oracle $A$, for every $k\geq 1$ there is a language in $Σ_{k}^{P,A}$ that is immune to $Π_{k}^{P,A}$, and, symmetrically, a language in $Π_{k}^{P,A}$ that is immune to $Σ_{k}^{P,A}$. We thus extend classical results due to Bennett and Gill and Vereshchagin. We accomplish this by developing a "rare-or-wrong" approach to strong separations. We highlight the flexibility of the approach by strengthening other separations with respect to a random oracle from the complexity-theoretic literature.

Authors: Gabriel Istrate

Rossman, Servedio, and Tan proved that the polynomial hierarchy is infinite relative to a random oracle. This paper strengthens this celebrated result by considering \emph{strong separations}, witnessed by immune languages. The main result of the paper shows that with respect to a random oracle the levels of the polynomial hierarchy separate with immunity. Specifically, with probability one over a random oracle $A$, for every $k\geq 1$ there is a language in $Σ_{k}^{P,A}$ that is immune to $Π_{k}^{P,A}$, and, symmetrically, a language in $Π_{k}^{P,A}$ that is immune to $Σ_{k}^{P,A}$. We thus extend classical results due to Bennett and Gill and Vereshchagin. We accomplish this by developing a "rare-or-wrong" approach to strong separations. We highlight the flexibility of the approach by strengthening other separations with respect to a random oracle from the complexity-theoretic literature.

Training Variational Quantum Algorithms Is NP-Hard, Even Locally

from arXiv: Computational Complexity

Authors: Dax Enshan Koh, Triscia Mundo, Iosif Sakos, Antonios Varvitsiotis

Variational quantum algorithms (VQAs) generally rely on classical optimization to train parameterized quantum circuits. This training seeks to minimize an objective function, and its efficiency is central to the practical success of these algorithms. However, globally minimizing such training objectives over the circuit parameters is known to be $\mathsf{NP}$-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly $\mathsf{NP}$-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within $\ell_p$-distance strictly less than $π/2$ of some local minimizer, for every $p\geq 1$. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly $\mathsf{NP}$-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.

Authors: Dax Enshan Koh, Triscia Mundo, Iosif Sakos, Antonios Varvitsiotis

Variational quantum algorithms (VQAs) generally rely on classical optimization to train parameterized quantum circuits. This training seeks to minimize an objective function, and its efficiency is central to the practical success of these algorithms. However, globally minimizing such training objectives over the circuit parameters is known to be $\mathsf{NP}$-hard, limiting the prospect of general guarantees for efficient training. In this Letter, we prove that even the weaker task of finding a local minimum of such VQA training objectives is strongly $\mathsf{NP}$-hard, including when the objective admits efficient classical evaluation. Moreover, we show that this hardness persists even for the task of finding a parameter vector within $\ell_p$-distance strictly less than $π/2$ of some local minimizer, for every $p\geq 1$. Our central technical result is that approximating a local minimizer of a Hermitian trigonometric polynomial is strongly $\mathsf{NP}$-hard. By explicitly constructing quantum circuits whose training objectives reproduce these hard instances, we obtain a polynomial-time reduction to VQA training. Our results establish a fundamental computational barrier to variational quantum training: even reaching the vicinity of a local minimum remains hard in the worst case.

Gap Amplification for Local Hamiltonians with Combinatorial Soundness

from arXiv: Computational Complexity

Authors: Mitali Bafna, Quynh T. Nguyen, Tina Zhang

The quantum PCP conjecture is one of the major open problems in quantum complexity theory. It has resisted attack in part because many primitives used in the proof of the classical PCP theorem, such as locality-preserving gap amplification and alphabet reduction, have no obvious quantum analogues due to quantum no-cloning. Locality-preserving gap amplification is a procedure that takes as input a local Hamiltonian problem instance and produces a new instance with a larger promise gap, without increasing the locality of the Hamiltonian, and instead moderately increasing its local qudit dimension. Obtaining this kind of control over the locality during gap amplification is critical to the success of many known strategies for proving the classical PCP theorem. In this work, we put forth the first known viable template for quantum locality-preserving gap amplification, and we prove that our procedure amplifies the combinatorial gap of local Hamiltonians. Our work introduces a new framework for reasoning about quantum gap amplification in terms of fault-tolerant computation, and illuminates a route toward importing one of the central ingredients in classical PCPs into the quantum setting. In particular, we build upon ideas from the recent classical PCP of Bafna--Minzer--Vyas based on high-dimensional expanders, and the circuit-to-Hamiltonian construction of Anshu--Breuckmann--Nguyen, in addition to several recent advances in quantum coding theory.

Authors: Mitali Bafna, Quynh T. Nguyen, Tina Zhang

The quantum PCP conjecture is one of the major open problems in quantum complexity theory. It has resisted attack in part because many primitives used in the proof of the classical PCP theorem, such as locality-preserving gap amplification and alphabet reduction, have no obvious quantum analogues due to quantum no-cloning. Locality-preserving gap amplification is a procedure that takes as input a local Hamiltonian problem instance and produces a new instance with a larger promise gap, without increasing the locality of the Hamiltonian, and instead moderately increasing its local qudit dimension. Obtaining this kind of control over the locality during gap amplification is critical to the success of many known strategies for proving the classical PCP theorem. In this work, we put forth the first known viable template for quantum locality-preserving gap amplification, and we prove that our procedure amplifies the combinatorial gap of local Hamiltonians. Our work introduces a new framework for reasoning about quantum gap amplification in terms of fault-tolerant computation, and illuminates a route toward importing one of the central ingredients in classical PCPs into the quantum setting. In particular, we build upon ideas from the recent classical PCP of Bafna--Minzer--Vyas based on high-dimensional expanders, and the circuit-to-Hamiltonian construction of Anshu--Breuckmann--Nguyen, in addition to several recent advances in quantum coding theory.

Group Action and Spin Representation Lead to a Holant* Complexity Dichotomy on Domain Size 3

from arXiv: Computational Complexity

Authors: Jin-Yi Cai, Jin Soo Ihm

We prove a complexity dichotomy theorem for $\mathrm{Holant}^*$ problems over complex-valued symmetric constraint functions $\mathcal{F}$ on domain size $3$. We give a decidable tractability criterion and prove that if $\mathcal{F}$ satisfies the criterion, then $\mathrm{Holant}^*(\mathcal{F})$ is solvable in polynomial time, and otherwise it is #P-hard. This is the first Holant dichotomy for a set of complex-valued constraint functions on higher domains. We show that complex-valued constraint functions have a rich structure not observed in real-valued constraint functions. This structure is only revealed when we analyze them in a bipartite Holant setting with a non-standard bilinear form and provides the backbone to the proof of the dichotomy. We use group actions and the spin representation of $\mathrm{SL}(2, \mathbb{C})$ in $\mathrm{SO}(3, \mathbb{C})$ and the generalized orthogonal group to facilitate this analysis. We also introduce $\textit{frames}$ and $\textit{shells}$. Frames linearize the group action and provide a unified framework for proving #P-hardness when used in conjunction with shells. Furthermore, we characterize the lower dimensional constraint functions by $\textit{annihilators}$, which makes it possible to analyze $\textit{essentially Boolean domain}$ functions in domain size $3$.

Authors: Jin-Yi Cai, Jin Soo Ihm

We prove a complexity dichotomy theorem for $\mathrm{Holant}^*$ problems over complex-valued symmetric constraint functions $\mathcal{F}$ on domain size $3$. We give a decidable tractability criterion and prove that if $\mathcal{F}$ satisfies the criterion, then $\mathrm{Holant}^*(\mathcal{F})$ is solvable in polynomial time, and otherwise it is #P-hard. This is the first Holant dichotomy for a set of complex-valued constraint functions on higher domains. We show that complex-valued constraint functions have a rich structure not observed in real-valued constraint functions. This structure is only revealed when we analyze them in a bipartite Holant setting with a non-standard bilinear form and provides the backbone to the proof of the dichotomy. We use group actions and the spin representation of $\mathrm{SL}(2, \mathbb{C})$ in $\mathrm{SO}(3, \mathbb{C})$ and the generalized orthogonal group to facilitate this analysis. We also introduce $\textit{frames}$ and $\textit{shells}$. Frames linearize the group action and provide a unified framework for proving #P-hardness when used in conjunction with shells. Furthermore, we characterize the lower dimensional constraint functions by $\textit{annihilators}$, which makes it possible to analyze $\textit{essentially Boolean domain}$ functions in domain size $3$.

A Fine-Grained Dichotomy for Bounded-Variable Query Evaluation: The Calculus of Relations, a Boolean Modal Logic, and One-Variable Counting Logic

from arXiv: Computational Complexity

Authors: Yoshiki Nakamura, Yuya Uezato

We study the fine-grained complexity of evaluating Boolean bounded-variable first-order queries over sparse relational structures. For every fixed $k \ge 2$, every relational signature, and every fragment between $k$-variable primitive positive ($\mathrm{PP}^{k}$) and first-order ($\mathrm{FO}^{k}$) logic, we prove, assuming the Sparse MAX-$3$-SAT hypothesis, a dichotomy theorem for evaluation in $O(m^{k-\varepsilon})$ time, where $m$ is the number of tuples in the input structure. The only tractable cases fall into three families: (1) three-variable fragments, (2) two-variable fragments, and (3) fragments over unary signatures. On the hard side, for every fixed $k \ge 4$ and every $\varepsilon > 0$, there is a fixed sentence $\varphi_\varepsilon$ in $\mathrm{PP}^{k}$ over a single binary relation symbol, depending on $\varepsilon$ but not on the input structure, whose evaluation cannot be performed in $O(m^{k-\varepsilon})$ time. For the tractable cases, we show that the evaluation problem can be solved in $2^{O(|\varphi|)} \cdot m^{k-\varepsilon}$ time for some $\varepsilon > 0$. Moreover, every tractable fragment reduces, with a $2^{O(|\varphi|)}$ blowup in formula size under DAG representations, to one of the following query languages: (1) Tarski's calculus of relations, (2) a new Boolean modal logic for sparse model checking, and (3) one-variable counting logic.

Authors: Yoshiki Nakamura, Yuya Uezato

We study the fine-grained complexity of evaluating Boolean bounded-variable first-order queries over sparse relational structures. For every fixed $k \ge 2$, every relational signature, and every fragment between $k$-variable primitive positive ($\mathrm{PP}^{k}$) and first-order ($\mathrm{FO}^{k}$) logic, we prove, assuming the Sparse MAX-$3$-SAT hypothesis, a dichotomy theorem for evaluation in $O(m^{k-\varepsilon})$ time, where $m$ is the number of tuples in the input structure. The only tractable cases fall into three families: (1) three-variable fragments, (2) two-variable fragments, and (3) fragments over unary signatures. On the hard side, for every fixed $k \ge 4$ and every $\varepsilon > 0$, there is a fixed sentence $\varphi_\varepsilon$ in $\mathrm{PP}^{k}$ over a single binary relation symbol, depending on $\varepsilon$ but not on the input structure, whose evaluation cannot be performed in $O(m^{k-\varepsilon})$ time. For the tractable cases, we show that the evaluation problem can be solved in $2^{O(|\varphi|)} \cdot m^{k-\varepsilon}$ time for some $\varepsilon > 0$. Moreover, every tractable fragment reduces, with a $2^{O(|\varphi|)}$ blowup in formula size under DAG representations, to one of the following query languages: (1) Tarski's calculus of relations, (2) a new Boolean modal logic for sparse model checking, and (3) one-variable counting logic.

Strictly Unfriendly $k$-Partitions: Sharp Degree Thresholds and ETH-Based Lower Bounds

from arXiv: Computational Complexity

Authors: Sanjay Jain, Frank Stephan, Haoyun Tang

We present a complete complexity classification and fine-grained analysis for the Strictly Unfriendly $k$-Partition problem ($\text{SU}k\text{P}$), which asks whether the vertices of a graph can be partitioned into $k$ classes such that every vertex has strictly more neighbors in each of the other $k-1$ classes than in its own. We first establish a sharp tractability-intractability threshold with respect to the maximum degree $Δ$: for $k \in \{2, 3\}$, $\text{SU}k\text{P}$ is solvable in polynomial time when $Δ\le 2$, but becomes $\mathbf{NP}$-hard and ETH-hard immediately on subcubic graphs ($Δ= 3$), resolving the degree limitations in prior work and establishing subcubic graphs as the precise frontier of intractability. Furthermore, under the Exponential Time Hypothesis (ETH), we establish the first fine-grained lower bounds via direct reductions from $(3,3)$-SAT. On general graphs, we establish a uniform lower bound across all partition parameters $k \ge 2$, revealing a striking complexity convergence where the core exponential complexity remains invariant despite technical divergences in gadget constructions. On subcubic graphs, we formally quantify the "cost of sparsity," deriving explicit lower bound constants to demonstrate how enforced structural degree restrictions degrade reduction efficiency.

Authors: Sanjay Jain, Frank Stephan, Haoyun Tang

We present a complete complexity classification and fine-grained analysis for the Strictly Unfriendly $k$-Partition problem ($\text{SU}k\text{P}$), which asks whether the vertices of a graph can be partitioned into $k$ classes such that every vertex has strictly more neighbors in each of the other $k-1$ classes than in its own. We first establish a sharp tractability-intractability threshold with respect to the maximum degree $Δ$: for $k \in \{2, 3\}$, $\text{SU}k\text{P}$ is solvable in polynomial time when $Δ\le 2$, but becomes $\mathbf{NP}$-hard and ETH-hard immediately on subcubic graphs ($Δ= 3$), resolving the degree limitations in prior work and establishing subcubic graphs as the precise frontier of intractability. Furthermore, under the Exponential Time Hypothesis (ETH), we establish the first fine-grained lower bounds via direct reductions from $(3,3)$-SAT. On general graphs, we establish a uniform lower bound across all partition parameters $k \ge 2$, revealing a striking complexity convergence where the core exponential complexity remains invariant despite technical divergences in gadget constructions. On subcubic graphs, we formally quantify the "cost of sparsity," deriving explicit lower bound constants to demonstrate how enforced structural degree restrictions degrade reduction efficiency.

Nonuniform QCPH Collapse Implies QCPH Collapse

from arXiv: Computational Complexity

Authors: Jeremy Ahrens Huang

Despite the importance of the non-uniform Polynomial-Time Hierarchy (PH/poly) in classical complexity understanding the collapse conditions of the Polynomial-Time Hierarchy (PH), a quantum equivalent of PH/poly has yet to be studied in the literature. We introduce the non-uniform computational Quantum Polynomial-Time Hierarchy (QCPH/mpoly), the quantum equivalent of PH/poly, and show that it collapses if and only if QCPH, the quantum equivalent of PH introduced by Gharibian et al. (comput. complex. 2022), also collapses. We also show that QCPH collapses if coQCMA is contained in QCMA/mpoly. These results are analogous to those of Yap (TCS 1983) commonly used to invoke the collapse of PH in classical complexity. QCPH/mpoly is analogous to QCPH with non-uniform quantum verifier circuits.

Authors: Jeremy Ahrens Huang

Despite the importance of the non-uniform Polynomial-Time Hierarchy (PH/poly) in classical complexity understanding the collapse conditions of the Polynomial-Time Hierarchy (PH), a quantum equivalent of PH/poly has yet to be studied in the literature. We introduce the non-uniform computational Quantum Polynomial-Time Hierarchy (QCPH/mpoly), the quantum equivalent of PH/poly, and show that it collapses if and only if QCPH, the quantum equivalent of PH introduced by Gharibian et al. (comput. complex. 2022), also collapses. We also show that QCPH collapses if coQCMA is contained in QCMA/mpoly. These results are analogous to those of Yap (TCS 1983) commonly used to invoke the collapse of PH in classical complexity. QCPH/mpoly is analogous to QCPH with non-uniform quantum verifier circuits.

A Hand-Checkable Proof That Two Hidden ReLU Layers Compute the Maximum of Six Numbers

from arXiv: Computational Complexity

Authors: Dimitrios Myrisiotis

Exactly computing the maximum function is a standard test case for studying depth in ReLU networks. Two hidden layers are known to suffice for up to twelve inputs through computer-assisted constructions. For six real inputs, we give an explicit hexagon identity whose local structure yields a self-contained analytical proof of this depth bound. The identity was found by computer-assisted search; we prove it through explicit cancellations that can be checked entirely by hand, without executing a verification program. The identity also yields an explicit network with hidden widths $17$ and $41$, zero biases, and rational weights.

Authors: Dimitrios Myrisiotis

Exactly computing the maximum function is a standard test case for studying depth in ReLU networks. Two hidden layers are known to suffice for up to twelve inputs through computer-assisted constructions. For six real inputs, we give an explicit hexagon identity whose local structure yields a self-contained analytical proof of this depth bound. The identity was found by computer-assisted search; we prove it through explicit cancellations that can be checked entirely by hand, without executing a verification program. The identity also yields an explicit network with hidden widths $17$ and $41$, zero biases, and rational weights.

The sublevel Flood bifiltration: towards scalable 2-parameter persistent homology

from arXiv: Computational Geometry

Authors: Mattéo Clémot, Julie Digne, Julien Tierny

Multiparameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel offset bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

Authors: Mattéo Clémot, Julie Digne, Julien Tierny

Multiparameter persistent homology is a rapidly developing branch of topological data analysis that improves the robustness of single-parameter persistent homology to outliers, while still capturing the metric characteristics of the data. However, a notable limitation is its lack of scalability. In this paper, we introduce a novel approach for efficiently computing 2-parameter persistent homology on large point sets. Our work extends the Flood filtration, originally developed for single-parameter persistence. Our construction, called the sublevel Flood bifiltration, offers a scalable approximation of the sublevel offset bifiltration. We show that it benefits from theoretical stability properties and describe how to compute it efficiently. We demonstrate the performance of our approach in classification tasks on low-dimensional synthetic datasets, where density awareness is critical, as well as on real-world time series datasets.

Min-Max Uniform Circle Formation by Asynchronous Mobile Robots

from arXiv: Computational Geometry

Authors: Animesh Maiti, Prakhar Shukla, Subhash Bhagat

Given a set of point robots $\mathcal{R}$ in the Euclidean plane and a target circle $\mathbf C$ enclosing all robot positions, the \textsc{Min-Max Uniform Circle Formation (MMUCF)} problem requires the robots to move to distinct positions on $\mathbf C$ such that the final configuration forms a regular $n$-gon while minimizing the maximum distance traveled by any robot. Uniform circle formation is a fundamental coordination task in swarm robotics with applications in perimeter monitoring, surveillance, boundary coverage, and pattern formation. The literature does not address the optimization of the maximum individual displacement during the formation process. In this work, we study the min--max versions of the circle formation and uniform circle formation problems, where the goal is to minimize the maximum distance traveled by any robot. We consider these problems under the $\mathcal{ASYNC}$ model, where robots are autonomous, anonymous, identical, homogeneous, oblivious, and silent, and operate under the \textit{Look--Compute--Move} model with non-rigid motion. We first give necessary conditions for a deterministic solution and then present deterministic, distributed, and collision-free algorithms that form a circle and a uniform circle in finite time while minimizing the maximum movement. The algorithms ensure that robots reach distinct positions on the circle and, in the uniform case, equally spaced positions on $\mathbf C$ under the considered model.

Authors: Animesh Maiti, Prakhar Shukla, Subhash Bhagat

Given a set of point robots $\mathcal{R}$ in the Euclidean plane and a target circle $\mathbf C$ enclosing all robot positions, the \textsc{Min-Max Uniform Circle Formation (MMUCF)} problem requires the robots to move to distinct positions on $\mathbf C$ such that the final configuration forms a regular $n$-gon while minimizing the maximum distance traveled by any robot. Uniform circle formation is a fundamental coordination task in swarm robotics with applications in perimeter monitoring, surveillance, boundary coverage, and pattern formation. The literature does not address the optimization of the maximum individual displacement during the formation process. In this work, we study the min--max versions of the circle formation and uniform circle formation problems, where the goal is to minimize the maximum distance traveled by any robot. We consider these problems under the $\mathcal{ASYNC}$ model, where robots are autonomous, anonymous, identical, homogeneous, oblivious, and silent, and operate under the \textit{Look--Compute--Move} model with non-rigid motion. We first give necessary conditions for a deterministic solution and then present deterministic, distributed, and collision-free algorithms that form a circle and a uniform circle in finite time while minimizing the maximum movement. The algorithms ensure that robots reach distinct positions on the circle and, in the uniform case, equally spaced positions on $\mathbf C$ under the considered model.

Budget-Constrained Fault-Tolerant Mutual Visibility for Autonomous Robots under the Mobility Fault Model

from arXiv: Computational Geometry

Authors: Prakhar Shukla, Animesh Maiti, Shivam Kumar, Subhash Bhagat

We investigate the mutual visibility problem for a swarm of $n\ge3$ autonomous mobile robots under the budget-constrained mobility fault model. The robots are opaque, so if three robots are collinear, the middle robot obstructs the visibility between the other two. Each robot is assigned a finite movement budget, reflecting its limited energy, that bounds the total distance it may traverse during the execution. Moreover, an arbitrary number of robots may become permanently immobile due to mobility faults. The objective is to design a distributed algorithm that enables the non-faulty robots to coordinate their movements so that, within a finite time, every non-faulty robot attains unobstructed visibility of all robots in the system, including the faulty ones, while respecting the prescribed movement budget. We consider luminous robots operating under the $\mathsf{SSYNC}$ model with non-rigid movements, without any agreement on their local coordinate systems, and equipped only with a {\it common fixed reference point}. We present a deterministic distributed algorithm that solves the problem despite an arbitrary number of mobility faults. The algorithm guarantees mutual visibility for the non-faulty robots, respects the movement budget of every robot, provides collision-free movements for the robots, and uses only 12 light colors.

Authors: Prakhar Shukla, Animesh Maiti, Shivam Kumar, Subhash Bhagat

We investigate the mutual visibility problem for a swarm of $n\ge3$ autonomous mobile robots under the budget-constrained mobility fault model. The robots are opaque, so if three robots are collinear, the middle robot obstructs the visibility between the other two. Each robot is assigned a finite movement budget, reflecting its limited energy, that bounds the total distance it may traverse during the execution. Moreover, an arbitrary number of robots may become permanently immobile due to mobility faults. The objective is to design a distributed algorithm that enables the non-faulty robots to coordinate their movements so that, within a finite time, every non-faulty robot attains unobstructed visibility of all robots in the system, including the faulty ones, while respecting the prescribed movement budget. We consider luminous robots operating under the $\mathsf{SSYNC}$ model with non-rigid movements, without any agreement on their local coordinate systems, and equipped only with a {\it common fixed reference point}. We present a deterministic distributed algorithm that solves the problem despite an arbitrary number of mobility faults. The algorithm guarantees mutual visibility for the non-faulty robots, respects the movement budget of every robot, provides collision-free movements for the robots, and uses only 12 light colors.

Adaptive Bregman Alternating Projections for Feasible Gromov-Wasserstein Learning

from arXiv: Computational Geometry

Authors: Aoran Zhang, César A. Uribe

The Gromov-Wasserstein (GW) problem compares structured distributions without requiring a shared feature space or known correspondences, but its nonconvex objective and coupled marginal constraints make computation challenging. Bregman alternating projected gradient (BAPG) uses inexpensive alternating row and column updates, yet its fixed-penalty relaxation leaves a persistent feasibility gap. We propose Adaptive KL-BAPG (A-KL-BAPG), which combines a finite fixed-penalty burn-in with a guarded increasing-penalty phase. At each tail iteration, the method reuses BAPG's alternating updates and backtracks a delayed-power step until a Sinkhorn-inspired projective-diameter safeguard is satisfied. We prove finite termination of the backtracking at each iteration and show that the feasibility gap vanishes asymptotically. We further establish a best-iterate $O(1/\log N)$ bound for the weighted squared corrected residual and, under a support regularity condition, the existence of a stationary accumulation point for the original GW problem. This distinguishes A-KL-BAPG from fixed-penalty BAPG, whose stationarity guarantees are given for the relaxed problem. Experiments show that A-KL-BAPG achieves a favorable balance of accuracy, objective value, feasibility, and stationarity relative to BAPG variants, projection-based methods, and task-specific baselines. For synthetic and real graph alignment problems, it closely matches the accuracy and objective value of fixed-penalty KL-BAPG while reducing the marginal feasibility gap by 62-99% and the projected stationarity residual by 28-98%. Heterogeneous domain adaptation experiments show a similar pattern: A-KL-BAPG maintains comparable target accuracy and objective values while achieving better feasibility and stationarity than fixed-penalty KL-BAPG.

Authors: Aoran Zhang, César A. Uribe

The Gromov-Wasserstein (GW) problem compares structured distributions without requiring a shared feature space or known correspondences, but its nonconvex objective and coupled marginal constraints make computation challenging. Bregman alternating projected gradient (BAPG) uses inexpensive alternating row and column updates, yet its fixed-penalty relaxation leaves a persistent feasibility gap. We propose Adaptive KL-BAPG (A-KL-BAPG), which combines a finite fixed-penalty burn-in with a guarded increasing-penalty phase. At each tail iteration, the method reuses BAPG's alternating updates and backtracks a delayed-power step until a Sinkhorn-inspired projective-diameter safeguard is satisfied. We prove finite termination of the backtracking at each iteration and show that the feasibility gap vanishes asymptotically. We further establish a best-iterate $O(1/\log N)$ bound for the weighted squared corrected residual and, under a support regularity condition, the existence of a stationary accumulation point for the original GW problem. This distinguishes A-KL-BAPG from fixed-penalty BAPG, whose stationarity guarantees are given for the relaxed problem. Experiments show that A-KL-BAPG achieves a favorable balance of accuracy, objective value, feasibility, and stationarity relative to BAPG variants, projection-based methods, and task-specific baselines. For synthetic and real graph alignment problems, it closely matches the accuracy and objective value of fixed-penalty KL-BAPG while reducing the marginal feasibility gap by 62-99% and the projected stationarity residual by 28-98%. Heterogeneous domain adaptation experiments show a similar pattern: A-KL-BAPG maintains comparable target accuracy and objective values while achieving better feasibility and stationarity than fixed-penalty KL-BAPG.

Truly Subquadratic 3SUM and Truly Subcubic APSP via Triangles in Sparse Lopsided Graphs

from arXiv: Data Structures and Algorithms

Authors: Josh Alman, Virginia Vassilevska Williams

We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n^{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N^{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N^2/D^{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Schönhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n^{\varepsilon}$ vertices for $\varepsilon<0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

Authors: Josh Alman, Virginia Vassilevska Williams

We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n^{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix--Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N^{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N^2/D^{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Schönhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n^{\varepsilon}$ vertices for $\varepsilon<0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

An FPRAS for Counting Common Bases of Two Matroids

from arXiv: Data Structures and Algorithms

Authors: Xiaoyu Chen, Kuikui Liu

We design the first polynomial-time algorithms for approximately counting and almost uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the study of quantum many-body systems, and beyond. Our approach has two key ingredients. $\bullet$ We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "$\textit{soft intersection}$" satisfies a Poincare inequality uniformly over all external fields. $\bullet$ We solve a dual maximum entropy convex program to compute external fields under which the hard constraint is satisfied with high probability under the soft intersection measure. We bound this success probability directly using the uniform Poincare inequality and smallness of the gradient norm. $\textbf{AI Disclosure}$ GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper. A more complete discussion is included in the acknowledgments.

Authors: Xiaoyu Chen, Kuikui Liu

We design the first polynomial-time algorithms for approximately counting and almost uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the study of quantum many-body systems, and beyond. Our approach has two key ingredients. $\bullet$ We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "$\textit{soft intersection}$" satisfies a Poincare inequality uniformly over all external fields. $\bullet$ We solve a dual maximum entropy convex program to compute external fields under which the hard constraint is satisfied with high probability under the soft intersection measure. We bound this success probability directly using the uniform Poincare inequality and smallness of the gradient norm. $\textbf{AI Disclosure}$ GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper. A more complete discussion is included in the acknowledgments.

A Uniform Algorithm for Strict NP on Bounded-Treedepth Graphs

from arXiv: Data Structures and Algorithms

Authors: Thomas Depian, Robert Ganian, Jakob Greilhuber, Marlene Gründel, Simon Wietheger

A classical well-quasi-ordering result guarantees the existence of non-uniform linear-time algorithms for all problems in Strict NP on relational structures of bounded treedepth; however, this provides neither a procedure for constructing these algorithms nor computable bounds on their parameter dependence. We turn this existential result into a uniform algorithmic metatheorem. Given a Strict NP sentence $\varphi$ and a relational structure $\mathcal{R}$, our algorithm decides whether $\mathcal{R}\models\varphi$ in time $f(|\varphi|, td(\mathcal{R})) \cdot |\mathcal{R}|$ for a computable function $f$, where the treedepth of $\mathcal{R}$ is measured on the Gaifman graph. The algorithm also constructs witness relations, with the polynomial exponent depending on their arity, and provides a unified framework for settling hereditary graph problems parameterized by treedepth. We also present several applications - among others, our result resolves open questions on the fixed-parameter tractability of computing the stack number, queue number, track number and twin-width parameterized by treedepth.

Authors: Thomas Depian, Robert Ganian, Jakob Greilhuber, Marlene Gründel, Simon Wietheger

A classical well-quasi-ordering result guarantees the existence of non-uniform linear-time algorithms for all problems in Strict NP on relational structures of bounded treedepth; however, this provides neither a procedure for constructing these algorithms nor computable bounds on their parameter dependence. We turn this existential result into a uniform algorithmic metatheorem. Given a Strict NP sentence $\varphi$ and a relational structure $\mathcal{R}$, our algorithm decides whether $\mathcal{R}\models\varphi$ in time $f(|\varphi|, td(\mathcal{R})) \cdot |\mathcal{R}|$ for a computable function $f$, where the treedepth of $\mathcal{R}$ is measured on the Gaifman graph. The algorithm also constructs witness relations, with the polynomial exponent depending on their arity, and provides a unified framework for settling hereditary graph problems parameterized by treedepth. We also present several applications - among others, our result resolves open questions on the fixed-parameter tractability of computing the stack number, queue number, track number and twin-width parameterized by treedepth.

Robust subspace designs and the power of a unique small quantum witness

from arXiv: Data Structures and Algorithms

Authors: Simon Apers, Roman Edenhofer, Benjamin Mathieu-Bloise, Partha Mukhopadhyay

The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie close to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.Comput.Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As further applications, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $\mathsf{C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.

Authors: Simon Apers, Roman Edenhofer, Benjamin Mathieu-Bloise, Partha Mukhopadhyay

The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie close to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.Comput.Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As further applications, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $\mathsf{C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.

Matching with Multiple Bottlenecks: Parameterized Complexity and Approximation

from arXiv: Data Structures and Algorithms

Authors: Jonas Friemel, Tilo Hoitz, Phillip Keldenich, Arne Schmidt

We consider a matching problem in which the cost of each edge is a vector with $k$ components. The cost of a matching is the sum of the bottlenecks over all components, and we ask whether there is a perfect matching of cost at most some value $Z$. This type of matching has applications in heavily synchronized job-shop scheduling problems and in reconfiguration problems, where movement is restricted to a single direction per step. In this paper, we analyze the problem from a parameterized complexity perspective and provide various results including FPT-membership for parameters $k$ and $Z$ combined, as well as W[P]-membership and W[SAT]-hardness for each of the two parameters individually. The reduction also implies para-NP-hardness parameterized by either maximum degree or treewidth. We further show hardness of approximation within a super-logarithmic factor for the optimization variant and provide a $k/d$-approximation algorithm for any constant $d \leq k$ as well as an efficient approximation scheme parameterized by $k$. With parameter $Z$, we show that no FPT-time $F(Z)$-approximation algorithm is possible for any computable function $F$, unless W[1] = FPT.

Authors: Jonas Friemel, Tilo Hoitz, Phillip Keldenich, Arne Schmidt

We consider a matching problem in which the cost of each edge is a vector with $k$ components. The cost of a matching is the sum of the bottlenecks over all components, and we ask whether there is a perfect matching of cost at most some value $Z$. This type of matching has applications in heavily synchronized job-shop scheduling problems and in reconfiguration problems, where movement is restricted to a single direction per step. In this paper, we analyze the problem from a parameterized complexity perspective and provide various results including FPT-membership for parameters $k$ and $Z$ combined, as well as W[P]-membership and W[SAT]-hardness for each of the two parameters individually. The reduction also implies para-NP-hardness parameterized by either maximum degree or treewidth. We further show hardness of approximation within a super-logarithmic factor for the optimization variant and provide a $k/d$-approximation algorithm for any constant $d \leq k$ as well as an efficient approximation scheme parameterized by $k$. With parameter $Z$, we show that no FPT-time $F(Z)$-approximation algorithm is possible for any computable function $F$, unless W[1] = FPT.

W[1]-Hardness of Upper Clique Transversal

from arXiv: Data Structures and Algorithms

Authors: Pascal J. Gollin, Tesshu Hanaka, Ekkehard Köhler, Martin Milanič, Yushi Uno

A clique transversal of a graph is a set of vertices intersecting every maximal clique. We prove that deciding whether a graph has an inclusion-wise minimal clique transversal of size at least $k$ is W[1]-hard when parameterized by $k$.

Authors: Pascal J. Gollin, Tesshu Hanaka, Ekkehard Köhler, Martin Milanič, Yushi Uno

A clique transversal of a graph is a set of vertices intersecting every maximal clique. We prove that deciding whether a graph has an inclusion-wise minimal clique transversal of size at least $k$ is W[1]-hard when parameterized by $k$.

Quantum Submodular Maximization

from arXiv: Data Structures and Algorithms

Authors: Yonggang Jiang, Xiaoming Sun, Penghui Yao, Zekun Ye, Jialin Zhang, Zhijie Zhang

We study the quantum query complexity of maximizing a non-negative submodular function, considering both the unconstrained setting and, for monotone functions, a cardinality constraint $k$ on an $n$-element ground set. In the exact reversible digital value-oracle model, our unconstrained algorithm achieves an expected $(1/2-\varepsilon)$-approximation using only $O_\varepsilon(\log n)$ queries. In contrast, any classical randomized algorithm that attains a fixed expected ratio above $1/4$ requires $Ω(n/\log n)$ queries (Li, Feldman, Kazemi, and Karbasi, 2022), establishing an exponential separation in query complexity. For cardinality-constrained maximization, we give a bounded-error quantum algorithm that achieves a $(1-1/e-\varepsilon)$-approximation using $\widetilde O_\varepsilon(\min\{\sqrt n,n/k\})$ queries. When $k=o(n)$, our algorithm achieves at least a quadratic speedup up to logarithmic factors over classical randomized algorithms (Mirzasoleiman, Badanidiyuru, Karbasi, Vondrák, and Krause, 2015; Peng and Rubinstein, 2025). Moreover, when $k=cn$ for any fixed rational $c<1-1/e-\varepsilon$, the query complexity reduces to $O_{\varepsilon,c}(\log n)$, yielding an exponential separation from the classical $Ω(n/\log n)$ lower bound (Li, Feldman, Kazemi, and Karbasi, 2022). We further prove quantum lower bounds of $\exp(Ω(\varepsilon^2n))$ queries for achieving a ratio beyond $1/2+\varepsilon$ without constraints, and $\exp(Ω(\varepsilon^2k))$ queries for exceeding $1-1/e+\varepsilon$ when $k/n\le\varepsilon$. These barriers demonstrate that quantum computation offers no exponential speedup at these approximation thresholds.

Authors: Yonggang Jiang, Xiaoming Sun, Penghui Yao, Zekun Ye, Jialin Zhang, Zhijie Zhang

We study the quantum query complexity of maximizing a non-negative submodular function, considering both the unconstrained setting and, for monotone functions, a cardinality constraint $k$ on an $n$-element ground set. In the exact reversible digital value-oracle model, our unconstrained algorithm achieves an expected $(1/2-\varepsilon)$-approximation using only $O_\varepsilon(\log n)$ queries. In contrast, any classical randomized algorithm that attains a fixed expected ratio above $1/4$ requires $Ω(n/\log n)$ queries (Li, Feldman, Kazemi, and Karbasi, 2022), establishing an exponential separation in query complexity. For cardinality-constrained maximization, we give a bounded-error quantum algorithm that achieves a $(1-1/e-\varepsilon)$-approximation using $\widetilde O_\varepsilon(\min\{\sqrt n,n/k\})$ queries. When $k=o(n)$, our algorithm achieves at least a quadratic speedup up to logarithmic factors over classical randomized algorithms (Mirzasoleiman, Badanidiyuru, Karbasi, Vondrák, and Krause, 2015; Peng and Rubinstein, 2025). Moreover, when $k=cn$ for any fixed rational $c<1-1/e-\varepsilon$, the query complexity reduces to $O_{\varepsilon,c}(\log n)$, yielding an exponential separation from the classical $Ω(n/\log n)$ lower bound (Li, Feldman, Kazemi, and Karbasi, 2022). We further prove quantum lower bounds of $\exp(Ω(\varepsilon^2n))$ queries for achieving a ratio beyond $1/2+\varepsilon$ without constraints, and $\exp(Ω(\varepsilon^2k))$ queries for exceeding $1-1/e+\varepsilon$ when $k/n\le\varepsilon$. These barriers demonstrate that quantum computation offers no exponential speedup at these approximation thresholds.

Locality Sensitive Hashing for p-Exponential Kernels with Applications to Density Estimation

from arXiv: Data Structures and Algorithms

Authors: Barak Gorodissky, Tal Wagner

A kernel $k(x,y)$ is LSHable if there exists a locality sensitive hashing scheme $H$ such that $k(x,y)=\Pr_{h\sim H}[h(x)=h(y)]$ for all $x,y$. This notion plays a key role in efficient kernel methods in high dimensions. In this work, we show that the $p$-exponential kernel $k(x,y)=\exp(-\lVert x-y \rVert_p)$ is LSHable in bounded regions for all $1

Authors: Barak Gorodissky, Tal Wagner

A kernel $k(x,y)$ is LSHable if there exists a locality sensitive hashing scheme $H$ such that $k(x,y)=\Pr_{h\sim H}[h(x)=h(y)]$ for all $x,y$. This notion plays a key role in efficient kernel methods in high dimensions. In this work, we show that the $p$-exponential kernel $k(x,y)=\exp(-\lVert x-y \rVert_p)$ is LSHable in bounded regions for all $1

Improved Upper and Lower Bounds for Quantum Convex-Body Volume Estimation

from arXiv: Data Structures and Algorithms

Authors: Ruizhe Zhang

Estimating the volume of a high-dimensional convex body is a fundamental problem in theoretical computer science. Given a quantum membership oracle for a convex body $K\subset\mathbb{R}^d$, we show that estimating $\mathrm{vol}(K)$ to relative error $\varepsilon$ takes $\widetilde O(d^{5/2}+d^{3/2}/\varepsilon)$ queries. This improves the previous quantum upper bound $\widetilde O(d^3+d^{9/4}/\varepsilon)$ of Chakrabarti et al. (ACM TQC 2023) and Cornelissen--Hamoudi (SODA 2023), and achieves a larger quantum speedup over the currently best classical upper bound $\widetilde O(d^{7/2}+d^3/\varepsilon^2)$ of Jia et al. (JACM 2026). Our quantum algorithm combines two new ingredients: an input-dependent effective spectral-gap analysis of quantum walks based on uniform classical warm-start mixing bounds, and a single quantum estimator for products of normalizer ratios in simulated annealing based on classical bridge sampling. We also prove an $Ω(d)$ quantum query lower bound for constant relative error, improving the previous $Ω(\sqrt d)$ lower bound.

Authors: Ruizhe Zhang

Estimating the volume of a high-dimensional convex body is a fundamental problem in theoretical computer science. Given a quantum membership oracle for a convex body $K\subset\mathbb{R}^d$, we show that estimating $\mathrm{vol}(K)$ to relative error $\varepsilon$ takes $\widetilde O(d^{5/2}+d^{3/2}/\varepsilon)$ queries. This improves the previous quantum upper bound $\widetilde O(d^3+d^{9/4}/\varepsilon)$ of Chakrabarti et al. (ACM TQC 2023) and Cornelissen--Hamoudi (SODA 2023), and achieves a larger quantum speedup over the currently best classical upper bound $\widetilde O(d^{7/2}+d^3/\varepsilon^2)$ of Jia et al. (JACM 2026). Our quantum algorithm combines two new ingredients: an input-dependent effective spectral-gap analysis of quantum walks based on uniform classical warm-start mixing bounds, and a single quantum estimator for products of normalizer ratios in simulated annealing based on classical bridge sampling. We also prove an $Ω(d)$ quantum query lower bound for constant relative error, improving the previous $Ω(\sqrt d)$ lower bound.

Fair Diversity Maximization via Local Search

from arXiv: Data Structures and Algorithms

Authors: Mohammad Ansari, Sina Azizeddin, AmirMohammad Bandari, Pouria Mahmoudkhan, Hamid Zarabi-Zadeh

Diversity maximization is a fundamental optimization problem with applications in machine learning, data summarization, information retrieval, and recommendation systems. In many such applications, the data are partitioned into groups, and the selected subset must satisfy prescribed group quotas. We study Fair Diversity Maximization: given a set of points in a metric space partitioned into $m$ groups, the goal is to select exactly $k_i$ points from each group $i$ while maximizing the minimum pairwise distance among the selected points. The best previously known approximation guarantee is $m+1$, which grows linearly with the number of groups. We show that this dependence on $m$ is not fundamental. We present a new local-search framework that yields a $4$-approximation for any constant number of groups, with no restrictions on the metric space or on the size of the selected set. To the best of our knowledge, this is the first constant-factor approximation whose guarantee is independent of the number of groups in this general setting. Our framework maintains all group quotas exactly while progressively eliminating violations of the diversity objective. We further develop a specialized algorithm for two groups that achieves a $2$-approximation, improving the previous best factor of $3$. This factor is optimal: unless $\mathrm{P}=\mathrm{NP}$, no polynomial-time algorithm can achieve an approximation factor strictly better than $2$, even for the unconstrained case.

Authors: Mohammad Ansari, Sina Azizeddin, AmirMohammad Bandari, Pouria Mahmoudkhan, Hamid Zarabi-Zadeh

Diversity maximization is a fundamental optimization problem with applications in machine learning, data summarization, information retrieval, and recommendation systems. In many such applications, the data are partitioned into groups, and the selected subset must satisfy prescribed group quotas. We study Fair Diversity Maximization: given a set of points in a metric space partitioned into $m$ groups, the goal is to select exactly $k_i$ points from each group $i$ while maximizing the minimum pairwise distance among the selected points. The best previously known approximation guarantee is $m+1$, which grows linearly with the number of groups. We show that this dependence on $m$ is not fundamental. We present a new local-search framework that yields a $4$-approximation for any constant number of groups, with no restrictions on the metric space or on the size of the selected set. To the best of our knowledge, this is the first constant-factor approximation whose guarantee is independent of the number of groups in this general setting. Our framework maintains all group quotas exactly while progressively eliminating violations of the diversity objective. We further develop a specialized algorithm for two groups that achieves a $2$-approximation, improving the previous best factor of $3$. This factor is optimal: unless $\mathrm{P}=\mathrm{NP}$, no polynomial-time algorithm can achieve an approximation factor strictly better than $2$, even for the unconstrained case.

CV-QAOA: Efficient Low-Depth Quantum Optimization of Continuous Variables

from arXiv: Data Structures and Algorithms

Authors: Sriram Bharadwaj, Di Luo, Leo Zhou

We study a Continuous-Variable Quantum Approximate Optimization Algorithm (CV-QAOA) for high-dimensional continuous optimization. Our formulation extends an earlier CV-QAOA proposal with a variationally optimized initial state and recovers the convergence guarantees of Quantum Hamiltonian Descent (QHD) in the high-depth limit. We prove rigorous performance guarantees of CV-QAOA on several families of cost functions. First, we show $d$-step CV-QAOA minimizes any $d$-dimensional strictly convex quadratic function with $2d$ quantum queries to the cost function. We then analyze a family of nonconvex "Rotated Double Well" (RDW) functions with $2^d$ local minima introduced by arXiv:2311.00811. While prior work showed QHD reaches its global minimum with $\tilde O(d^3)$ queries, we prove that 1-step CV-QAOA solves RDW with just two quantum queries. Although general-purpose classical solvers need superpolynomial time for RDW and structure-awareness can reduce the cost to polynomial time, we show that the 1-step CV-QAOA protocol can be efficiently dequantized, and that a gradient-aligned line search succeeds with $O(d)$ queries, nearly matching the information-theoretic $Ω(d/\log d)$ query lower bound. To move beyond the dequantizable regime, we introduce a ``Rotated Square Well'' (RSW) problem, whose globally flat landscape suppresses useful local gradient information. For this family, we show that an adiabatic evolution simulated by CV-QAOA can reach the global minimum using $d^{o(1)}$ queries. On the other hand, any classical algorithm that learn the hidden rotation in RSW provably requires $Ω(d^2/\log d)$ queries, a bound we nearly match with an explicit $Θ(d^2\log d)$-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.

Authors: Sriram Bharadwaj, Di Luo, Leo Zhou

We study a Continuous-Variable Quantum Approximate Optimization Algorithm (CV-QAOA) for high-dimensional continuous optimization. Our formulation extends an earlier CV-QAOA proposal with a variationally optimized initial state and recovers the convergence guarantees of Quantum Hamiltonian Descent (QHD) in the high-depth limit. We prove rigorous performance guarantees of CV-QAOA on several families of cost functions. First, we show $d$-step CV-QAOA minimizes any $d$-dimensional strictly convex quadratic function with $2d$ quantum queries to the cost function. We then analyze a family of nonconvex "Rotated Double Well" (RDW) functions with $2^d$ local minima introduced by arXiv:2311.00811. While prior work showed QHD reaches its global minimum with $\tilde O(d^3)$ queries, we prove that 1-step CV-QAOA solves RDW with just two quantum queries. Although general-purpose classical solvers need superpolynomial time for RDW and structure-awareness can reduce the cost to polynomial time, we show that the 1-step CV-QAOA protocol can be efficiently dequantized, and that a gradient-aligned line search succeeds with $O(d)$ queries, nearly matching the information-theoretic $Ω(d/\log d)$ query lower bound. To move beyond the dequantizable regime, we introduce a ``Rotated Square Well'' (RSW) problem, whose globally flat landscape suppresses useful local gradient information. For this family, we show that an adiabatic evolution simulated by CV-QAOA can reach the global minimum using $d^{o(1)}$ queries. On the other hand, any classical algorithm that learn the hidden rotation in RSW provably requires $Ω(d^2/\log d)$ queries, a bound we nearly match with an explicit $Θ(d^2\log d)$-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.

Finding Gaussian Structure in Bosonic States

from arXiv: Data Structures and Algorithms

Authors: Alvan Arulandu, Sitan Chen, Ziyun Chen, Jerry Li, Eric Ma

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $ρ$, the goal is to output a pure Gaussian state whose infidelity with $ρ$ is at most $\mathrm{opt} + ε$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/ε$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + ε$ or $\mathrm{opt} < c - ε$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/ε)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

Authors: Alvan Arulandu, Sitan Chen, Ziyun Chen, Jerry Li, Eric Ma

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $ρ$, the goal is to output a pure Gaussian state whose infidelity with $ρ$ is at most $\mathrm{opt} + ε$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/ε$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/ε)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + ε$ or $\mathrm{opt} < c - ε$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/ε)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

Quantum 1-PCA with Pauli Measurements in Nearly Linear Time

from arXiv: Data Structures and Algorithms

Authors: Dutch Hansen, Jerry Li

We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $λ$ and spectral gap at least $Δ> 0$, we give an algorithm that recovers the leading eigenvector to fidelity at least $1 - \varepsilon$ with high probability using $\tilde{O}\left( {2^n \cdot η^2} / {Δ^3 \varepsilon^3}\right)$ copies and $\tilde{O}((2^n/Δ\varepsilon) \cdot \operatorname{poly}(η/Δ\varepsilon))$ time, where $η= \max (1 - λ, \varepsilon)$. All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. $\varepsilon = Ω(η)$, our runtime and copy complexity become $\tilde{O} (2^n / \varepsilon)$. This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption $η= 0$, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.

Authors: Dutch Hansen, Jerry Li

We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $λ$ and spectral gap at least $Δ> 0$, we give an algorithm that recovers the leading eigenvector to fidelity at least $1 - \varepsilon$ with high probability using $\tilde{O}\left( {2^n \cdot η^2} / {Δ^3 \varepsilon^3}\right)$ copies and $\tilde{O}((2^n/Δ\varepsilon) \cdot \operatorname{poly}(η/Δ\varepsilon))$ time, where $η= \max (1 - λ, \varepsilon)$. All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. $\varepsilon = Ω(η)$, our runtime and copy complexity become $\tilde{O} (2^n / \varepsilon)$. This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption $η= 0$, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.

Polynomial-time classical algorithms for mean-field models up to the glass transition

from arXiv: Data Structures and Algorithms

Authors: Alexander Schmidhuber, Alexander Zlokapa

The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.

Authors: Alexander Schmidhuber, Alexander Zlokapa

The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.

Approximating Random Walks in $\widetilde{O}(\log n + \log^2 κ)$ Space for $κ$-Conditioned Graphs

from arXiv: Data Structures and Algorithms

Authors: Junzhao Yang

For $κ>1$, a directed graph is $κ$-conditioned if it is $κ$-mixing and its stationary distribution is approximated by the uniform distribution within a factor of $κ$. We present a deterministic algorithm that approximates the stationary distribution of a $κ$-conditioned graph to inverse polynomial relative error in $O((\log n + \log^2 κ) \log \log n)$ space. In the regime $κ= \exp(Θ(\log^αn))$ for any $α\in (0, 2/3)$, our result improves the best-known $O(\log n \sqrt{\log κ} / \sqrt{\log \log n})$ space bound for approximating $κ$-step random walks in general directed graphs by [Hoza, RANDOM 2021]. We release this preliminary version due to recent rumors of LLM-based progress on related problems and uncertainty about when those results may appear. Further implementation details will be provided in a subsequent version.

Authors: Junzhao Yang

For $κ>1$, a directed graph is $κ$-conditioned if it is $κ$-mixing and its stationary distribution is approximated by the uniform distribution within a factor of $κ$. We present a deterministic algorithm that approximates the stationary distribution of a $κ$-conditioned graph to inverse polynomial relative error in $O((\log n + \log^2 κ) \log \log n)$ space. In the regime $κ= \exp(Θ(\log^αn))$ for any $α\in (0, 2/3)$, our result improves the best-known $O(\log n \sqrt{\log κ} / \sqrt{\log \log n})$ space bound for approximating $κ$-step random walks in general directed graphs by [Hoza, RANDOM 2021]. We release this preliminary version due to recent rumors of LLM-based progress on related problems and uncertainty about when those results may appear. Further implementation details will be provided in a subsequent version.

Random Order in Quantum Streaming: Replenishment and Robust Lower Bounds

from arXiv: Data Structures and Algorithms

Authors: Nadezhda Voronova

How can random order change the role of quantum memory in streaming? Later classical input can restore the usefulness of a quantum state consumed by earlier queries. We call this replenishment. We construct an artificial problem based on Hidden Matching, with repeated coordinate data and online matching requests. It admits a one-pass quantum algorithm using polylogarithmic space in uniformly random order, but unconditionally requires polynomial space both classically in random order and quantumly when all updates precede the requests. To prove the quantum lower bound, we strengthen the consumability bounds of Gilboa, Jain, and McClean for Multiple Hidden Matching. Without prior entanglement, any quantum encoding supporting $r$ independent matching requests requires $Ω(r)$ qubits for $r\le N^{1/2-δ}$ and every fixed $δ\in(0,1/2)$, even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through $r=Θ(\sqrt N)$. We also adapt Kallaugher's triangle-counting algorithm to uniformly random streams in which every edge is repeated equally often. Rebuilding the quantum sketch and resampling the classical estimator improve its expected-space bound in suitable parameter regimes. Finally, we extend the robust Noisy Gap Cycle framework of Assadi and Sundaresan to quantum streaming. A quantum communication lower bound for Block Hidden XOR yields an $Ω(n)$ space lower bound in random edge order for large enough cycles, with consequences for several graph problems. Thus random order can enable replenishment of small quantum representations, while substantial space requirements persist for other tasks.

Authors: Nadezhda Voronova

How can random order change the role of quantum memory in streaming? Later classical input can restore the usefulness of a quantum state consumed by earlier queries. We call this replenishment. We construct an artificial problem based on Hidden Matching, with repeated coordinate data and online matching requests. It admits a one-pass quantum algorithm using polylogarithmic space in uniformly random order, but unconditionally requires polynomial space both classically in random order and quantumly when all updates precede the requests. To prove the quantum lower bound, we strengthen the consumability bounds of Gilboa, Jain, and McClean for Multiple Hidden Matching. Without prior entanglement, any quantum encoding supporting $r$ independent matching requests requires $Ω(r)$ qubits for $r\le N^{1/2-δ}$ and every fixed $δ\in(0,1/2)$, even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through $r=Θ(\sqrt N)$. We also adapt Kallaugher's triangle-counting algorithm to uniformly random streams in which every edge is repeated equally often. Rebuilding the quantum sketch and resampling the classical estimator improve its expected-space bound in suitable parameter regimes. Finally, we extend the robust Noisy Gap Cycle framework of Assadi and Sundaresan to quantum streaming. A quantum communication lower bound for Block Hidden XOR yields an $Ω(n)$ space lower bound in random edge order for large enough cycles, with consequences for several graph problems. Thus random order can enable replenishment of small quantum representations, while substantial space requirements persist for other tasks.

Near-Optimal Oracle Bounds for Isotropic Rounding

from arXiv: Data Structures and Algorithms

Authors: Yang P. Liu, Richard Peng, Alicia Stepin, Colin Tang

We give optimal bounds, up to polylogarithmic factors, for rounding convex bodies to near-isotropic position. A convex body $B(0,r)\subseteq K\subseteq B(0,R)$ in $\R^n$ can be rounded using $\Ot(n^3)$ membership queries; the upper bound extends to logconcave distributions. We prove a matching $\Omegat(n^3)$ lower bound, even when $R/r=n^{O(1)}$. The key lemma states that if $B(0,1)\subseteq K$ and $\Cov(\Unif(K))\preceqκI_n$, with $κ\ge1$, then approximate uniform sampling from an initial density bounded by a constant times the uniform density uses $\Ot(n^2\sqrtκ)$ expected membership queries. We prove this by simulating reflected kinetic dynamics using the analysis of Eberle and Lörler~\cite{EL26:journal}. Combining this sampler with the ideas of Jia, Laddha, Lee, and Vempala~\cite{JLLV26:journal} for rounding well-rounded bodies yields our $\Ot(n^3)$ query rounding algorithm. We also give a counterexample to an ellipsoid-growth conjecture from previous papers on this topic.

Authors: Yang P. Liu, Richard Peng, Alicia Stepin, Colin Tang

We give optimal bounds, up to polylogarithmic factors, for rounding convex bodies to near-isotropic position. A convex body $B(0,r)\subseteq K\subseteq B(0,R)$ in $\R^n$ can be rounded using $\Ot(n^3)$ membership queries; the upper bound extends to logconcave distributions. We prove a matching $\Omegat(n^3)$ lower bound, even when $R/r=n^{O(1)}$. The key lemma states that if $B(0,1)\subseteq K$ and $\Cov(\Unif(K))\preceqκI_n$, with $κ\ge1$, then approximate uniform sampling from an initial density bounded by a constant times the uniform density uses $\Ot(n^2\sqrtκ)$ expected membership queries. We prove this by simulating reflected kinetic dynamics using the analysis of Eberle and Lörler~\cite{EL26:journal}. Combining this sampler with the ideas of Jia, Laddha, Lee, and Vempala~\cite{JLLV26:journal} for rounding well-rounded bodies yields our $\Ot(n^3)$ query rounding algorithm. We also give a counterexample to an ellipsoid-growth conjecture from previous papers on this topic.

Optimal compression with quantum retrieval

from arXiv: Data Structures and Algorithms

Authors: Shyam Dhamapurkar, Mohit Garg, Manaswi Paraashar, Jaikumar Radhakrishnan

We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.

Authors: Shyam Dhamapurkar, Mohit Garg, Manaswi Paraashar, Jaikumar Radhakrishnan

We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.

Faster high-accuracy multicommodity flow in dense graphs

from arXiv: Data Structures and Algorithms

Authors: Chenxin Dai, Alicia Stepin, Colin Tang

We give a fast algorithm for solving min-cost $k$-commodity flow. The basic idea is to construct an auxiliary linear program that has low rank and whose minimum value is at most $1/k$ times the minimum value of the original problem (thus, solving this auxiliary linear program will make at least $1/k$ fraction of progress in the original problem). Low-rank linear programs can be solved quickly using black-box techniques. Thus, our algorithm runs in time $\tilde{O}(\operatorname{poly}(k)(n^{2.5}+m\sqrt{n}))$ on a directed graph with $n$ vertices and $m$ edges. We do not rely on any fast matrix multiplication.

Authors: Chenxin Dai, Alicia Stepin, Colin Tang

We give a fast algorithm for solving min-cost $k$-commodity flow. The basic idea is to construct an auxiliary linear program that has low rank and whose minimum value is at most $1/k$ times the minimum value of the original problem (thus, solving this auxiliary linear program will make at least $1/k$ fraction of progress in the original problem). Low-rank linear programs can be solved quickly using black-box techniques. Thus, our algorithm runs in time $\tilde{O}(\operatorname{poly}(k)(n^{2.5}+m\sqrt{n}))$ on a directed graph with $n$ vertices and $m$ edges. We do not rely on any fast matrix multiplication.

A Decremental Algorithm for Checking the Possibility of Braess Paradox in Dynamic Nets

from arXiv: Data Structures and Algorithms

Authors: Dario Fiorenza, Daniele Gorla, Ivano Salvo

Braess paradox originates when latency at Wardrop equilibrium in traffic networks decreases because of removing edges. The graph-theoretic property of networks suffering from the Braess paradox was called vulnerability by Roughgarden in 2006; it was then characterized and algorithmically checked both for undirected and for directed nets. In this paper, we provide a decremental algorithm of linear amortized complexity to check vulnerability for dynamically evolving networks. The basic idea of our dynamic algorithm is to use a static algorithm that marks some edges as irrelevant for the vulnerability of the graph and ignore those edges for all subsequent runs of the decremental procedure. To get a linear amortized cost for every edge remotion, we also provide a new version of such a static algorithm that improves its complexity from O(n m^2) to O(m^2), that is in turn aligned with the cost of the best state-of-the-art static algorithm for vulnerability.

Authors: Dario Fiorenza, Daniele Gorla, Ivano Salvo

Braess paradox originates when latency at Wardrop equilibrium in traffic networks decreases because of removing edges. The graph-theoretic property of networks suffering from the Braess paradox was called vulnerability by Roughgarden in 2006; it was then characterized and algorithmically checked both for undirected and for directed nets. In this paper, we provide a decremental algorithm of linear amortized complexity to check vulnerability for dynamically evolving networks. The basic idea of our dynamic algorithm is to use a static algorithm that marks some edges as irrelevant for the vulnerability of the graph and ignore those edges for all subsequent runs of the decremental procedure. To get a linear amortized cost for every edge remotion, we also provide a new version of such a static algorithm that improves its complexity from O(n m^2) to O(m^2), that is in turn aligned with the cost of the best state-of-the-art static algorithm for vulnerability.

Sharp dimensional analysis of midpoint methods for Langevin sampling

from arXiv: Data Structures and Algorithms

Authors: Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S. Zhang

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $π\propto e^{-V}$, where $0 \prec αI\preceq\nabla^2V\preceqβI$ and $κ=β/α$. To achieve $\sqrtα\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(κ^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(κ^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

Authors: Fan Chen, Sinho Chewi, Jianfeng Lu, Matthew S. Zhang

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $π\propto e^{-V}$, where $0 \prec αI\preceq\nabla^2V\preceqβI$ and $κ=β/α$. To achieve $\sqrtα\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(κ^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(κ^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.

Hiding a Vertex from the Temporal Explorer: A Lower Bound for Degree-Bounded Temporal Graphs

from arXiv: Data Structures and Algorithms

Authors: Daniele Carnevale

A temporal graph is a sequence of graphs on a common set of $n$ vertices, its snapshots, one for each time step. An agent, knowing the entire sequence in advance, may at each time step wait or move along an edge of the current snapshot, and the temporal graph is explored once the agent has visited every vertex. We consider always-connected temporal graphs, in which every snapshot is connected, and ask how long exploration can be forced to take when the underlying graph, the union of all snapshots, has maximum degree at most~$Δ$. Two lower bounds were known in this setting, $Ω(Δn)$ and $Ω(n\log n)$, realised by different constructions. We establish a stronger lower bound, answering a question of Bastide, Groenland, Michel and Rambaud. Specifically, for every $n$ and $Δ$ with $Δ_0\leΔ\le n-1$ we construct an always-connected temporal graph on $n$ vertices with underlying maximum degree at most~$Δ$ that cannot be explored in fewer than $γ\,Δn\,(1+\log(n/Δ))$ time steps from any start vertex, where $γ>0$ and $Δ_0$ are absolute constants. The construction is deterministic, and every snapshot is a spanning tree with exactly one vertex of degree greater than three. Time is divided into phases. In each phase, a rotating-cycle gadget prevents the agent from reaching more than half of the trees attached to it. Between phases, we reassign target vertices among these trees using walks on a constant-degree expander, and a Gray code order of the targets keeps the underlying degree bounded. The reassignment guarantees that, for any walk of the agent, some target remains unvisited throughout all phases.

Authors: Daniele Carnevale

A temporal graph is a sequence of graphs on a common set of $n$ vertices, its snapshots, one for each time step. An agent, knowing the entire sequence in advance, may at each time step wait or move along an edge of the current snapshot, and the temporal graph is explored once the agent has visited every vertex. We consider always-connected temporal graphs, in which every snapshot is connected, and ask how long exploration can be forced to take when the underlying graph, the union of all snapshots, has maximum degree at most~$Δ$. Two lower bounds were known in this setting, $Ω(Δn)$ and $Ω(n\log n)$, realised by different constructions. We establish a stronger lower bound, answering a question of Bastide, Groenland, Michel and Rambaud. Specifically, for every $n$ and $Δ$ with $Δ_0\leΔ\le n-1$ we construct an always-connected temporal graph on $n$ vertices with underlying maximum degree at most~$Δ$ that cannot be explored in fewer than $γ\,Δn\,(1+\log(n/Δ))$ time steps from any start vertex, where $γ>0$ and $Δ_0$ are absolute constants. The construction is deterministic, and every snapshot is a spanning tree with exactly one vertex of degree greater than three. Time is divided into phases. In each phase, a rotating-cycle gadget prevents the agent from reaching more than half of the trees attached to it. Between phases, we reassign target vertices among these trees using walks on a constant-degree expander, and a Gray code order of the targets keeps the underlying degree bounded. The reassignment guarantees that, for any walk of the agent, some target remains unvisited throughout all phases.

On the Parameterized Complexity of Conflict-Free Edge Cut in Undirected Graphs

from arXiv: Data Structures and Algorithms

Authors: Sourav Das, Ashwin Jacob, Arpit Kumar, Diptapriyo Majumdar

In this paper, we study CONFLICT-FREE EDGE CUT (CF-CUT), which is a recently introduced conflict-free version of the MIN-CUT problem that asks to find the minimum number of edges to disconnect a connected graph. The CF-CUT takes as input a connected undirected graph G = (V, E), a conflict graph $\widehat{G}$ such that $E(G) = V(\widehat{G})$, and the objective is to decide whether there exists $F \subseteq E(G)$ such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. Rauch et. al. [IPL-2025] proved that CF-CUT is NP-Complete and also provided some results on the parameterized complexity of CF-CUT. A related variant MIN CONFLICT-FREE EDGE CUT (MIN-CF-CUT) takes a connected graph $G$, a conflict graph $\widehat{G}$ such that $V(\widehat{G}) = E(G)$, and an integer $k$ as input and asks if there is a set $F$ of at most $k$ edges such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. In this paper, we extend the work of Rauch et al. [IPL-2025] and provide a systematic study on the CF-CUT and MIN-CF-CUT from the perspective of parameterized complexity and polynomial kernelization. We prove that CF-CUT is NP-hard when the dissociation number of the input graph is at most two. We also complement it by proving that CF-CUT is poly-time solvable when the dissociation number of an input graph is at most one. Additionally, for MIN-CF-CUT, we consider both solution size and various structural parameters of the input graph as parameters, and provide fixed-parameter tractability and W[1]-hardness results when the conflict graph is restricted to various graph classes. We also prove that unless NP $\subseteq$ coNP/poly, MIN-CF-CUT admits no polynomial kernel when parameterized by the vertex cover number of the input graph; and also when parameterized by the sum of the solution size and the vertex integrity of the input graph.

Authors: Sourav Das, Ashwin Jacob, Arpit Kumar, Diptapriyo Majumdar

In this paper, we study CONFLICT-FREE EDGE CUT (CF-CUT), which is a recently introduced conflict-free version of the MIN-CUT problem that asks to find the minimum number of edges to disconnect a connected graph. The CF-CUT takes as input a connected undirected graph G = (V, E), a conflict graph $\widehat{G}$ such that $E(G) = V(\widehat{G})$, and the objective is to decide whether there exists $F \subseteq E(G)$ such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. Rauch et. al. [IPL-2025] proved that CF-CUT is NP-Complete and also provided some results on the parameterized complexity of CF-CUT. A related variant MIN CONFLICT-FREE EDGE CUT (MIN-CF-CUT) takes a connected graph $G$, a conflict graph $\widehat{G}$ such that $V(\widehat{G}) = E(G)$, and an integer $k$ as input and asks if there is a set $F$ of at most $k$ edges such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. In this paper, we extend the work of Rauch et al. [IPL-2025] and provide a systematic study on the CF-CUT and MIN-CF-CUT from the perspective of parameterized complexity and polynomial kernelization. We prove that CF-CUT is NP-hard when the dissociation number of the input graph is at most two. We also complement it by proving that CF-CUT is poly-time solvable when the dissociation number of an input graph is at most one. Additionally, for MIN-CF-CUT, we consider both solution size and various structural parameters of the input graph as parameters, and provide fixed-parameter tractability and W[1]-hardness results when the conflict graph is restricted to various graph classes. We also prove that unless NP $\subseteq$ coNP/poly, MIN-CF-CUT admits no polynomial kernel when parameterized by the vertex cover number of the input graph; and also when parameterized by the sum of the solution size and the vertex integrity of the input graph.

Improved Sublinear Algorithms for Maximal Independent Set and Metric Steiner Forest

from arXiv: Data Structures and Algorithms

Authors: Sepideh Mahabadi, Jakub Tarnawski

In this work we consider the Maximal Independent Set (MIS) problem and the metric Steiner Forest problem in the sublinear time setting, under the adjacency/distance matrix query model. First, we give an algorithm that estimates the size of an MIS up to a multiplicative factor of $(1+\eps)$ using $\tO(n^{4/3}/\eps^2)$ queries. This improves the best previous algorithm by Mahabadi, Roghani, Tarnawski, and Vakilian (SODA 2026), which had a query complexity of $\tO(n^{3/2}/\eps^2)$. Via a reduction from that work, this would automatically imply the same improvement for the problem of estimating the metric Steiner Forest cost up to an $O(\log n)$ factor. However, as our second contribution, we consider the Steiner Forest problem directly and provide an algorithm with $\tO(n)$ query complexity that is very simple and does not proceed via MIS.

Authors: Sepideh Mahabadi, Jakub Tarnawski

In this work we consider the Maximal Independent Set (MIS) problem and the metric Steiner Forest problem in the sublinear time setting, under the adjacency/distance matrix query model. First, we give an algorithm that estimates the size of an MIS up to a multiplicative factor of $(1+\eps)$ using $\tO(n^{4/3}/\eps^2)$ queries. This improves the best previous algorithm by Mahabadi, Roghani, Tarnawski, and Vakilian (SODA 2026), which had a query complexity of $\tO(n^{3/2}/\eps^2)$. Via a reduction from that work, this would automatically imply the same improvement for the problem of estimating the metric Steiner Forest cost up to an $O(\log n)$ factor. However, as our second contribution, we consider the Steiner Forest problem directly and provide an algorithm with $\tO(n)$ query complexity that is very simple and does not proceed via MIS.

Near-Exact Computation of Independent Chip Model Placement Probabilities for Thousands of Players

from arXiv: Data Structures and Algorithms

Authors: Wataru Inariba

The Independent Chip Model (ICM) converts the chip stacks of the players remaining in a poker tournament into finishing-place probabilities and prize equities. It is the standard model in tournament solvers, but its definition considers all possible finishing orders, and exact computation has been regarded as intractable for large fields. The same mathematics appears in other fields; for instance, the ICM is a Plackett-Luce ranking model with stacks as weights. We present DE-ICM, a deterministic algorithm that computes the placement probabilities of all $n$ players for all paid places in $O(M n^{2})$ time, where the number $M$ of quadrature nodes is fixed for the admissible inputs and the target accuracy. Its numerical scheme allows the target to be set close to the accuracy of double-precision arithmetic. The algorithm evaluates a classical integral representation in which, conditional on a player's exponential clock, the number of players ahead is Poisson-binomial. A double-exponential quadrature on a single grid evaluates every integral; its truncation errors have closed-form bounds, and the step size, which controls the discretization error, follows from the field size. A numerically stable deconvolution then recovers each player's leave-one-out coefficients in $O(n)$ time from one shared product. On exactly solvable instances with up to 4,000 players, the worst observed relative error of an equity is $4.0 \times 10^{-14}$ and the worst absolute error of a placement probability $1.1 \times 10^{-14}$. The full placement matrix for 1,000 players takes 0.2 seconds on one CPU core.

Authors: Wataru Inariba

The Independent Chip Model (ICM) converts the chip stacks of the players remaining in a poker tournament into finishing-place probabilities and prize equities. It is the standard model in tournament solvers, but its definition considers all possible finishing orders, and exact computation has been regarded as intractable for large fields. The same mathematics appears in other fields; for instance, the ICM is a Plackett-Luce ranking model with stacks as weights. We present DE-ICM, a deterministic algorithm that computes the placement probabilities of all $n$ players for all paid places in $O(M n^{2})$ time, where the number $M$ of quadrature nodes is fixed for the admissible inputs and the target accuracy. Its numerical scheme allows the target to be set close to the accuracy of double-precision arithmetic. The algorithm evaluates a classical integral representation in which, conditional on a player's exponential clock, the number of players ahead is Poisson-binomial. A double-exponential quadrature on a single grid evaluates every integral; its truncation errors have closed-form bounds, and the step size, which controls the discretization error, follows from the field size. A numerically stable deconvolution then recovers each player's leave-one-out coefficients in $O(n)$ time from one shared product. On exactly solvable instances with up to 4,000 players, the worst observed relative error of an equity is $4.0 \times 10^{-14}$ and the worst absolute error of a placement probability $1.1 \times 10^{-14}$. The full placement matrix for 1,000 players takes 0.2 seconds on one CPU core.

Fast mixing of the SYK model at high temperatures

from arXiv: Data Structures and Algorithms

Authors: Yiyi Cai, Yongtao Zhan, Alexander Zlokapa

The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions and uses an especially simple SYK analysis based on a comparison to classical dynamics.

Authors: Yiyi Cai, Yongtao Zhan, Alexander Zlokapa

The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions and uses an especially simple SYK analysis based on a comparison to classical dynamics.

Prime factorisation of stable-matching instances: uniqueness, simultaneous products, and an exact census

from arXiv: Data Structures and Algorithms

Authors: Yoshiteru Ishida

Every balanced instance of the stable marriage problem with strict complete preferences has a unique finest partition into prime blocks, and that single partition simultaneously factors three different structures: the reachable execution digraph as a Cartesian product, the proposal-prefix antimatroid as a direct sum, and the stable-matching lattice as a direct product. The converse fails, and fails at every size from two on: two explicit families share the identical Boolean-cube execution while one is maximally decomposable with a single stable matching and the other is prime with n. Uniqueness yields an exact census, a recursion counting the prime instances at every size, under which exactly 88,478,208 of the 110,075,314,176 profiles with four agents on each side are decomposable and the decomposable fraction is asymptotically n! / n^(2n). The blocks are characterised as the square components of the mutual-rank filtration, so the partition is computable in polynomial time and the factorisation is a tool rather than only a fact.

Authors: Yoshiteru Ishida

Every balanced instance of the stable marriage problem with strict complete preferences has a unique finest partition into prime blocks, and that single partition simultaneously factors three different structures: the reachable execution digraph as a Cartesian product, the proposal-prefix antimatroid as a direct sum, and the stable-matching lattice as a direct product. The converse fails, and fails at every size from two on: two explicit families share the identical Boolean-cube execution while one is maximally decomposable with a single stable matching and the other is prime with n. Uniqueness yields an exact census, a recursion counting the prime instances at every size, under which exactly 88,478,208 of the 110,075,314,176 profiles with four agents on each side are decomposable and the decomposable fraction is asymptotically n! / n^(2n). The blocks are characterised as the square components of the mutual-rank filtration, so the partition is computable in polynomial time and the factorisation is a tool rather than only a fact.

The Infectious Vaccination Problem: a variant of Firefighting with Spreading Defence

from arXiv: Data Structures and Algorithms

Authors: Jessica Enright, Melissa A. Huggan, Ethan Hunter-Frankland, Margaret-Ellen Messinger, Dylan Pearson

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

Authors: Jessica Enright, Melissa A. Huggan, Ethan Hunter-Frankland, Margaret-Ellen Messinger, Dylan Pearson

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

Guanaco: A Global-Uniformity Algorithm for Near-Submodular-Width Conjunctive Query Evaluation

from arXiv: Data Structures and Algorithms

Authors: Mahmoud Abo Khamis, Hubie Chen

We present Guanaco, an algorithm for performing conjunctive query evaluation where, for each Boolean conjunctive query, and positive epsilon, the algorithm achieves polynomial time with exponent equal to the submodular width plus epsilon. The algorithm and its running time generalize smoothly to general conjunctive queries. We believe the algorithm and its analysis to be notably simple, indeed, together we believe they form a highly simple argument that conjunctive query evaluation can be performed in essentially submodular width time. In the case of Boolean conjunctive queries, the algorithm is based on interleaving three simple primitives: a subroutine for establishing a form of consistency; a subroutine for establishing global uniformity, which, briefly speaking, partitions relations as needed to control discrepancies between average degree and maximum degree; and, a simple step that joins pairs of existing relations to form new relations.

Authors: Mahmoud Abo Khamis, Hubie Chen

We present Guanaco, an algorithm for performing conjunctive query evaluation where, for each Boolean conjunctive query, and positive epsilon, the algorithm achieves polynomial time with exponent equal to the submodular width plus epsilon. The algorithm and its running time generalize smoothly to general conjunctive queries. We believe the algorithm and its analysis to be notably simple, indeed, together we believe they form a highly simple argument that conjunctive query evaluation can be performed in essentially submodular width time. In the case of Boolean conjunctive queries, the algorithm is based on interleaving three simple primitives: a subroutine for establishing a form of consistency; a subroutine for establishing global uniformity, which, briefly speaking, partitions relations as needed to control discrepancies between average degree and maximum degree; and, a simple step that joins pairs of existing relations to form new relations.

CT-Miner: Fast and Coarse-Grained Time-Series Pattern Mining via Cartesian Trees

from arXiv: Data Structures and Algorithms

Authors: Hyundong Jin, Hyunki Hong, Yo-Sub Han

Time series often contain recurring structural patterns, and efficiently mining such patterns into compact representations is essential for scalable analysis of long sequences. Cartesian tree (CT) equivalence provides a well-established structural abstraction that preserves hierarchical order structure while discarding exact values and fine-grained ordinal variations. By grouping multiple ordinal patterns into a shared structural form, CT equivalence offers a principled way to compress recurring temporal structure. However, mining frequent CT-equivalent patterns at scale remains computationally expensive. A naive pairwise approach repeatedly constructs and counts CT representations over subsequences, requiring $O(n^4)$ time for a sequence of length $n$, which severely limits its applicability to long sequences. We propose a new Cartesian pattern mining algorithm based on a Cartesian suffix tree that compactly organizes CT-equivalent subsequences and reuses shared structural information. Our method reduces exhaustive CT-pattern occurrence collection from $O(n^4)$ to $O(n^2)$ time, and we formally prove the correctness and complexity bounds. We further show that this computational gain translates into effective compact representations. Across diverse time-series datasets, a small set of mined CT patterns preserves meaningful clustering structure, and comparisons with finer-grained order-preserving representations show that CT equivalence reduces redundant ordinal distinctions under limited feature budgets. Our implementation is available at github.com/hyundong98/CT-Miner .

Authors: Hyundong Jin, Hyunki Hong, Yo-Sub Han

Time series often contain recurring structural patterns, and efficiently mining such patterns into compact representations is essential for scalable analysis of long sequences. Cartesian tree (CT) equivalence provides a well-established structural abstraction that preserves hierarchical order structure while discarding exact values and fine-grained ordinal variations. By grouping multiple ordinal patterns into a shared structural form, CT equivalence offers a principled way to compress recurring temporal structure. However, mining frequent CT-equivalent patterns at scale remains computationally expensive. A naive pairwise approach repeatedly constructs and counts CT representations over subsequences, requiring $O(n^4)$ time for a sequence of length $n$, which severely limits its applicability to long sequences. We propose a new Cartesian pattern mining algorithm based on a Cartesian suffix tree that compactly organizes CT-equivalent subsequences and reuses shared structural information. Our method reduces exhaustive CT-pattern occurrence collection from $O(n^4)$ to $O(n^2)$ time, and we formally prove the correctness and complexity bounds. We further show that this computational gain translates into effective compact representations. Across diverse time-series datasets, a small set of mined CT patterns preserves meaningful clustering structure, and comparisons with finer-grained order-preserving representations show that CT equivalence reduces redundant ordinal distinctions under limited feature budgets. Our implementation is available at https://github.com/hyundong98/CT-Miner .

Multilevel Dynamic Thinning for Matroid Secretary

from arXiv: Data Structures and Algorithms

Authors: Dennis Joyce

In the matroid secretary problem, weighted elements arrive in random order, and an online algorithm must irrevocably accept elements forming a high-weight independent set. Dynamic Thinning is a recent, conceptually simple $3.1462$-competitive algorithm for the matroid secretary problem that maintains a random reference set. Given this reference set, the elements in its maximum-weight independent subset have been accepted independently with a time-dependent probability. We extend this approach by replacing the single reference set with a finite hierarchy of nested reference sets. For every fixed $\eps>0$, the resulting algorithm is $(e+\eps)$-probability-competitive for arbitrary matroids, with $O_\eps(n^2)$ queries in the worst case.

Authors: Dennis Joyce

In the matroid secretary problem, weighted elements arrive in random order, and an online algorithm must irrevocably accept elements forming a high-weight independent set. Dynamic Thinning is a recent, conceptually simple $3.1462$-competitive algorithm for the matroid secretary problem that maintains a random reference set. Given this reference set, the elements in its maximum-weight independent subset have been accepted independently with a time-dependent probability. We extend this approach by replacing the single reference set with a finite hierarchy of nested reference sets. For every fixed $\eps>0$, the resulting algorithm is $(e+\eps)$-probability-competitive for arbitrary matroids, with $O_\eps(n^2)$ queries in the worst case.

A Factor-5 Conversion to Internal Collage Systems

from arXiv: Data Structures and Algorithms

Authors: Simone Faro

A collage system is a grammar-based compression model that extends straight-line programs with repetition and substring truncation. Internal collage systems additionally require every nonterminal to be structurally reachable from the start symbol. Migita, Uehata, and I (CPM 2026) showed that any collage system of size $m$ can be converted into an internal one generating the same string with size at most $9m$, and left the improvement of this constant as an open problem. We show that a simple refinement of their top-down conversion reduces the bound to $5m$. The proof combines three elementary ideas: canonicalization of generated truncations that remain aligned with a target endpoint; a repetition decomposition that absorbs every complete copy of the repetition base into a maximal core; and monotonicity of structural reachability, which prevents an input truncation rule from both becoming structurally reachable and later acting as a hidden target at which endpoint alignment is lost. The conversion runs in deterministic $O(m^2)$ worst-case time in the stated unit-cost random-access machine model. Consequently, for every string, the minimum size of an internal collage system is at most five times the minimum size of a general collage system.

Authors: Simone Faro

A collage system is a grammar-based compression model that extends straight-line programs with repetition and substring truncation. Internal collage systems additionally require every nonterminal to be structurally reachable from the start symbol. Migita, Uehata, and I (CPM 2026) showed that any collage system of size $m$ can be converted into an internal one generating the same string with size at most $9m$, and left the improvement of this constant as an open problem. We show that a simple refinement of their top-down conversion reduces the bound to $5m$. The proof combines three elementary ideas: canonicalization of generated truncations that remain aligned with a target endpoint; a repetition decomposition that absorbs every complete copy of the repetition base into a maximal core; and monotonicity of structural reachability, which prevents an input truncation rule from both becoming structurally reachable and later acting as a hidden target at which endpoint alignment is lost. The conversion runs in deterministic $O(m^2)$ worst-case time in the stated unit-cost random-access machine model. Consequently, for every string, the minimum size of an internal collage system is at most five times the minimum size of a general collage system.

Exact Optimal Transport by Matching

from arXiv: Data Structures and Algorithms

Authors: Dmitry Kamenetsky

Balanced discrete optimal transport between n sources and n targets of unit mass is exactly the minimum-cost assignment problem-a bipartite perfect matching-and is therefore solvable exactly by industrial matching engines in milliseconds to seconds. We ask when the exact approach beats the standard approximate alternatives, entropic Sinkhorn and its accelerated variant Greenkhorn, and make the sparse-exact side certified by a textbook LP dual-feasibility clip. Three contributions. (i) Measurement: on dense 2-D instances, exact matching (Jonker-Volgenant) is faster and strictly more accurate than either approximate method throughout the moderate-n regime (0.01 s at n=500 to 11.5 s at n=8000); reaching a 1% quality target on the same hardware requires roughly 10-80 min for Greenkhorn (factors 4e2-6e4 over exact; plain Sinkhorn is 20-650x slower still), a rough power-law projection beyond the measured range. Greenkhorn's measured speedup over plain Sinkhorn is only 1.0-1.5x on most converged cells. (ii) A simple kNN-pool gap certificate: given a pool matching and its Blossom dual, a one-pass O(n^2) clip produces a dense-feasible lower bound; combined with the Sinkhorn dual potential (valid at every iterate, not just at convergence), the bound is valid on all 45 measured configurations and tightens monotonically with k. (iii) A multi-robot task-allocation sanity check where the discrete plan is the deliverable: per-round exact assignment costs 0.1-68 ms, while a Sinkhorn-plus-hardening pipeline costs 0.12-15.9 s and accumulates 6-27% extra travel over 15 rounds. The Sinkhorn family's large-n dense regime is acknowledged and left untouched. Code, data, and results under MIT: github.com/dimkadimon/OT-Blossom.

Authors: Dmitry Kamenetsky

Balanced discrete optimal transport between n sources and n targets of unit mass is exactly the minimum-cost assignment problem-a bipartite perfect matching-and is therefore solvable exactly by industrial matching engines in milliseconds to seconds. We ask when the exact approach beats the standard approximate alternatives, entropic Sinkhorn and its accelerated variant Greenkhorn, and make the sparse-exact side certified by a textbook LP dual-feasibility clip. Three contributions. (i) Measurement: on dense 2-D instances, exact matching (Jonker-Volgenant) is faster and strictly more accurate than either approximate method throughout the moderate-n regime (0.01 s at n=500 to 11.5 s at n=8000); reaching a 1% quality target on the same hardware requires roughly 10-80 min for Greenkhorn (factors 4e2-6e4 over exact; plain Sinkhorn is 20-650x slower still), a rough power-law projection beyond the measured range. Greenkhorn's measured speedup over plain Sinkhorn is only 1.0-1.5x on most converged cells. (ii) A simple kNN-pool gap certificate: given a pool matching and its Blossom dual, a one-pass O(n^2) clip produces a dense-feasible lower bound; combined with the Sinkhorn dual potential (valid at every iterate, not just at convergence), the bound is valid on all 45 measured configurations and tightens monotonically with k. (iii) A multi-robot task-allocation sanity check where the discrete plan is the deliverable: per-round exact assignment costs 0.1-68 ms, while a Sinkhorn-plus-hardening pipeline costs 0.12-15.9 s and accumulates 6-27% extra travel over 15 rounds. The Sinkhorn family's large-n dense regime is acknowledged and left untouched. Code, data, and results under MIT: https://github.com/dimkadimon/OT-Blossom.

Optimal Convergence of Iterative Methods for Datalogo

from arXiv: Data Structures and Algorithms

Authors: Simon Frisk, Sungjin Im, Paraschos Koutris, Benjamin Moseley, Hung Ngo, Kirk Pruhs, Hangdong Zhao

$\mathsf{Datalog}^\circ$ has been introduced as an extension to Datalog that increases expressiveness, yet retains simple least fixpoint semantics and admits optimization techniques such as semi-naïve evaluation and demand transformation. $\mathsf{Datalog}^\circ$ accomplishes this by generalizing the {\em or} and {\em and} operators of Datalog to addition and multiplication over a semiring. Finding a (minimal) fixpoint of a $\mathsf{Datalog}^\circ$ program is equivalent to finding a solution to a system of polynomial equations over the underlying semiring. Solving these systems of polynomial equations is not only a fundamental problem in the theory of $\mathsf{Datalog}^\circ$, but also has many applications in computer science, such as in databases, program analysis, and optimization. This paper resolves a key open problem in the theory of $\mathsf{Datalog}^\circ$: we prove a tight upper bound on the number of steps until convergence of iterative methods for solving these polynomial equation systems over a commutative $p$-stable semiring. In particular, we show that the number of steps until convergence is $O((p+1)n)$ where $n$ is the output size of the $\mathsf{Datalog}^\circ$ program. As the number of steps until convergence is only a proxy for the runtime of $\mathsf{Datalog}^\circ$ evaluation, we also consider the number of semiring operations used and show that, for a natural class of algorithms, $O((p+1)mn)$ is a tight upper bound, where $m$ is the maximum number of semiring operations per iteration.

Authors: Simon Frisk, Sungjin Im, Paraschos Koutris, Benjamin Moseley, Hung Ngo, Kirk Pruhs, Hangdong Zhao

$\mathsf{Datalog}^\circ$ has been introduced as an extension to Datalog that increases expressiveness, yet retains simple least fixpoint semantics and admits optimization techniques such as semi-naïve evaluation and demand transformation. $\mathsf{Datalog}^\circ$ accomplishes this by generalizing the {\em or} and {\em and} operators of Datalog to addition and multiplication over a semiring. Finding a (minimal) fixpoint of a $\mathsf{Datalog}^\circ$ program is equivalent to finding a solution to a system of polynomial equations over the underlying semiring. Solving these systems of polynomial equations is not only a fundamental problem in the theory of $\mathsf{Datalog}^\circ$, but also has many applications in computer science, such as in databases, program analysis, and optimization. This paper resolves a key open problem in the theory of $\mathsf{Datalog}^\circ$: we prove a tight upper bound on the number of steps until convergence of iterative methods for solving these polynomial equation systems over a commutative $p$-stable semiring. In particular, we show that the number of steps until convergence is $O((p+1)n)$ where $n$ is the output size of the $\mathsf{Datalog}^\circ$ program. As the number of steps until convergence is only a proxy for the runtime of $\mathsf{Datalog}^\circ$ evaluation, we also consider the number of semiring operations used and show that, for a natural class of algorithms, $O((p+1)mn)$ is a tight upper bound, where $m$ is the maximum number of semiring operations per iteration.

Geometry-Dependent Approximation for Non-Monotone $k$-Submodular Maximization

from arXiv: Data Structures and Algorithms

Authors: Vaneet Aggarwal

We study nonnegative, non-monotone $k$-submodular maximization with $k\ge2$ labels under support constraints, and show how the certified approximation coefficient improves as the support region permits more uniform selection. For a compact convex down-closed support region $P\subseteq[0,1]^n$, the diagonal level $ζ(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ ranges from $ζ=0$, which carries no geometric promise, to $ζ=1$, which is unrestricted support. Our main structural result is a comparator-uniform linearization of the multilinear extension, built from an objective-independent action and a comparator-independent update field. For $k\ge3$, its validity reduces, independently of the number of elements, to four polynomial inequalities of degree at most three in one or two variables, only one of which depends on $k$. Explicit parameter choices give a nondecreasing certified profile $\underlineα_k(ζ)$, in closed form on all of $[0,1]$ when $k=2$. At $ζ=0$ we certify $0.4456\ldots$ for $k=2$ and $0.4541\ldots$ for every $k\ge3$, improving the recent $\sqrt2-1$ guarantee for one matroid or one knapsack, as well as the $1/3$-type guarantees for a fixed number of budgets; at $ζ=1$ we certify $1/2$ for $k=2$, $(\sqrt{17}-3)/2$ for $k=3,4$, and $k/(2k-1)$ for $k\ge5$, whose excess over $1/2$ is of order $1/k$ rather than the previous $1/k^2$. Value-retaining rounding transfers these guarantees to matroid and knapsack constraints, and the same field yields $O(\sqrt T)$ approximate regret online under gradient or post-decision value feedback.

Authors: Vaneet Aggarwal

We study nonnegative, non-monotone $k$-submodular maximization with $k\ge2$ labels under support constraints, and show how the certified approximation coefficient improves as the support region permits more uniform selection. For a compact convex down-closed support region $P\subseteq[0,1]^n$, the diagonal level $ζ(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ ranges from $ζ=0$, which carries no geometric promise, to $ζ=1$, which is unrestricted support. Our main structural result is a comparator-uniform linearization of the multilinear extension, built from an objective-independent action and a comparator-independent update field. For $k\ge3$, its validity reduces, independently of the number of elements, to four polynomial inequalities of degree at most three in one or two variables, only one of which depends on $k$. Explicit parameter choices give a nondecreasing certified profile $\underlineα_k(ζ)$, in closed form on all of $[0,1]$ when $k=2$. At $ζ=0$ we certify $0.4456\ldots$ for $k=2$ and $0.4541\ldots$ for every $k\ge3$, improving the recent $\sqrt2-1$ guarantee for one matroid or one knapsack, as well as the $1/3$-type guarantees for a fixed number of budgets; at $ζ=1$ we certify $1/2$ for $k=2$, $(\sqrt{17}-3)/2$ for $k=3,4$, and $k/(2k-1)$ for $k\ge5$, whose excess over $1/2$ is of order $1/k$ rather than the previous $1/k^2$. Value-retaining rounding transfers these guarantees to matroid and knapsack constraints, and the same field yields $O(\sqrt T)$ approximate regret online under gradient or post-decision value feedback.

Conjunctive Queries with Negation: Beyond Signed-Acyclicity

from arXiv: Data Structures and Algorithms

Authors: Simon Frisk, Paraschos Koutris

We study constant-delay enumeration for conjunctive queries with negation ($\texttt{CQ}^{\neg}$). Prior work defined \emph{signed-acyclicity}, which characterizes the class of queries where linear preprocessing time is achievable, but little has been known beyond this. We introduce a new hypergraph width measure for $\texttt{CQ}^{\neg}$, the \emph{signed fractional hypertree width} ($\textsf{sfhw}$), defined by requiring that a single variable order simultaneously handle every subset of the negative atoms. We show that $\textsf{sfhw}$ strictly generalizes signed-acyclicity (recovered when $\textsf{sfhw} = 1$) and fractional hypertree width (recovered on queries without negation). Our main algorithmic result is a variable-elimination algorithm that achieves constant-delay enumeration on input $I$ for any full $\texttt{CQ}^{\neg}$ query with preprocessing time $O(|I|^{\textsf{sfhw}})$. We further show that using multiple variable orders can improve this bound, exhibiting an $O(|I|^{3/2})$ algorithm for $k$-cycle queries with at least one negative edge.

Authors: Simon Frisk, Paraschos Koutris

We study constant-delay enumeration for conjunctive queries with negation ($\texttt{CQ}^{\neg}$). Prior work defined \emph{signed-acyclicity}, which characterizes the class of queries where linear preprocessing time is achievable, but little has been known beyond this. We introduce a new hypergraph width measure for $\texttt{CQ}^{\neg}$, the \emph{signed fractional hypertree width} ($\textsf{sfhw}$), defined by requiring that a single variable order simultaneously handle every subset of the negative atoms. We show that $\textsf{sfhw}$ strictly generalizes signed-acyclicity (recovered when $\textsf{sfhw} = 1$) and fractional hypertree width (recovered on queries without negation). Our main algorithmic result is a variable-elimination algorithm that achieves constant-delay enumeration on input $I$ for any full $\texttt{CQ}^{\neg}$ query with preprocessing time $O(|I|^{\textsf{sfhw}})$. We further show that using multiple variable orders can improve this bound, exhibiting an $O(|I|^{3/2})$ algorithm for $k$-cycle queries with at least one negative edge.

Vertex Cover Problem with Advice

from arXiv: Data Structures and Algorithms

Authors: Lucas de Oliveira Silva, Lehilton Lelis Chaves Pedrosa

We study the minimum Vertex Cover problem under two non-adaptive offline advice models, where predictions about a fixed optimum solution are provided once as part of the input. These forms of prediction were introduced independently by Cohen-Addad, d'Orsi, Gupta, Lee, and Panigrahi (2024) and by Ghoshal, Makarychev, and Makarychev (2025). In Partial Predictions, independently revealed vertices come with correct membership labels. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(2-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. In Noisy Predictions, every vertex instead receives a mutually independent noisy membership label of bias $\varepsilon$. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(1-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. Our Noisy Predictions bound matches the leading asymptotic improvement below $2$ obtained by Aamand, Chen, Gollapudi, Silwal, and Wu (2025), despite using only one noisy label per vertex rather than a separate independent label for each endpoint of each incident edge. Both our bounds hold as $\varepsilon$ goes to $0$ and are strictly below $2$, the optimal approximation threshold for minimum Vertex Cover without advice under the Unique Games Conjecture, as shown by Khot and Regev (2008).

Authors: Lucas de Oliveira Silva, Lehilton Lelis Chaves Pedrosa

We study the minimum Vertex Cover problem under two non-adaptive offline advice models, where predictions about a fixed optimum solution are provided once as part of the input. These forms of prediction were introduced independently by Cohen-Addad, d'Orsi, Gupta, Lee, and Panigrahi (2024) and by Ghoshal, Makarychev, and Makarychev (2025). In Partial Predictions, independently revealed vertices come with correct membership labels. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(2-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. In Noisy Predictions, every vertex instead receives a mutually independent noisy membership label of bias $\varepsilon$. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(1-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. Our Noisy Predictions bound matches the leading asymptotic improvement below $2$ obtained by Aamand, Chen, Gollapudi, Silwal, and Wu (2025), despite using only one noisy label per vertex rather than a separate independent label for each endpoint of each incident edge. Both our bounds hold as $\varepsilon$ goes to $0$ and are strictly below $2$, the optimal approximation threshold for minimum Vertex Cover without advice under the Unique Games Conjecture, as shown by Khot and Regev (2008).

Monday, October 05

TR26-230 | Robust subspace designs and the power of a unique small quantum witness | Simon Apers, Roman Edenhofer, Benjamin Mathieu-Bloise, Partha Mukhopadhyay

from ECCC Papers
The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie \emph{close} to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.~Comput.~Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As a further application, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.
The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce \emph{robust subspace designs}. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie \emph{close} to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.~Comput.~Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As a further application, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.

TR26-229 | A polynomial scaling window for random $k$-SAT and a proof of the satisfiability conjecture | Gaia Carenini

from ECCC Papers
We prove that the scaling window of random $k$-SAT has width $O(n^{1/2+1/k})$, a polynomial improvement over our previous bound of $O(n/\log n)$. Combined with a result of Abbe and Montanari, this establishes the satisfiability conjecture for every fixed $k\geq 3$.
We prove that the scaling window of random $k$-SAT has width $O(n^{1/2+1/k})$, a polynomial improvement over our previous bound of $O(n/\log n)$. Combined with a result of Abbe and Montanari, this establishes the satisfiability conjecture for every fixed $k\geq 3$.

QuICS Hartree Postdoctoral Fellowships at Joint Center for Quantum Information and Computer Science (QuICS) at QuICS/University of Maryland (apply by December 1, 2026)

from CCI: jobs
The Joint Center for Quantum Information and Computer Science (QuICS, quics.umd.edu) is seeking exceptional candidates for the QuICS Hartree Postdoctoral Fellowships in Quantum Information and Computer Science. Apply at: umd.wd1.myworkdayjobs.com/en-US/UMCP/job/QuICS-Hartree-Postdoctoral-Fellow-1_JR104987 Website: umd.wd1.myworkdayjobs.com/en-US/UMCP/job/QuICS-Hartree-Postdoctoral-Fellow-1_JR104987 Email: quics-coordinator@umiacs.umd.edu

The Joint Center for Quantum Information and Computer Science (QuICS, http://quics.umd.edu) is seeking exceptional candidates for the QuICS Hartree Postdoctoral Fellowships in Quantum Information and Computer Science. Apply at: https://umd.wd1.myworkdayjobs.com/en-US/UMCP/job/QuICS-Hartree-Postdoctoral-Fellow-1_JR104987

Website: https://umd.wd1.myworkdayjobs.com/en-US/UMCP/job/QuICS-Hartree-Postdoctoral-Fellow-1_JR104987
Email: quics-coordinator@umiacs.umd.edu

By shacharlovett

A Theory of Nested Cascading in Directed Logic

from arXiv: Computational Complexity

Authors: Ihar Babushkin, Oliver Melchert, Ayhan Demircan, Uwe Morgner

In directed logic (DL), electronically controlled optical elements serve as photonic gates. Such electro-optical elements are of mixed nature: they have two inputs--electronic and optical--but only one, optical, output. This makes cascading such gates without repeated conversion between optical and electronic representations cumbersome. This problem can be largely overcome using the nested cascading scheme proposed by Shamir and Hardy [Opt. Express, 17, 150 (2007)]. Although promising as a solution to the cascading problem, the Shamir-Hardy scheme has so far been proved or implemented only for a few simplest cases. Here, we develop a general rigorous theory of nested cascading, valid for an arbitrary number of gates. We propose a variant of the algorithm which easily extendable to large number of gates, and rigorously prove its validity. Furthermore, we analyze how nested cascading scales with the size of the corresponding Boolean formula. We show that good (linear) scalability is guaranteed in many important cases, while the average scaling with respect to the corresponding Boolean formulas is only moderately polynomial, with an exponent of approximately 3/2. Yet, the worst-case scaling remains exponential with respect to more general Boolean circuits allowing sharing and reuse of intermediate results.

Authors: Ihar Babushkin, Oliver Melchert, Ayhan Demircan, Uwe Morgner

In directed logic (DL), electronically controlled optical elements serve as photonic gates. Such electro-optical elements are of mixed nature: they have two inputs--electronic and optical--but only one, optical, output. This makes cascading such gates without repeated conversion between optical and electronic representations cumbersome. This problem can be largely overcome using the nested cascading scheme proposed by Shamir and Hardy [Opt. Express, 17, 150 (2007)]. Although promising as a solution to the cascading problem, the Shamir-Hardy scheme has so far been proved or implemented only for a few simplest cases. Here, we develop a general rigorous theory of nested cascading, valid for an arbitrary number of gates. We propose a variant of the algorithm which easily extendable to large number of gates, and rigorously prove its validity. Furthermore, we analyze how nested cascading scales with the size of the corresponding Boolean formula. We show that good (linear) scalability is guaranteed in many important cases, while the average scaling with respect to the corresponding Boolean formulas is only moderately polynomial, with an exponent of approximately 3/2. Yet, the worst-case scaling remains exponential with respect to more general Boolean circuits allowing sharing and reuse of intermediate results.

The complexity of entangled graph colouring via polymorphisms

from arXiv: Computational Complexity

Authors: Eric Culf

Constraint satisfaction problems (CSPs) with operator assignments to the variables provide a well-structured setting to study the decision complexity of the entangled value of classes of nonlocal games. Due to the CSP dichotomy theorem, the complexity of constraint satisfaction problems with classical assignments can be fully understood by studying the symmetries of the CSP, in terms of the polymorphisms of the underlying relational structure. In this work, we show that the polymorphism-based reductions between CSPs can be generalised to gap-preserving reductions between entangled CSPs based on an entangled analogue of the polymorphisms. This reduction allows us to show undecidability of entangled graph colouring with more than three colours, a problem that has proved resistant to prior hardness reductions based on commutativity gadgets.

Authors: Eric Culf

Constraint satisfaction problems (CSPs) with operator assignments to the variables provide a well-structured setting to study the decision complexity of the entangled value of classes of nonlocal games. Due to the CSP dichotomy theorem, the complexity of constraint satisfaction problems with classical assignments can be fully understood by studying the symmetries of the CSP, in terms of the polymorphisms of the underlying relational structure. In this work, we show that the polymorphism-based reductions between CSPs can be generalised to gap-preserving reductions between entangled CSPs based on an entangled analogue of the polymorphisms. This reduction allows us to show undecidability of entangled graph colouring with more than three colours, a problem that has proved resistant to prior hardness reductions based on commutativity gadgets.

How to Have a Sensitive Debate: An Instance-Optimal Protocol for AI Debate

from arXiv: Computational Complexity

Authors: Jiawei Li, Zhiyang Xun, Lijie Chen, Jonah Brown-Cohen

As powerful AI systems reach and sometimes surpass the abilities of human experts across a range of cognitively demanding tasks, the problem of accurate oversight and supervision of these systems has become increasingly urgent. One promising approach is AI debate, which seeks to leverage a debate between two powerful AIs to break complex questions down into simpler claims that can be easily judged directly. Theoretical work on debate has formalized this intuition in the language of computational complexity theory, where the goal is to design protocols (i.e., rules of the debate game) that provide rigorous guarantees on correctness for judging solutions to complex problems with limited supervision. Specifically, the current best protocol has been shown to work for all problems that have sufficiently stable decompositions into subproblems. In this paper, we design a new protocol for this same class of problems that improves on the prior work in several ways. First, correctness holds in a worst-case rather than an average-case sense. Second, being honest and correct is a dominant-strategy equilibrium for both debaters, rather than a Stackelberg equilibrium. Finally, we prove black-box lower bounds, showing that our new protocol is instance-wise optimal. That is, no protocol for this class of problems can outperform ours while making only black-box queries to human judgments. We obtain these results by relating the notion of stable problem decompositions to the concept of fractional block sensitivity from query complexity.

Authors: Jiawei Li, Zhiyang Xun, Lijie Chen, Jonah Brown-Cohen

As powerful AI systems reach and sometimes surpass the abilities of human experts across a range of cognitively demanding tasks, the problem of accurate oversight and supervision of these systems has become increasingly urgent. One promising approach is AI debate, which seeks to leverage a debate between two powerful AIs to break complex questions down into simpler claims that can be easily judged directly. Theoretical work on debate has formalized this intuition in the language of computational complexity theory, where the goal is to design protocols (i.e., rules of the debate game) that provide rigorous guarantees on correctness for judging solutions to complex problems with limited supervision. Specifically, the current best protocol has been shown to work for all problems that have sufficiently stable decompositions into subproblems. In this paper, we design a new protocol for this same class of problems that improves on the prior work in several ways. First, correctness holds in a worst-case rather than an average-case sense. Second, being honest and correct is a dominant-strategy equilibrium for both debaters, rather than a Stackelberg equilibrium. Finally, we prove black-box lower bounds, showing that our new protocol is instance-wise optimal. That is, no protocol for this class of problems can outperform ours while making only black-box queries to human judgments. We obtain these results by relating the notion of stable problem decompositions to the concept of fractional block sensitivity from query complexity.

Two-Sided Product Expanding Codes via Rademacher Matrices

from arXiv: Computational Complexity

Authors: Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner

We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

Authors: Eshan Chattopadhyay, Noam Ringach, Nicholas Spooner

We give a proof of the existence of two-sided product expanding codes which, unlike the earlier result of Kalachev and Panteleev (FOCS, 2025), does not rely on explicit constructions of asymptotically optimal locally testable codes ($c^3$-LTCs). For every fixed number of component codes and dimensions whose rates are bounded away from zero and one, we show that independent random linear codes over a sufficiently large prime field have constant two-sided product expansion with probability tending to one. The tradeoff is that our proof requires the characteristic to grow with the block length, while [KP25] takes extensions of $\mathbb{F}_2$. To replace the use of $c^3$-LTCs, we develop several new techniques that we view as interesting in their own right. Instead of working directly over finite fields, we work over the reals and use random Rademacher matrices for the generator matrices of the component codes. From here, we we show that the extendability of $\varepsilon$-closed sets can be reduced to controlling the operator norm of sparse restrictions of carefully chosen Gram matrices. Applying the trace power method to bound this norm reduces to bounding the number of possible labelings of certain closed walks on bipartite graphs, which we bound using the sparsity of the operators and bounds on the number of equivalence classes of weak Wigner words from Anderson and Zeitouni (Probab. Theory Relat. Fields., 2006), which were originally applied to band matrices. Due to our techniques not relying on explicit $c^3$-LTCs, we believe that they form a promising starting point towards showing the existence of product expanding tensor codes where the component codes have non-trivial automorphism groups and coboundary expanding codes that could be used as the local codes of non-cubical complexes, such as simplicial complexes.

Non-Malleable Affine Extractors with Small Error and Complexity Lower Bounds

from arXiv: Computational Complexity

Authors: Xin Li, Yan Zhong

We construct explicit non-malleable affine extractors for every constant entropy rate, with linear output length and exponentially small error, against any fixed number of affine tamperings without fixed points. For every fixed $0<η<1$ and $t$, we also obtain entropy threshold $C_{η,t}n/\log n$, output length $\lfloor n^{1-η}\rfloor$, and error $2^{-n^{1-η}}$ against $t$ tamperings. Our extractors, as well as the directional affine extractors of Li and Zhong (CCC 2024), yield explicit Boolean functions with correlation $2^{-Ω(n)}$ against weakly read-once linear branching programs of size $2^{Ω(n)}$. For non-oblivious decision trees, we prove linear depth lower bounds for queries of each fixed degree $r\ge2$. Applying Li's sumset extractor (FOCS 2023) gives depth $Ω_δ((n/\ell)\log\ell)$ for growing locality $\ell\le n^{1-δ}$, where $0<δ<1$ is fixed. In the same range, directional affine extractors give correlation $2^{-Ω(n/\sqrt\ell)}$ against local trees of depth $c(n/\ell)\log\ell/\log\log\ell$, for a sufficiently small constant $c>0$. Our extractors derandomize the lossless lifting of Efremenko and Itsykson (STOC 2026). For every fixed $0<ξ<1$, this gives explicit polynomial-size unsatisfiable CNFs on $N$ variables whose $\mathrm{Res}(\oplus)$ refutations of resolution depth at most $N$ require size at least $2^{(1-ξ)N}$. Separately, parity substitutions give polynomial-size CNFs on $N$ variables with polynomial-size ordinary-resolution proofs for which every $\mathrm{Res}(\oplus)$ refutation of size $S$ and depth $d$ satisfies $d\log(2S)=Ω(N^2)$. This removes the $\log^2 N$ loss in the tradeoff of Itsykson, Podolskii, and Shekhovtsov (CCC 2026).

Authors: Xin Li, Yan Zhong

We construct explicit non-malleable affine extractors for every constant entropy rate, with linear output length and exponentially small error, against any fixed number of affine tamperings without fixed points. For every fixed $0<η<1$ and $t$, we also obtain entropy threshold $C_{η,t}n/\log n$, output length $\lfloor n^{1-η}\rfloor$, and error $2^{-n^{1-η}}$ against $t$ tamperings. Our extractors, as well as the directional affine extractors of Li and Zhong (CCC 2024), yield explicit Boolean functions with correlation $2^{-Ω(n)}$ against weakly read-once linear branching programs of size $2^{Ω(n)}$. For non-oblivious decision trees, we prove linear depth lower bounds for queries of each fixed degree $r\ge2$. Applying Li's sumset extractor (FOCS 2023) gives depth $Ω_δ((n/\ell)\log\ell)$ for growing locality $\ell\le n^{1-δ}$, where $0<δ<1$ is fixed. In the same range, directional affine extractors give correlation $2^{-Ω(n/\sqrt\ell)}$ against local trees of depth $c(n/\ell)\log\ell/\log\log\ell$, for a sufficiently small constant $c>0$. Our extractors derandomize the lossless lifting of Efremenko and Itsykson (STOC 2026). For every fixed $0<ξ<1$, this gives explicit polynomial-size unsatisfiable CNFs on $N$ variables whose $\mathrm{Res}(\oplus)$ refutations of resolution depth at most $N$ require size at least $2^{(1-ξ)N}$. Separately, parity substitutions give polynomial-size CNFs on $N$ variables with polynomial-size ordinary-resolution proofs for which every $\mathrm{Res}(\oplus)$ refutation of size $S$ and depth $d$ satisfies $d\log(2S)=Ω(N^2)$. This removes the $\log^2 N$ loss in the tradeoff of Itsykson, Podolskii, and Shekhovtsov (CCC 2026).

Separating QMA from QCIP with a Classical Oracle, or, the Power of Quantum Proofs over Classical Interaction for Quantum Verifiers

from arXiv: Computational Complexity

Authors: Alper Cakan

Whether some problems require quantum proofs has been a central question in quantum complexity (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC 2007). Recently, breakthrough work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), followed by a simpler separation due to Bostanci, Huang, and Vaikuntanathan (FOCS 2026), established a classical oracle separation between $\mathsf{QMA}$ and $\mathsf{QCMA}$. However, while they are not in $\mathsf{QCMA}$, the problems used in both separations still lie in $\mathsf{AM}$: they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class $\mathsf{QCIP}$, introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle $\mathcal{O}$ such that $\mathsf{QMA}^{\mathcal{O}}\not\subseteq\mathsf{QCIP}^{\mathcal{O}}$, thus showing that some languages indeed require quantum proofs, with no classical replacements. Since $\mathsf{QCMA}=\mathsf{QCIP}[1]$, this strengthens the earlier $\mathsf{QMA}$--$\mathsf{QCMA}$ separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.

Authors: Alper Cakan

Whether some problems require quantum proofs has been a central question in quantum complexity (Aharonov and Naveh, 2002; Aaronson and Kuperberg, CCC 2007). Recently, breakthrough work of Bostanci, Haferkamp, Nirkhe, and Zhandry (STOC 2026), followed by a simpler separation due to Bostanci, Huang, and Vaikuntanathan (FOCS 2026), established a classical oracle separation between $\mathsf{QMA}$ and $\mathsf{QCMA}$. However, while they are not in $\mathsf{QCMA}$, the problems used in both separations still lie in $\mathsf{AM}$: they admit a two-message public-coin proof system with a classical verifier. In this work, we ask whether some problems truly require quantum proofs. More formally, we consider the complexity class $\mathsf{QCIP}$, introduced by Buhrman, Le Gall, and Weggemans (2024), where an efficient quantum verifier interacts with an unbounded prover over a classical channel for an arbitrary polynomial number of rounds. We construct a classical oracle $\mathcal{O}$ such that $\mathsf{QMA}^{\mathcal{O}}\not\subseteq\mathsf{QCIP}^{\mathcal{O}}$, thus showing that some languages indeed require quantum proofs, with no classical replacements. Since $\mathsf{QCMA}=\mathsf{QCIP}[1]$, this strengthens the earlier $\mathsf{QMA}$--$\mathsf{QCMA}$ separations, which now follow as a special case of our result. As a technical contribution, we extend to the complexity theory setting the techniques developed by Cakan, Goyal, and Shmueli (CRYPTO 2026) in the context of cryptography for analyzing classically interacting quantum machines. We believe this may be of independent interest.

Exponential quantum space advantage in random data streams

from arXiv: Computational Complexity

Authors: Adam Bouland, Matthew Ding, Siddhartha Jain

We show two unconditional quantum space advantages in the random-order streaming model. First, we show the Yamakawa--Zhandry Code Intersection problem admits exponential quantum advantage in the streaming model when its inputs are streamed in random order. This means quantum computers exhibit exponential space advantage even when simply receiving $(x,f(x))$ pairs for a uniformly random function $f$ in a uniformly random order. Our lower bound is shown using density-restoring partitions as in the work of Göös, Gur, Jain, and Li (STOC 2025) combined with a convex potential, similar to the work of Raz (J.ACM 2018) on parity learning and its generalization by Garg, Raz, and Tal (STOC 2018). Second, we use our framework to show quantum space advantage for the Optimal Polynomial Intersection (OPI) problem in certain regimes via a streaming version of the Decoded Quantum Interferometry algorithm (Nature 2025; arXiv:2510.10967). In particular, we show that for degree $d$ and $n$ evaluation points, attaining $1/2 + Ω(\sqrt{d/n})$ fraction of satisfied OPI constraints via streaming requires $Ω(n)$ classical bits of memory but only $O(d\log n)$ qubits. This yields provable quantum advantage in a "low-rate" regime when the number of evaluation points is much larger than the degree, with a space advantage that can be as large as exponential in certain parameter settings.

Authors: Adam Bouland, Matthew Ding, Siddhartha Jain

We show two unconditional quantum space advantages in the random-order streaming model. First, we show the Yamakawa--Zhandry Code Intersection problem admits exponential quantum advantage in the streaming model when its inputs are streamed in random order. This means quantum computers exhibit exponential space advantage even when simply receiving $(x,f(x))$ pairs for a uniformly random function $f$ in a uniformly random order. Our lower bound is shown using density-restoring partitions as in the work of Göös, Gur, Jain, and Li (STOC 2025) combined with a convex potential, similar to the work of Raz (J.ACM 2018) on parity learning and its generalization by Garg, Raz, and Tal (STOC 2018). Second, we use our framework to show quantum space advantage for the Optimal Polynomial Intersection (OPI) problem in certain regimes via a streaming version of the Decoded Quantum Interferometry algorithm (Nature 2025; arXiv:2510.10967). In particular, we show that for degree $d$ and $n$ evaluation points, attaining $1/2 + Ω(\sqrt{d/n})$ fraction of satisfied OPI constraints via streaming requires $Ω(n)$ classical bits of memory but only $O(d\log n)$ qubits. This yields provable quantum advantage in a "low-rate" regime when the number of evaluation points is much larger than the degree, with a space advantage that can be as large as exponential in certain parameter settings.

No Size-Preserving Amplification with Quantum Advice

from arXiv: Computational Complexity

Authors: Shih-Han Hung, Han-Hsuan Lin

Marriott and Watrous showed that quantum Merlin--Arthur games admit generic error reduction without increasing witness size [Computational Complexity, 2005]. In this work, we show that this state-size-preserving amplification property does not hold for polynomial-time quantum computation with quantum advice. In particular, we present decision problems for which even a vanishing additive error reduction requires longer advice. More precisely, for every polynomially bounded advice length $m(n)\geq n^4$ and every error bound $\varepsilon(n)$ that stays below $1/2$ by at least an inverse polynomial, there is a positive function $δ$ with $δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)$ such that $\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}m$; for constant $\varepsilon$ the gap is $O(\sqrt{\log m/m})$. Here, $\mathsf{BQP}_\varepsilon/\mathsf{q}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a polynomial-time quantum algorithm with an $m(n)$-qubit advice state that only depends on the input length $n$. We show this by proving a stronger separation $\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m$, where $\mathsf{P}_{\varepsilon}/\mathsf{r}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a deterministic polynomial-time algorithm with an $m(n)$-bit advice string sampled from a distribution that depends only on $n$.

Authors: Shih-Han Hung, Han-Hsuan Lin

Marriott and Watrous showed that quantum Merlin--Arthur games admit generic error reduction without increasing witness size [Computational Complexity, 2005]. In this work, we show that this state-size-preserving amplification property does not hold for polynomial-time quantum computation with quantum advice. In particular, we present decision problems for which even a vanishing additive error reduction requires longer advice. More precisely, for every polynomially bounded advice length $m(n)\geq n^4$ and every error bound $\varepsilon(n)$ that stays below $1/2$ by at least an inverse polynomial, there is a positive function $δ$ with $δ(n)=O\bigl(\min\{(\log m/m)^{1/4},\ \sqrt{\log m/m}\,/(1/2-\varepsilon(n))\}\bigr)$ such that $\mathsf{BQP}_{\varepsilon}/\mathsf{q}m \subsetneq \mathsf{BQP}_{\varepsilon + δ}/\mathsf{q}m$; for constant $\varepsilon$ the gap is $O(\sqrt{\log m/m})$. Here, $\mathsf{BQP}_\varepsilon/\mathsf{q}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a polynomial-time quantum algorithm with an $m(n)$-qubit advice state that only depends on the input length $n$. We show this by proving a stronger separation $\mathsf{P}_{\varepsilon+δ}/\mathsf{r} m \not\subset \mathsf{BQP}_{\varepsilon}/\mathsf{q} m$, where $\mathsf{P}_{\varepsilon}/\mathsf{r}m$ is the class of languages recognizable with error at most $\varepsilon(n)$ by a deterministic polynomial-time algorithm with an $m(n)$-bit advice string sampled from a distribution that depends only on $n$.

The Reeb Transform

from arXiv: Computational Geometry

Authors: Erin Chambers, Shankha Shubhra Mukherjee, Katharine Turner

Reeb Transforms offer a compact representation of how shapes in $\mathbb{R}^n$ change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in $\mathbb{R}^d$ and on surfaces in $\mathbb{R}^3$. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in $\mathbb{R}^3$, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.

Authors: Erin Chambers, Shankha Shubhra Mukherjee, Katharine Turner

Reeb Transforms offer a compact representation of how shapes in $\mathbb{R}^n$ change when sliced along varying directions. We define the Reeb Transform as the family of Reeb graphs induced by height functions along all directions in the unit sphere. In this paper, we develop a rigorous treatment of Reeb Transforms for o-minimal definable sets, with particular emphasis on one-dimensional stratified spaces embedded in $\mathbb{R}^d$ and on surfaces in $\mathbb{R}^3$. We establish injectivity of the Reeb Transform in multiple settings, including compact surfaces in $\mathbb{R}^3$, capturing the essential topological features needed to uniquely reconstruct such surfaces. However, in dimensions above three, the Reeb Transform ceases to be injective, indicating the limitations of this descriptor in higher-dimensional settings.

Point and Line Nearest-Neighbor Searching in 3-Space

from arXiv: Computational Geometry

Authors: Pankaj K. Agarwal, Esther Ezra, Micha Sharir

This paper presents data structures for nearest-neighbor (NN) searching problems involving points, lines, segments, and triangles in 3-space, achieving significantly better performance than the previously best-known results for these problems. For example, we present a linear-size data structure for answering NN queries with lines or segments amid $n$ points in 3-space with $O^*(n^{1/2})$ query time (where the $O^*(\cdot)$ notation hides subpolynomial factors). We also present a data structure of $O^*(n^4)$ size that answers such queries in $O^*(1)$ time. For the converse problem, in which we seek the nearest neighbor of a query point amid $n$ lines, segments, or triangles in 3-space, we present a linear-size data structure with $O^*(n^{2/3})$ query time. These results constitute a significant improvement over previous solutions. We obtain improved solutions for the two extreme regimes of (near-)linear storage and of fast query time. These results also yield trade-off bounds between the query time and the size of the data structure. Our results rely on several combinatorial and algorithmic results on arrangements of surfaces in 3-space and 4-space, particularly on recent results on vertical decompositions of substructures in such arrangements established by the authors.

Authors: Pankaj K. Agarwal, Esther Ezra, Micha Sharir

This paper presents data structures for nearest-neighbor (NN) searching problems involving points, lines, segments, and triangles in 3-space, achieving significantly better performance than the previously best-known results for these problems. For example, we present a linear-size data structure for answering NN queries with lines or segments amid $n$ points in 3-space with $O^*(n^{1/2})$ query time (where the $O^*(\cdot)$ notation hides subpolynomial factors). We also present a data structure of $O^*(n^4)$ size that answers such queries in $O^*(1)$ time. For the converse problem, in which we seek the nearest neighbor of a query point amid $n$ lines, segments, or triangles in 3-space, we present a linear-size data structure with $O^*(n^{2/3})$ query time. These results constitute a significant improvement over previous solutions. We obtain improved solutions for the two extreme regimes of (near-)linear storage and of fast query time. These results also yield trade-off bounds between the query time and the size of the data structure. Our results rely on several combinatorial and algorithmic results on arrangements of surfaces in 3-space and 4-space, particularly on recent results on vertical decompositions of substructures in such arrangements established by the authors.

Hamiltonian locality testing and certification do not achieve the Heisenberg limit

from arXiv: Data Structures and Algorithms

Authors: Francisco Escudero Gutiérrez, Junseo Lee, Sebastian Zur

We establish lower bounds for Hamiltonian property testing with access to the time-evolution operator but not its inverse. Each experiment may query the time-evolution operator multiple times, and distances between Hamiltonians are measured in the normalized Frobenius norm. In this model, we show that testing whether a Hamiltonian is $k$-local or $\varepsilon$-far from every $k$-local Hamiltonian requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Kallaugher and Liang (TQC'25). We also prove that testing whether an unknown Hamiltonian equals a target Hamiltonian or is $\varepsilon$-far from it requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Sinha and Tong (2025). These are the first lower bounds for natural problems in Hamiltonian learning and testing that rule out Heisenberg-limited scaling of $1/\varepsilon$. As a third result, we show that amplitude estimation to precision $\varepsilon$ requires $Ω(1/\varepsilon^2)$ total time evolution, recovering the result of Tang and Wright (QIP'26) in the continuous-time query model. All three results follow from the hardness of distinguishing the zero Hamiltonian from a suitably chosen ensemble of random Hamiltonians. We establish this hardness by adapting the continuous-time adversary method to forward Hamiltonian evolution.

Authors: Francisco Escudero Gutiérrez, Junseo Lee, Sebastian Zur

We establish lower bounds for Hamiltonian property testing with access to the time-evolution operator but not its inverse. Each experiment may query the time-evolution operator multiple times, and distances between Hamiltonians are measured in the normalized Frobenius norm. In this model, we show that testing whether a Hamiltonian is $k$-local or $\varepsilon$-far from every $k$-local Hamiltonian requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Kallaugher and Liang (TQC'25). We also prove that testing whether an unknown Hamiltonian equals a target Hamiltonian or is $\varepsilon$-far from it requires $Ω(1/\varepsilon^2)$ total evolution time, matching the upper bound of Sinha and Tong (2025). These are the first lower bounds for natural problems in Hamiltonian learning and testing that rule out Heisenberg-limited scaling of $1/\varepsilon$. As a third result, we show that amplitude estimation to precision $\varepsilon$ requires $Ω(1/\varepsilon^2)$ total time evolution, recovering the result of Tang and Wright (QIP'26) in the continuous-time query model. All three results follow from the hardness of distinguishing the zero Hamiltonian from a suitably chosen ensemble of random Hamiltonians. We establish this hardness by adapting the continuous-time adversary method to forward Hamiltonian evolution.

Hamiltonian Eigenvalue Transformation by Tridiagonal Gadgets

from arXiv: Data Structures and Algorithms

Authors: Arthur Braida, Joseph Cunningham, Jérémie Roland

An analog device implements a local Hamiltonian H, but a polynomial P (H) is in general not local, so the device cannot implement it. We carry P (H), to any prescribed accuracy, as the action of a single time-independent local Hamiltonian on an explicitly described invariant subspace. Short chains of ancilla qubits are attached to H. A chain of 2m sites has a unique isolated eigenvalue that is an analytic function of the input vanishing to order exactly 2m, because the input must cross the chain and come back before it can shift the energy at the far end. Chains of different lengths therefore form a triangular family, and a weighted sum of them reproduces a prescribed polynomial term by term. Even chains give the even part and odd chains the odd part, so any polynomial is reached. We prove uniqueness, analyticity and coefficient bounds uniform in the chain length for inputs H of norm below any r < 1. Where the circuit model pays degree 2l in 2l sequential oracle calls, the result here is one Hamiltonian of locality two more than that of H, which pays the degree in O(l^2 + log2^(1/eps)) ancillas and in energy scale. As an application we filter a marked eigenstate of a local Hamiltonian. Composing a synthesised square along the Chebyshev doubling identity works, but its energy scale grows quasi-polynomially with the degree. Iterating instead the exact eigenvalue branch of the two-site chain k times gives a filter on k + 1 ancilla qubits whose error decays geometrically in k, at an energy scale polynomial in the inverse passband width and independent of k, the locality growing by one per stage. In adiabatic optimisation with a rank-one driver, the construction replaces that non-local driver, by a Hamiltonian with O(log n) ancillas and O(log n)- body terms, at an energy scale polynomial in n. Simulations including every ancilla reproduce the spectrum of the ideal algorithm.

Authors: Arthur Braida, Joseph Cunningham, Jérémie Roland

An analog device implements a local Hamiltonian H, but a polynomial P (H) is in general not local, so the device cannot implement it. We carry P (H), to any prescribed accuracy, as the action of a single time-independent local Hamiltonian on an explicitly described invariant subspace. Short chains of ancilla qubits are attached to H. A chain of 2m sites has a unique isolated eigenvalue that is an analytic function of the input vanishing to order exactly 2m, because the input must cross the chain and come back before it can shift the energy at the far end. Chains of different lengths therefore form a triangular family, and a weighted sum of them reproduces a prescribed polynomial term by term. Even chains give the even part and odd chains the odd part, so any polynomial is reached. We prove uniqueness, analyticity and coefficient bounds uniform in the chain length for inputs H of norm below any r < 1. Where the circuit model pays degree 2l in 2l sequential oracle calls, the result here is one Hamiltonian of locality two more than that of H, which pays the degree in O(l^2 + log2^(1/eps)) ancillas and in energy scale. As an application we filter a marked eigenstate of a local Hamiltonian. Composing a synthesised square along the Chebyshev doubling identity works, but its energy scale grows quasi-polynomially with the degree. Iterating instead the exact eigenvalue branch of the two-site chain k times gives a filter on k + 1 ancilla qubits whose error decays geometrically in k, at an energy scale polynomial in the inverse passband width and independent of k, the locality growing by one per stage. In adiabatic optimisation with a rank-one driver, the construction replaces that non-local driver, by a Hamiltonian with O(log n) ancillas and O(log n)- body terms, at an energy scale polynomial in n. Simulations including every ancilla reproduce the spectrum of the ideal algorithm.

Monge matrix searching for lot sizing with piecewise-concave production costs

from arXiv: Data Structures and Algorithms

Authors: Kleitos Papadopoulos

We study single-item lot sizing with piecewise-concave production and holding--backlog costs over $T$ periods. Production has $m$ common positive finite breakpoints, and stock costs and domains have $K$ common finite boundary levels. For fixed $m,K$, we give an exact deterministic algorithm using $\OmK(T^{m+2}α(T+2))$ arithmetic operations and a Las Vegas algorithm using $\OmK(T^{m+2})$ expected operations, where $α$ is the inverse Ackermann function. The algorithms retain nonlinear concave cost pieces and cover backlogging, common-grid stock limits, and explicit terminal-stock policies. A structural decomposition at stock boundary levels yields breakpoint-only prefix and suffix tables. Batching transitions by a designated production period produces implicit matrices that are Monge double staircases on each production piece. Established matrix searches, merged state orders, and backward evaluation provide the bounds. A simpler rectangle/SMAWK variant requires $\OmK(T^{m+2}\log(T+2))$ operations. On 300 no-backlogging benchmark cases, its objectives agree exactly with the predecessor dynamic program and an inventory-state oracle. Geometric-mean paired time ratios are \GeoUncap\ for uncapacitated cases and \GeoCap\ for capacitated cases, with ratios above one favoring matrix searching. A 640-case base-model validation suite and a separate 624-case generalized-model suite pass. The timing study does not benchmark the generalized models or the stronger staircase routines. All bounds assume exact arithmetic, constant-time cost queries, and explicit closed-domain concavity conditions.

Authors: Kleitos Papadopoulos

We study single-item lot sizing with piecewise-concave production and holding--backlog costs over $T$ periods. Production has $m$ common positive finite breakpoints, and stock costs and domains have $K$ common finite boundary levels. For fixed $m,K$, we give an exact deterministic algorithm using $\OmK(T^{m+2}α(T+2))$ arithmetic operations and a Las Vegas algorithm using $\OmK(T^{m+2})$ expected operations, where $α$ is the inverse Ackermann function. The algorithms retain nonlinear concave cost pieces and cover backlogging, common-grid stock limits, and explicit terminal-stock policies. A structural decomposition at stock boundary levels yields breakpoint-only prefix and suffix tables. Batching transitions by a designated production period produces implicit matrices that are Monge double staircases on each production piece. Established matrix searches, merged state orders, and backward evaluation provide the bounds. A simpler rectangle/SMAWK variant requires $\OmK(T^{m+2}\log(T+2))$ operations. On 300 no-backlogging benchmark cases, its objectives agree exactly with the predecessor dynamic program and an inventory-state oracle. Geometric-mean paired time ratios are \GeoUncap\ for uncapacitated cases and \GeoCap\ for capacitated cases, with ratios above one favoring matrix searching. A 640-case base-model validation suite and a separate 624-case generalized-model suite pass. The timing study does not benchmark the generalized models or the stronger staircase routines. All bounds assume exact arithmetic, constant-time cost queries, and explicit closed-domain concavity conditions.

A Shortest Augmenting Path Algorithm for Linear Matroid Parity

from arXiv: Data Structures and Algorithms

Authors: Kou Hamada, Satoru Iwata

The matroid parity problem serves as a fundamental framework that generalizes both graph matching and matroid intersection. Although the general version is intractable, Lovász (1981) developed a polynomial-time algorithm for the linear matroid parity problem, assuming the availability of matrix representations. Subsequently, Gabow and Stallmann (1986) presented an augmenting path algorithm, which has long been recognized as one of the fastest deterministic algorithms. Since shortest augmenting paths improved algorithms for graph matching (Micali & Vazirani, 1980) and linear matroid intersection (Cunningham, 1986), extending these techniques to linear matroid parity appears to be a natural progression. However, such an algorithm has remained elusive for four decades. In this paper, we present the first shortest augmenting path algorithm for linear matroid parity. Our approach synthesizes the augmenting path algorithm of Gabow and Stallmann with the synchronized blossom formation of the Micali$\unicode{8211}$Vazirani framework. Our key technical contributions are threefold: (i) a linear-algebraic argument that bounds the lengths of shortest augmenting paths for linear matroid parity, which generalizes Cunningham's bound for linear matroid intersection; (ii) an a priori characterization of shortest search paths through lower bounds on their lengths; and (iii) an extension of the structural properties for graph matching established by Izumi, Kitamura, and Yamaguchi (2025) to the linear matroid parity setting. Our algorithm deterministically solves the linear matroid parity problem in ${\rm O}(nr^2\log r)$ time, where $n$ is the ground set size and $r$ is the matroid rank. By incorporating fast matrix multiplication, this complexity can be further reduced to ${\rm O}(nr^2)$. These results improve upon the long-standing deterministic bounds of ${\rm O}(nr^3)$ and ${\rm O}(nr^ω)$.

Authors: Kou Hamada, Satoru Iwata

The matroid parity problem serves as a fundamental framework that generalizes both graph matching and matroid intersection. Although the general version is intractable, Lovász (1981) developed a polynomial-time algorithm for the linear matroid parity problem, assuming the availability of matrix representations. Subsequently, Gabow and Stallmann (1986) presented an augmenting path algorithm, which has long been recognized as one of the fastest deterministic algorithms. Since shortest augmenting paths improved algorithms for graph matching (Micali & Vazirani, 1980) and linear matroid intersection (Cunningham, 1986), extending these techniques to linear matroid parity appears to be a natural progression. However, such an algorithm has remained elusive for four decades. In this paper, we present the first shortest augmenting path algorithm for linear matroid parity. Our approach synthesizes the augmenting path algorithm of Gabow and Stallmann with the synchronized blossom formation of the Micali$\unicode{8211}$Vazirani framework. Our key technical contributions are threefold: (i) a linear-algebraic argument that bounds the lengths of shortest augmenting paths for linear matroid parity, which generalizes Cunningham's bound for linear matroid intersection; (ii) an a priori characterization of shortest search paths through lower bounds on their lengths; and (iii) an extension of the structural properties for graph matching established by Izumi, Kitamura, and Yamaguchi (2025) to the linear matroid parity setting. Our algorithm deterministically solves the linear matroid parity problem in ${\rm O}(nr^2\log r)$ time, where $n$ is the ground set size and $r$ is the matroid rank. By incorporating fast matrix multiplication, this complexity can be further reduced to ${\rm O}(nr^2)$. These results improve upon the long-standing deterministic bounds of ${\rm O}(nr^3)$ and ${\rm O}(nr^ω)$.

The Power of Flexible Budgets in Adwords

from arXiv: Data Structures and Algorithms

Authors: Suho Kang, Rajan Udwani

Search advertising platforms routinely spend beyond an advertiser's average daily budget on high-traffic days, so long as total spending over the month stays within the monthly budget. Motivated by this practice, we study a $D$-day generalization of the Adwords problem (Mehta et al. 2007), where each advertiser $i$ has a nominal (average) daily budget $B_i$ and a total horizon (monthly) budget $DB_i$. Given a flexibility parameter $δ$, the platform may spend at most $δB_i$ on advertiser $i$ on any single day, subject to the horizon spending limit of $DB_i$. We quantify the power of $δ$-flexible budgets by benchmarking against the inflexible offline optimum, which may spend at most $B_i$ on advertiser $i$ on each day. We show that no amount of flexibility helps direct generalizations of the classical algorithm of Mehta et al. (2007). By contrast, for every fixed $δ$, we design an algorithm whose competitive ratio converges to $1-e^{-δ}$ as $D\to\infty$, and we show that this is asymptotically optimal. Perhaps surprisingly, this matches the optimal competitive ratio in a more permissive setting where the algorithm receives a fresh spending limit of $δB_i$ each day and may spend up to $δD B_i$ over the horizon. Along the way, we characterize the exact optimal competitive ratio for every pair $(D,δ)$ on high-traffic instances, where the offline benchmark exhausts every advertiser's budget on every day.

Authors: Suho Kang, Rajan Udwani

Search advertising platforms routinely spend beyond an advertiser's average daily budget on high-traffic days, so long as total spending over the month stays within the monthly budget. Motivated by this practice, we study a $D$-day generalization of the Adwords problem (Mehta et al. 2007), where each advertiser $i$ has a nominal (average) daily budget $B_i$ and a total horizon (monthly) budget $DB_i$. Given a flexibility parameter $δ$, the platform may spend at most $δB_i$ on advertiser $i$ on any single day, subject to the horizon spending limit of $DB_i$. We quantify the power of $δ$-flexible budgets by benchmarking against the inflexible offline optimum, which may spend at most $B_i$ on advertiser $i$ on each day. We show that no amount of flexibility helps direct generalizations of the classical algorithm of Mehta et al. (2007). By contrast, for every fixed $δ$, we design an algorithm whose competitive ratio converges to $1-e^{-δ}$ as $D\to\infty$, and we show that this is asymptotically optimal. Perhaps surprisingly, this matches the optimal competitive ratio in a more permissive setting where the algorithm receives a fresh spending limit of $δB_i$ each day and may spend up to $δD B_i$ over the horizon. Along the way, we characterize the exact optimal competitive ratio for every pair $(D,δ)$ on high-traffic instances, where the offline benchmark exhausts every advertiser's budget on every day.

Job Scheduling with Battery Recharging Constraints

from arXiv: Data Structures and Algorithms

Authors: Rudransh Kumar, Nima Nasiri, Jared Paul, Sathish Gopalakrishnan

A battery-powered device, such as a delivery drone that works from a depot, must stop to recharge between jobs, and the time spent recharging delays every job that follows. Scheduling models that fix the duration of a recharge do not describe a device whose recharge takes longer when it acquires more energy. We study a single device that executes a batch of non-preemptive jobs with known execution times, energy demands, and optional deadlines. Under \emph{partial recharging} the device may acquire any amount of energy between jobs; under \emph{complete recharging} every recharge fills the battery. Four objectives and four relationships between execution time and energy demand give 32 variants. When acquiring $q$ units of energy takes $q$ time units, we show that 14 of the 16 partial-recharging variants are polynomial, and we give a tight 2-approximation for the average completion time, one of the two NP-hard variants. Under complete recharging, the four variants with equal energy demands are polynomial and the other 12 are NP-hard; for makespan we give a $5/4$-approximation. When each recharge also incurs a fixed \emph{setup time} $h$, the equal-energy variants remain polynomial and the other 24 are strongly NP-hard if $h$ is part of the input. For these we give exact algorithms that are exponential only in the number of jobs, and approximation algorithms for makespan and, when the battery starts empty, for the weighted average completion time. Experiments on synthetic and trace-derived job sets compare the algorithms with exact optima. The model is offline and deterministic, and we have not validated the schedules on hardware.

Authors: Rudransh Kumar, Nima Nasiri, Jared Paul, Sathish Gopalakrishnan

A battery-powered device, such as a delivery drone that works from a depot, must stop to recharge between jobs, and the time spent recharging delays every job that follows. Scheduling models that fix the duration of a recharge do not describe a device whose recharge takes longer when it acquires more energy. We study a single device that executes a batch of non-preemptive jobs with known execution times, energy demands, and optional deadlines. Under \emph{partial recharging} the device may acquire any amount of energy between jobs; under \emph{complete recharging} every recharge fills the battery. Four objectives and four relationships between execution time and energy demand give 32 variants. When acquiring $q$ units of energy takes $q$ time units, we show that 14 of the 16 partial-recharging variants are polynomial, and we give a tight 2-approximation for the average completion time, one of the two NP-hard variants. Under complete recharging, the four variants with equal energy demands are polynomial and the other 12 are NP-hard; for makespan we give a $5/4$-approximation. When each recharge also incurs a fixed \emph{setup time} $h$, the equal-energy variants remain polynomial and the other 24 are strongly NP-hard if $h$ is part of the input. For these we give exact algorithms that are exponential only in the number of jobs, and approximation algorithms for makespan and, when the battery starts empty, for the weighted average completion time. Experiments on synthetic and trace-derived job sets compare the algorithms with exact optima. The model is offline and deterministic, and we have not validated the schedules on hardware.

Fine-Grained Analysis of SIMD-Based Hash Table Implementations

from arXiv: Data Structures and Algorithms

Authors: Cyril Nicaud, Pablo Rotondo

In recent years, several variants of classical hash table schemes have been developed by engineers in order to take advantage of the processor's internal parallelism using SIMD instructions, which make it possible to operate on multiple bytes simultaneously. At a small additional memory cost, this enables a significant speedup, making it a data structure that is increasingly popular in practice, when very high performance is required. In this article, we provide a detailed theoretical analysis of the dynamics of such hash tables. From a methodological standpoint, we use and adapt a technique developed by Wormald in the 1990s to study dynamic graphs. This approach, which can be adapted to many variants, enables us to accurately estimate the quantities of interest by capturing the dynamics of the data structure through systems of differential equations. Although complex, we provide an explicit description of the solutions of these systems, which can furthermore be efficiently approximated numerically. Our main results are stated with high probability, which is significantly more precise than average-case analyses, and they match experimental results remarkably well, even for hash tables of moderate size.

Authors: Cyril Nicaud, Pablo Rotondo

In recent years, several variants of classical hash table schemes have been developed by engineers in order to take advantage of the processor's internal parallelism using SIMD instructions, which make it possible to operate on multiple bytes simultaneously. At a small additional memory cost, this enables a significant speedup, making it a data structure that is increasingly popular in practice, when very high performance is required. In this article, we provide a detailed theoretical analysis of the dynamics of such hash tables. From a methodological standpoint, we use and adapt a technique developed by Wormald in the 1990s to study dynamic graphs. This approach, which can be adapted to many variants, enables us to accurately estimate the quantities of interest by capturing the dynamics of the data structure through systems of differential equations. Although complex, we provide an explicit description of the solutions of these systems, which can furthermore be efficiently approximated numerically. Our main results are stated with high probability, which is significantly more precise than average-case analyses, and they match experimental results remarkably well, even for hash tables of moderate size.

Simple analysis of an algorithm for multiple-source shortest paths in planar graphs

from arXiv: Data Structures and Algorithms

Authors: Philip N. Klein

This paper addresses the following problem: given a planar embedding graph, compute a representation of the shortest-path trees rooted at all the boundary nodes of the graph. Klein gave an $O(n \log n)$ algorithm for this problem; the algorithm subsequently became an essential ingredient in dozens of algorithms, addressing problems ranging from distance oracles to edit distance. However, the correctness and analysis in that original paper is complicated and messy and hard to understand. In this paper, we give a simple and clear analysis.

Authors: Philip N. Klein

This paper addresses the following problem: given a planar embedding graph, compute a representation of the shortest-path trees rooted at all the boundary nodes of the graph. Klein gave an $O(n \log n)$ algorithm for this problem; the algorithm subsequently became an essential ingredient in dozens of algorithms, addressing problems ranging from distance oracles to edit distance. However, the correctness and analysis in that original paper is complicated and messy and hard to understand. In this paper, we give a simple and clear analysis.

Normal-Form Correlation in Markov Games

from arXiv: Data Structures and Algorithms

Authors: Ioannis Anagnostides, Constantinos Daskalakis, Gabriele Farina, Noah Golowich, Tuomas Sandholm, Brian Hu Zhang

There has been a surge of recent work on correlated equilibrium concepts in Markov games. However, existing results focus on concepts weaker than normal-form correlated equilibria (NFCEs), leaving open the more challenging question of computing such equilibria, which goes back to the seminal work of Papadimitriou and Roughgarden (JACM'08). Here, we establish the first efficient algorithm for NFCEs in finite-horizon Markov games with a fixed number of players $n$. In particular, with $S$ states, horizon $H$, and at most $A$ actions per player, it computes an $ε$-NFCE in time $S(AH/ε)^{O(n)}$. This is the first algorithm polynomial in $1/ε$ and the description of the game for NFCEs in an interesting class of problems beyond the normal-form setting. Moreover, under the usual assumption that recommendations are independent across states, we show PPAD-completeness---that is, computational equivalence to Nash equilibria---either in many-player games or when the precision is exponentially small. The key idea behind our approach is to run backward induction on a sequence of auxiliary stage games, but with the twist that in each step we compute a constant-expectation correlated equilibrium. This is a natural refinement of correlated equilibrium in which the conditional expected payoff from obeying is independent of the recommendation. In fact, our reduction goes both ways, establishing an equivalence between constant-expectation CEs and NFCEs in Markov games. For a fixed number of players, we observe that a constant-expectation CE can be computed approximately by combining linear programming with suitable discretization. In contrast, it is PPAD-hard in i) polymatrix (many-player) games at constant precision, and ii) two-player games at exponentially small precision. The latter result follows from an unexpected connection to rank-2 two-player games.

Authors: Ioannis Anagnostides, Constantinos Daskalakis, Gabriele Farina, Noah Golowich, Tuomas Sandholm, Brian Hu Zhang

There has been a surge of recent work on correlated equilibrium concepts in Markov games. However, existing results focus on concepts weaker than normal-form correlated equilibria (NFCEs), leaving open the more challenging question of computing such equilibria, which goes back to the seminal work of Papadimitriou and Roughgarden (JACM'08). Here, we establish the first efficient algorithm for NFCEs in finite-horizon Markov games with a fixed number of players $n$. In particular, with $S$ states, horizon $H$, and at most $A$ actions per player, it computes an $ε$-NFCE in time $S(AH/ε)^{O(n)}$. This is the first algorithm polynomial in $1/ε$ and the description of the game for NFCEs in an interesting class of problems beyond the normal-form setting. Moreover, under the usual assumption that recommendations are independent across states, we show PPAD-completeness---that is, computational equivalence to Nash equilibria---either in many-player games or when the precision is exponentially small. The key idea behind our approach is to run backward induction on a sequence of auxiliary stage games, but with the twist that in each step we compute a constant-expectation correlated equilibrium. This is a natural refinement of correlated equilibrium in which the conditional expected payoff from obeying is independent of the recommendation. In fact, our reduction goes both ways, establishing an equivalence between constant-expectation CEs and NFCEs in Markov games. For a fixed number of players, we observe that a constant-expectation CE can be computed approximately by combining linear programming with suitable discretization. In contrast, it is PPAD-hard in i) polymatrix (many-player) games at constant precision, and ii) two-player games at exponentially small precision. The latter result follows from an unexpected connection to rank-2 two-player games.

The Plan Language of a Curriculum: A Formal Model and the Complexity of Degree Planning

from arXiv: Data Structures and Algorithms

Authors: Sherzod Turaev, Mary John, Mamoun Awad

We model an academic curriculum as a generator of a language of feasible study plans: prerequisites are monotone Boolean formulas in conjunctive normal form, degree requirements are credit-threshold covering constraints, and a study plan is a sequence of terms bounded by a per-term credit capacity. Within this model, we settle the complexity of the two natural planning objectives, the number of terms to a degree and the total credit load, and we isolate the structural commitment responsible for each source of hardness. Time to degree is polynomial whenever the per-term capacity is unbounded, for arbitrary disjunctive prerequisites and arbitrary electives, so disjunction never contributes to its hardness, yet it becomes strongly NP-hard as soon as capacity binds, even without any prerequisite. Load is complementary: disjunction and overlapping electives are each strongly NP-hard in isolation and their complexity does not depend on capacity, while load is polynomial on the conjunctive, mandatory fragment. The two objectives therefore have disjoint sources of hardness. We show that the delay-factor component of the standard curricular-complexity metric is a polynomially computable upper bound on time to degree, exact on the conjunctive fragment and loose elsewhere by a quantity we name the disjunctive slack, and we prove that program subsumption is coNP-complete and consensus prerequisite recovery is NP-complete. Instantiating the model on a corpus of twenty-two universities, we find that 88 percent of prerequisite-bearing courses are purely conjunctive and that capacity, not prerequisite logic, is the operative constraint on time to degree. The curriculum corpus is openly available (doi.org/10.5281/zenodo.22334674), and the analysis and figure-generation code accompany the paper.

Authors: Sherzod Turaev, Mary John, Mamoun Awad

We model an academic curriculum as a generator of a language of feasible study plans: prerequisites are monotone Boolean formulas in conjunctive normal form, degree requirements are credit-threshold covering constraints, and a study plan is a sequence of terms bounded by a per-term credit capacity. Within this model, we settle the complexity of the two natural planning objectives, the number of terms to a degree and the total credit load, and we isolate the structural commitment responsible for each source of hardness. Time to degree is polynomial whenever the per-term capacity is unbounded, for arbitrary disjunctive prerequisites and arbitrary electives, so disjunction never contributes to its hardness, yet it becomes strongly NP-hard as soon as capacity binds, even without any prerequisite. Load is complementary: disjunction and overlapping electives are each strongly NP-hard in isolation and their complexity does not depend on capacity, while load is polynomial on the conjunctive, mandatory fragment. The two objectives therefore have disjoint sources of hardness. We show that the delay-factor component of the standard curricular-complexity metric is a polynomially computable upper bound on time to degree, exact on the conjunctive fragment and loose elsewhere by a quantity we name the disjunctive slack, and we prove that program subsumption is coNP-complete and consensus prerequisite recovery is NP-complete. Instantiating the model on a corpus of twenty-two universities, we find that 88 percent of prerequisite-bearing courses are purely conjunctive and that capacity, not prerequisite logic, is the operative constraint on time to degree. The curriculum corpus is openly available (https://doi.org/10.5281/zenodo.22334674), and the analysis and figure-generation code accompany the paper.

Quantum Simulation on Riemannian Manifolds

from arXiv: Data Structures and Algorithms

Authors: Dylan Herman, Jacob Watkins, Guneykan Ozgul, Jiayu Shen, Brandon Augustino, Junhyung Lyle Kim, Shouvanik Chakrabarti

We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and $\ell_p$ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.

Authors: Dylan Herman, Jacob Watkins, Guneykan Ozgul, Jiayu Shen, Brandon Augustino, Junhyung Lyle Kim, Shouvanik Chakrabarti

We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace--Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace--Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright--Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and $\ell_p$ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.

Single-Sample Prophet Inequalities: A Combinatorial to Single-Item Reduction

from arXiv: Data Structures and Algorithms

Authors: Shuchi Chawla, Trung Dang

We study single-sample prophet inequalities for online combinatorial allocation. Our main contribution is a general reduction from combinatorial to single-item prophet inequalities for valuation classes admitting suitable supporting prices. The reduction uses a free-disposal value to separate buyer-side combinatorial constraints from item-side supply constraints, yielding a modular framework that applies in the stronger Game of Googol model. This framework yields a $\frac{1}{6\sqrt{3}}\approx\frac{1}{10.4}$-competitive single-sample prophet inequality and a $(β_{k-1}/4)$-competitive $k$-sample prophet inequality for XOS valuations, where $β_k$ is the competitive ratio of a $k$-sample single-item prophet inequality, improving upon the work of [DKL+24]. Both results extend directly to divisible resources with capped-XOS valuations. Along the way, we obtain new results for online free disposal and an optimal single-sample prophet inequality for fractional knapsack in the Game of Googol model.

Authors: Shuchi Chawla, Trung Dang

We study single-sample prophet inequalities for online combinatorial allocation. Our main contribution is a general reduction from combinatorial to single-item prophet inequalities for valuation classes admitting suitable supporting prices. The reduction uses a free-disposal value to separate buyer-side combinatorial constraints from item-side supply constraints, yielding a modular framework that applies in the stronger Game of Googol model. This framework yields a $\frac{1}{6\sqrt{3}}\approx\frac{1}{10.4}$-competitive single-sample prophet inequality and a $(β_{k-1}/4)$-competitive $k$-sample prophet inequality for XOS valuations, where $β_k$ is the competitive ratio of a $k$-sample single-item prophet inequality, improving upon the work of [DKL+24]. Both results extend directly to divisible resources with capped-XOS valuations. Along the way, we obtain new results for online free disposal and an optimal single-sample prophet inequality for fractional knapsack in the Game of Googol model.

Faster Sublinear Maximal Independent Set Size

from arXiv: Data Structures and Algorithms

Authors: Peter Kiss, Arash Kooroshnezhad

We give a sublinear-time algorithm for estimating the size of a maximal independent set in a graph using adjacency-query access with expected running time $\tilde{O}(n^{1+1/3})$, improving over the previous $\tilde{O}(n^{1+1/2})$ bound of Mahadabi et al. [MRTV26]. As a consequence of a reduction of [MRTV26], this also improves the running time for sublinear metric Steiner forest. We further show that a bi-criteria estimate of the $k$-center objective in general metrics can be obtained via a reduction to maximal independent set size estimation.

Authors: Peter Kiss, Arash Kooroshnezhad

We give a sublinear-time algorithm for estimating the size of a maximal independent set in a graph using adjacency-query access with expected running time $\tilde{O}(n^{1+1/3})$, improving over the previous $\tilde{O}(n^{1+1/2})$ bound of Mahadabi et al. [MRTV26]. As a consequence of a reduction of [MRTV26], this also improves the running time for sublinear metric Steiner forest. We further show that a bi-criteria estimate of the $k$-center objective in general metrics can be obtained via a reduction to maximal independent set size estimation.

Quantum algorithms for orthogonal polynomial transforms

from arXiv: Data Structures and Algorithms

Authors: Anupam Prakash, Shree Hari Sureshbabu, Dylan Herman, Shouvanik Chakrabarti

A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states $\sum_k α_{k} \ket{k}$ coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure $μ(x)$. We provide new discrete and continuous QOPTs for polynomial families in the Askey scheme extending the results for the quantum Hermite transform (Jain et al., STOC'26). Our efficient QOPT algorithms require time $O(\text{polylog}(N, 1/ε))$, where $N$ is the grid size and $ε$ is the error, for the discrete Charlier, Meixner and Krawtchouk transforms and for the integer-order Laguerre transform in the continuous setting. The efficient QOPTs are obtained by uncovering the links between Askey scheme polynomials and Gaussian quantum optical gates and developing a compilation framework for $SU(2)$ and $SU(1,1)$ optical gates on Cartesian grids extending the framework developed by Iyer et al. (arXiv:2602.15180). Further, we reduce the continuous Jacobi transform to the discrete Hahn transform and provide an $O\!\left(N\operatorname{polylog}((N+α+β+1)/ε)\right)$-time Hahn transform, a quadratic speedup over the naive implementation. This is based on a more efficient compilation of the Clebsch--Gordan transform for coupling $\mathrm{SU}(2)$ representations with spins $(j_{1}, j_{2})$. Finally, we develop a new framework for Laguerre transforms for all orders $ν>0$ by fast-forwarding the corresponding radial oscillator using a 3-term chirp decomposition and an efficient algorithm for the Quantum Hankel Transform on a logarithmic grid.

Authors: Anupam Prakash, Shree Hari Sureshbabu, Dylan Herman, Shouvanik Chakrabarti

A quantum orthogonal polynomial transform (QOPT) is an algorithmic primitive that maps a superposition of standard basis states $\sum_k α_{k} \ket{k}$ coherently to a basis of normalized univariate polynomials orthogonal with respect to a probability measure $μ(x)$. We provide new discrete and continuous QOPTs for polynomial families in the Askey scheme extending the results for the quantum Hermite transform (Jain et al., STOC'26). Our efficient QOPT algorithms require time $O(\text{polylog}(N, 1/ε))$, where $N$ is the grid size and $ε$ is the error, for the discrete Charlier, Meixner and Krawtchouk transforms and for the integer-order Laguerre transform in the continuous setting. The efficient QOPTs are obtained by uncovering the links between Askey scheme polynomials and Gaussian quantum optical gates and developing a compilation framework for $SU(2)$ and $SU(1,1)$ optical gates on Cartesian grids extending the framework developed by Iyer et al. (arXiv:2602.15180). Further, we reduce the continuous Jacobi transform to the discrete Hahn transform and provide an $O\!\left(N\operatorname{polylog}((N+α+β+1)/ε)\right)$-time Hahn transform, a quadratic speedup over the naive implementation. This is based on a more efficient compilation of the Clebsch--Gordan transform for coupling $\mathrm{SU}(2)$ representations with spins $(j_{1}, j_{2})$. Finally, we develop a new framework for Laguerre transforms for all orders $ν>0$ by fast-forwarding the corresponding radial oscillator using a 3-term chirp decomposition and an efficient algorithm for the Quantum Hankel Transform on a logarithmic grid.

Grid Theory and Polynomiality in Dynamic Lot-Sizing

from arXiv: Data Structures and Algorithms

Authors: El-Mehdi Mehiri, Nabil Absi, Elodie Suzanne

Why are some dynamic lot-sizing problems polynomial? We address this question by introducing Grid Theory, a structural framework based on cumulative production and the additive structure of production bounds. For a general single-item dynamic lot-sizing model with lower and upper production bounds, there exists an optimal extreme solution in which, within each regeneration interval, all but at most one production quantity lie on a boundary value. This induces additive grids, and the Main Grid Theorem establishes that an optimal cumulative production trajectory can be restricted to these discrete sets. Although the resulting grids may be exponentially large, we introduce the notion of additive dimension to capture production-bound profiles whose boundary sums admit a low-dimensional representation. We show that bounded additive dimension yields a polynomially constructible grid envelope and a polynomial time grid-based dynamic programming algorithm. The framework extends to separable concave costs and establishes polynomial solvability of several families, including constant capacities, minimum order quantities, a fixed number of capacity levels, fixed-degree polynomial capacities, periodic capacities, and piecewise polynomial capacities. In particular, polynomiality may hold even when the number of distinct capacity values grows with the planning horizon. Grid Theory thus identifies additive structure, rather than the number of distinct resource values, as a sufficient mechanism for polynomial solvability.

Authors: El-Mehdi Mehiri, Nabil Absi, Elodie Suzanne

Why are some dynamic lot-sizing problems polynomial? We address this question by introducing Grid Theory, a structural framework based on cumulative production and the additive structure of production bounds. For a general single-item dynamic lot-sizing model with lower and upper production bounds, there exists an optimal extreme solution in which, within each regeneration interval, all but at most one production quantity lie on a boundary value. This induces additive grids, and the Main Grid Theorem establishes that an optimal cumulative production trajectory can be restricted to these discrete sets. Although the resulting grids may be exponentially large, we introduce the notion of additive dimension to capture production-bound profiles whose boundary sums admit a low-dimensional representation. We show that bounded additive dimension yields a polynomially constructible grid envelope and a polynomial time grid-based dynamic programming algorithm. The framework extends to separable concave costs and establishes polynomial solvability of several families, including constant capacities, minimum order quantities, a fixed number of capacity levels, fixed-degree polynomial capacities, periodic capacities, and piecewise polynomial capacities. In particular, polynomiality may hold even when the number of distinct capacity values grows with the planning horizon. Grid Theory thus identifies additive structure, rather than the number of distinct resource values, as a sufficient mechanism for polynomial solvability.

Network Speed Scaling with Competitive Ratios Independent of the Network

from arXiv: Data Structures and Algorithms

Authors: Yash Khanna

In network speed scaling, jobs arrive over time at a network of servers whose speeds can be tuned, every job must be processed by the servers along one of its allowed routes, and the goal is to minimize the total flow time plus the total energy. For stochastic arrivals, the competitive ratio of Vaze and Nair depends on the network, through the lengths of the routes it uses. We show that this dependence can be removed: we give an algorithm, which routes the jobs by solving a convex program and runs every server at a fixed speed, whose competitive ratio depends only on the power functions; for $P(s)=s^2$, it is at most the golden ratio $\varphi\approx1.618$. The key idea is a lower bound on the optimal cost which, like the cost of our algorithm, is a sum over the servers of a function of each server's load, so the analysis reduces to a single server. We also show that the routing rule of Vaze and Nair can be a factor $Ω(L)$ away from optimal, where $L$ is the length of the longest route.

Authors: Yash Khanna

In network speed scaling, jobs arrive over time at a network of servers whose speeds can be tuned, every job must be processed by the servers along one of its allowed routes, and the goal is to minimize the total flow time plus the total energy. For stochastic arrivals, the competitive ratio of Vaze and Nair depends on the network, through the lengths of the routes it uses. We show that this dependence can be removed: we give an algorithm, which routes the jobs by solving a convex program and runs every server at a fixed speed, whose competitive ratio depends only on the power functions; for $P(s)=s^2$, it is at most the golden ratio $\varphi\approx1.618$. The key idea is a lower bound on the optimal cost which, like the cost of our algorithm, is a sum over the servers of a function of each server's load, so the analysis reduces to a single server. We also show that the routing rule of Vaze and Nair can be a factor $Ω(L)$ away from optimal, where $L$ is the length of the longest route.

Sunday, October 04

Tenure-track Faculty (Assistant Professor) at McGill University (apply by November 14, 2026)

from CCI: jobs
The School of Computer Science at McGill University (Montreal, Canada) invites applications for a tenure-track appointment at the rank of Assistant Professor. We are seeking candidates with a PhD (or close to finishing) in Computer Science and expertise at the intersection of quantum computing and computer science. Website: mcgill.wd3.myworkdayjobs.com/en-US/McGill_Careers/job/Tenure-Track-Faculty-Position-in-Computer-Science–Quantum-Computing_JR0000080250 Email: brigitte.pientka@mcgill.ca

The School of Computer Science at McGill University (Montreal, Canada) invites applications for a tenure-track appointment at the rank of Assistant Professor. We are seeking candidates with a PhD (or close to finishing) in Computer Science and expertise at the intersection of quantum computing and computer science.

Website: https://mcgill.wd3.myworkdayjobs.com/en-US/McGill_Careers/job/Tenure-Track-Faculty-Position-in-Computer-Science–Quantum-Computing_JR0000080250
Email: brigitte.pientka@mcgill.ca

By shacharlovett

My new course at UT Austin: AI Alignment Theory

from Scott Aaronson
This semester, I’ve been teaching a brand-new course, entitled CS395T AI Alignment Theory. Here’s the course description: The astounding progress of AI over the past decade has been accompanied by a rising fear: do we really understand how to align and control powerful AI systems—how to get them reliably to do what we wanted, or […]

This semester, I’ve been teaching a brand-new course, entitled CS395T AI Alignment Theory. Here’s the course description:

The astounding progress of AI over the past decade has been accompanied by a rising fear: do we really understand how to align and control powerful AI systems—how to get them reliably to do what we wanted, or would want them to do on reflection, rather than merely what we said? If we succeed at building general-purpose superhuman intelligences along the current paradigm, should we expect that development to go well for humanity? Can we modify the design, training, monitoring, or scaffolding of those intelligences to help ensure that it goes well? While there’s been a great deal of recent empirical work touching on these questions, this course will concentrate mainly on theoretical and mathematical foundations. As a warning, the theoretical foundations of AI alignment have not yet gelled into any one coherent body of results accepted as canonical by the field. Nevertheless, in this course, we’ll read and debate many of the conceptual and mathematical works that have been most influential in the AI alignment field, from both before and during the current LLM revolution. Student presentations, reports, and projects will play a central role.

I vividly remember encountering Eliezer Yudkowsky and his Sequences 20 years ago. I remember thinking: even if these people talk and act like crazy cultists, still, let me bend over backwards to be epistemically virtuous, and entertain their ideas on their merits, as very few academics would. Even if, of course, I ultimately end up rejecting the ideas, on the simple ground that powerful AI is such an absurdly remote prospect that it’s almost impossible to say anything useful about it today, outside the realm of speculative fiction.

For my failure to see what was coming, it seems like an appropriate punishment that I’m now, in 2026, effectively teaching a course on Yudkowsky Studies. And it’s the most important course I can teach.

Well, for some definition of “teach.” The thing about AI alignment is that there’s no textbook (though apparently ILIAD is working on one), no core of nontrivial theorems considered canonical by the field, no real body of mathematical theory at all. This makes it extremely different from the courses I’m used to teaching, like Quantum Information Science or Computability and Complexity.

So we’ve been running the course as a discussion seminar. Every session, a “rapporteur” presents an AI alignment research paper or other reading; then I and others ask questions and discuss. Some of the readings (like Omohundro on the “basic AI drives,” or Hadfield-Menell et al. on the off-switch game) predate the current LLM revolution, while others (like the METR report on the HuggingFace incident or Dario Amodei’s “We Must Pace the Frontier”) are so timely that they were only released while the course was underway. Most are somewhere in between.

I expected to have to make a case to students about why AI alignment is a pressing concern, why it’s no longer science fiction, etc. There was huge demand for the course, and while of course there’s a selection effect, the students who’ve shown up have been extremely engaged, sometimes criticizing the assigned papers for not taking existential risk seriously enough.

Perhaps unsurprisingly, we didn’t get that criticism about our very first assigned reading, which was Eliezer Yudkowsky’s 2022 essay AGI Ruin: A List of Lethalities—one the most canonical statements of what Eliezer believes and why that’s shorter than a book. Which brings me to the topic of the rest of this post! Our rapporteurs are not merely presenting the papers in class; they’re also submitting written reports about what the papers said, what their own thoughts were, and what were the highlights of the class discussion. And, with student permission, I’ll be sharing those reports on this blog!

So, without further ado, I present to you our first report, on Eliezer’s list of lethalities, by Tennyson Bardwell, who I thank for his work. Feel free to discuss in the comment section; some of the students might also chime in. Expect more reports here over the coming weeks.

“AGI Ruin: A List of Lethalities” by Eliezer Yudkowsky: Rapporteur Report by Tennyson Bardwell

UT Austin has a new Computer Science course this fall. Alongside familiar graduate-level classes such as Advanced Computer Networks and Convex Optimization sits CS 395T: AI Alignment Theory, taught by Scott Aaronson. This is one of a growing number of AI Alignment courses taught at academic institutions. Just as concerns over catastrophic consequences for misaligned AGI systems reach a broader public discourse, Eliezer Yudkowsky—one of the loudest voices in the field and author of the first assigned reading in Professor Aaronson’s course—is declaring the cause hopeless.

Thus, the students of AI Alignment Theory began their semester by reading a laundry list of critical problems in AI Alignment research, how failure to solve those problems will result in catastrophic consequences, and the reasons to be pessimistic about both past and future progress on these problems. The essay by Eliezer, titled AGI Ruin: A List of Lethalities and posted to his popular community-driven website LessWrong in 2022, is divided into three sections.

Section A roughly describes the magnitude of the AI Alignment problem. That is, the magnitude of the consequences for a complete failure to align an AGI system to human values before construction. It posits that AGI would quickly catch up to all human knowledge simply by learning from existing human productions (colloquially referred to as “eating the internet”) and then, nearly as quickly, begin to meaningfully surpass human knowledge. AlphaGo Zero is presented as a model both for how this might happen, and how it might be difficult to correctly predict beforehand. Many believed that AlphaGo’s success in the board game Go was chiefly attributed to its ability to learn from the extensive history of human-played games. Less than a year after AlphaGo beat the best human player, the successor system AlphaGo Zero surpassed the original AlphaGo. Unlike its predecessor, AlphaGo Zero was trained in just three days by exclusively playing against itself without seeing a single human game.

This quick ramp from AGI to super-intelligence would pose a different sort of problem than humans are generally used to dealing with. Unlike traditional problems in science and engineering, the consequence for a failed attempt might not leave room for another try. An intelligent entity with a misaligned goal would be well aware that it stands in opposition to humans, and might act deceitfully until in a position to act openly against humans without jeopardizing its own survival. Since most goals benefit from control of power and resources, it seems likely that nearly any goal-driven intelligence would have ample opportunity to be misaligned with human desires.

Section B describes reasons why, by default, any AGI that humans build using current methods is likely to be unaligned even if considerable attention is paid to the topic. This “current method” is gradient descent. That is, incremental progress with respect to some loss function which “punishes” a model for undesirable behavior. A notoriously elusive property of such trained models is the ability to generalize out of their training distributions. To train a primitive model to be aligned to humans might involve learning a great many behavioral rules. However, the sorts of rules needed to keep a drastically smarter agent in check might not always be relevant to simpler models (e.g., “do not emotionally dysregulate humans you speak with” might not be relevant to a simpler model that is less able to reliably get under the skin of humans it operates with, or which is assigned tasks in training which do not benefit from such anti-social behavior).

Eliezer focuses on the misalignment of humans with their creators (evolution or evolutionary pressures) as a critical data point for reasoning about misaligned intelligent systems. Despite being a generally slow process, evolution eventually created a runaway intelligent system (Homo sapiens) which proceeded to dominate the globe, decimate related species, and eventually (it is forecasted) effectuate population decline. That last development is arguably in opposition to the sole imperative demanded by evolution: to reproduce.

Section B also makes time for criticism of the most popular paths toward AI alignment, including interpretability (unworkable, and attempting to train on it evokes Goodhart’s law, incentivizing deceit), using multiple AIs to maintain a balance of power (it is not clear how multiple strong AIs unaligned with humanity results in better outcomes for the weak humans), and corrigibility (it seems impossible to motivate an AI system to effect outcomes without also motivating it to desire its own survival to effectuate said outcomes).

Section C describes a bleak state of affairs in which veterans in AI alignment are unsatisfied with current progress and do not have a plan to deliver tangible solutions before the advent of AGI systems. In particular, Eliezer describes recent results as showy but useless. He believes that even with additional funding, the lack of appropriate evaluation mechanisms will prevent the most effective researchers from rising to the top.

A summary of the landscape, as described by Eliezer, in the flowchart below.

Figure 1: A flow chart of (select) paths described by Eliezer in his essay. A common feature of this flow chart is that many “good states”—such as disabling a misbehaving AGI or choosing not to build an AGI—are not “final” states in the sense that they are not permanent solutions. Such a state merely represent the avoidance of a single potential disaster, rather than the emergence of a new stable world state. Hence, these nodes posses back-arrows.

Despite the bleak content, Eliezer’s colorful prose inspired a lively class discussion. Before this discussion started, a survey was taken of the class’s predictions for various outcomes of the AGI in the coming years (with the full results below in figure 2). This survey asked students for their opinion of a number of statements. Each of these individual statement, if true, would reduce concerns of catastrophic AI-driven disasters. For example, when asked “How much do you agree with the statement: Humans will choose to not build AGI” half of respondents said they strongly disagreed with high confidence (agreement = 1, confidence = 5). Students also generally disagreed with the statements:

  • “AGI will not be technically feasible in our lifetime”
  • “(hyper-)AGI will not make extremely obviously unethical decisions”
  • “No reason is individually sufficient, but taken together they provide justification to not fear AGI”

There was a divergence in responses regarding interpretability, corrigibility, and “other” AI alignment research. In the latter two cases, a plurality of respondents (about a quarter) agreed strongly with statements that such research would defang AGI (agreement = 4, confidence=4), while most other responses express various levels of agreement with low confidence. However, when asked about the likelihood of interpretability research defanging AI, the pessimistic voices were more united. A quarter of responses still expressed the same optimism, but roughly half expressed pessimism (agreement ≤ 2) with half of those expressing at least moderate confidence (confidence ≥ 4). Based on the following discussion, this might have been caused by more familiarity with interpretability research, including first-hand experience.

The only statement with general agreement was “(hyper-)AGI will understand human intentions better than we can code it.” However, it should be noted that no statement such as “AGI will respect human desires, as it understand them” was asked on the survey.

Figure 2: Class Survey Results; conducted before a class-wide discussion. Note that students were instructed to answer confidence = 1 when they had not previously considered the statement, to answer confidence = 3 when they felt there were strong arguments on both sides, and to answer confidence = 5 when they possessed well-considered resolve.

After the survey was completed, the results were displayed as an open discussion began. Similar to recent empirical research from frontier labs, interpretability research received more airtime than in Eliezer’s article. Students disagreed first about the definition of interpretability: whether it refers to the ability to interpret a model’s behavior solely by its weights, to interpration via repeated probing of the model in a sandbox, or whether it can also refer to the modern chain-of-thought traces. Regardless of how it was defined, however, participants were either pessimistic or very pessimistic about interpretability research broadly. One student criticized common misunderstandings of chain of thought. Rather than being a verbatim copy of the models internal dialog, it is instead a superficial summary of the complete thought state and routinely produced gibberish, such as rarely used Chinese characters in the middle of otherwise English reasoning.

A popular topic was the exact shape and speed of a recursive self-improvement loop. If it takes place slowly, then what might we learn from “near misses” such as the Hugging Face incident? The number of near misses we are able to learn from before AI possesses sufficient power to prevent further iterations could depend on this curve, with some students arguing that the sheer number of humans, as well as their default robustness in the physical world compared to AI systems means that AI-driven extinction events are still a long way off. Bolstering this “slow take-off” opinion are rumors that AI already plays a major role in model development which could be interpreted as the start of this process.

Some criticized a focus on “solving ethics” as a needlessly high bar that distracts from the more mundane tasks dominating AI alignment work. In particular, the student volunteer who presented this paper (and the author of this report) included a section on “Ethical Dilemmas” in their presentation. Among arguments against focusing on abstract moral philosophy, Professor Aaronson cites Eliezer to emphasize that any alignment at all is difficult, not just in morally gray cases:

When I say that alignment is difficult, I mean that in practice, using the techniques we actually have, “please don’t disassemble literally everyone with probability roughly 1” is an overly large ask that we are not on course to get.

In response, I argue that some examination of everyday decisions with a critical lens—such as telling white lies to loved ones or consuming animal products—can help disabuse us of the notion that goodness emerges in every sufficiently intelligent agent.

One of the most interesting discussions was about the difference between state-of-the-art LLMs and the theorized AI agents long discussed in rationalist discourse. Since current LLMs “mimic the human distribution,” they come preloaded with extensive understanding of human social norms and moral behavior. This makes constitutional alignment (the current practices of using system prompts to establish ground rules) extremely effective. This might either fundamentally change the orthogonality thesis, or provide a new tool to better approximate human judgment in complicated situations.

Of all the points made, the one I found most interesting was simply (paraphrased):

I think human-alignment is just very tractable

Here, “human-alignment” refers not to AI alignment with human values, but cooperation between different humans. More specifically, it refers to the ability for human societies to choose not to rush recklessly into larger-and-larger AI systems. In an academic course focused on the technical problem of AI alignment, this was a reminder to not completely discard policy discussions in the believe that they lack any value. After all, many destructive technologies have been previously contained by international agreements. Notable examples include nuclear weapons and engineered plagues. However, even this was a contentious topic. The main criticisms were (1) the extreme “dual-use” nature of AIs for both peaceful growth and warfare, and (2) the greater danger for AI escapes even after taking precautions to prevent it. However, in the interest of ending on an optimistic note—unlike the assigned reading—it is on this belief in human cooperation that I will leave you.

By Scott

How many humans does it take to make tech seem human? Do you want to be one of them?

from Computational Complexity

 

(This was written about 9 months ago.  Its not out of date... yet)


People think that AI is going to DESTROY some jobs and CREATE some jobs. It may be too early to tell if this is true. Even so, here are some thoughts. 

1) Who will win? Who will lose?

2) Historically in the long term society was better off after a tech change (e.g., we live longer now than we did in the farm-era). Will that happen here as well?

3) We have some sense of what kinds of jobs will be destroyed. But what kind will be created? Will they be interesting? See later in this blog for a job you might not have thought of. 

4) Many jobs will change.  If you are over X years old then think about how much technology has changed your job even before the AI revolution. (The value of X may vary depending on how high-tech you are.) 

For an intelligent view of the questions above, see here.

For my view of one aspect of this, read on. 


There is one job which has been either created or expanded by AI:

Annotator.

(See here for an ARTICLE about these jobs, from which I got most of the rest of this post. The word ARTICLE is in caps so when I refer to it later you'll know what I am referring to.) 

The job consists of looking at pictures and labeling things.

Here is a direct quote from the manual: 

LABEL real items that can be worn by real people.

Does Lady Gaga count as a real person? She sometimes wears dresses made of meat. Should that count?  See here for a real article about her and see here for the Weird Al parody of Born that Way. Note that this is Weird AL, not Weird Artificial Intelligence. (See here for the Weird AI for Weird AL problem.)


They do this to create data for AI.

a) The jobs don't pay well though there are some exceptions.

b) The jobs are boring though there are some exceptions (and that may depend on the worker). 

c) This job is needed because AI keeps running into edge cases. This may have happened with AI's attempts to solve my GROUP ONE-GROUP TWO prez-VP problem from a prior blog post here  or my baseball-brother-pitchers post here.

c) KEY: People in AI used to think this is a temporary thing and these jobs will soon be automated. This might not  be the case. There are SO MANY edge cases that AI encounters. The more we expect from AI the more edge cases there will be.

A quote from page 26 (of the ARTICLE pointed to above) which is informative if you can parse it. I think. 

Put another way, ChatGPT seems so human because it was trained by an AI that was mimicking humans who were rating an AI that was mimicking humans who were pretending to be a better version of an AI that was trained on human writing.


By gasarch

 

(This was written about 9 months ago.  Its not out of date... yet)


People think that AI is going to DESTROY some jobs and CREATE some jobs. It may be too early to tell if this is true. Even so, here are some thoughts. 

1) Who will win? Who will lose?

2) Historically in the long term society was better off after a tech change (e.g., we live longer now than we did in the farm-era). Will that happen here as well?

3) We have some sense of what kinds of jobs will be destroyed. But what kind will be created? Will they be interesting? See later in this blog for a job you might not have thought of. 

4) Many jobs will change.  If you are over X years old then think about how much technology has changed your job even before the AI revolution. (The value of X may vary depending on how high-tech you are.) 

For an intelligent view of the questions above, see here.

For my view of one aspect of this, read on. 


There is one job which has been either created or expanded by AI:

Annotator.

(See here for an ARTICLE about these jobs, from which I got most of the rest of this post. The word ARTICLE is in caps so when I refer to it later you'll know what I am referring to.) 

The job consists of looking at pictures and labeling things.

Here is a direct quote from the manual: 

LABEL real items that can be worn by real people.

Does Lady Gaga count as a real person? She sometimes wears dresses made of meat. Should that count?  See here for a real article about her and see here for the Weird Al parody of Born that Way. Note that this is Weird AL, not Weird Artificial Intelligence. (See here for the Weird AI for Weird AL problem.)


They do this to create data for AI.

a) The jobs don't pay well though there are some exceptions.

b) The jobs are boring though there are some exceptions (and that may depend on the worker). 

c) This job is needed because AI keeps running into edge cases. This may have happened with AI's attempts to solve my GROUP ONE-GROUP TWO prez-VP problem from a prior blog post here  or my baseball-brother-pitchers post here.

c) KEY: People in AI used to think this is a temporary thing and these jobs will soon be automated. This might not  be the case. There are SO MANY edge cases that AI encounters. The more we expect from AI the more edge cases there will be.

A quote from page 26 (of the ARTICLE pointed to above) which is informative if you can parse it. I think. 

Put another way, ChatGPT seems so human because it was trained by an AI that was mimicking humans who were rating an AI that was mimicking humans who were pretending to be a better version of an AI that was trained on human writing.


By gasarch

TR26-228 | Bitwise-Optimal Cryptography: From One-Wayness to Pseudorandomness and Target Collision Resistance | Benny Applebaum

from ECCC Papers
We study cryptographic primitives that are both locally computable (i.e., in $\mathrm{NC}^0$) and exponentially secure. For pseudorandom generators (PRGs) and universal one-way hash functions (UOWHFs), we further require linear stretch and linear compression, respectively, which is essentially the best one can hope for in this setting. Such primitives simultaneously achieve an extreme level of security and efficiency: each output bit inspects only a constant number of input bits, while each input bit buys a constant amount of security and expansion/shrinkage, in an amortized sense. We refer to such primitives as \emph{bitwise optimal}. We prove that bitwise-optimal PRGs and UOWHFs can be obtained from \emph{any} exponentially secure one-way function (OWF) in $\mathrm{NC}^0$, thereby establishing an equivalence between these three primitives in the bitwise-optimal regime. Notably, an analogous equivalence is not known in the exponential-security regime for general, unrestricted cryptographic primitives without imposing an additional regularity condition. Our results combine the machinery of the author (Applebaum, FOCS'17) with new structural results on locally computable functions that may be of independent interest. We prove a sparsification theorem that reduces the output length of any exponentially secure local OWF to $O(n)$ while preserving exponential hardness, an input-locality reduction that bounds the number of outputs affected by each input bit, and a structural theorem showing that functions with bounded input locality are typically almost regular. Together, these ingredients remove the regularity assumption required by previous constructions and yield a non-black-box transformation from exponentially secure OWFs in $\mathrm{NC}^0$ to bitwise-optimal PRGs and UOWHFs. As an additional contribution, we prove a projection-based compression theorem for locally samplable sources.
We study cryptographic primitives that are both locally computable (i.e., in $\mathrm{NC}^0$) and exponentially secure. For pseudorandom generators (PRGs) and universal one-way hash functions (UOWHFs), we further require linear stretch and linear compression, respectively, which is essentially the best one can hope for in this setting. Such primitives simultaneously achieve an extreme level of security and efficiency: each output bit inspects only a constant number of input bits, while each input bit buys a constant amount of security and expansion/shrinkage, in an amortized sense. We refer to such primitives as \emph{bitwise optimal}. We prove that bitwise-optimal PRGs and UOWHFs can be obtained from \emph{any} exponentially secure one-way function (OWF) in $\mathrm{NC}^0$, thereby establishing an equivalence between these three primitives in the bitwise-optimal regime. Notably, an analogous equivalence is not known in the exponential-security regime for general, unrestricted cryptographic primitives without imposing an additional regularity condition. Our results combine the machinery of the author (Applebaum, FOCS'17) with new structural results on locally computable functions that may be of independent interest. We prove a sparsification theorem that reduces the output length of any exponentially secure local OWF to $O(n)$ while preserving exponential hardness, an input-locality reduction that bounds the number of outputs affected by each input bit, and a structural theorem showing that functions with bounded input locality are typically almost regular. Together, these ingredients remove the regularity assumption required by previous constructions and yield a non-black-box transformation from exponentially secure OWFs in $\mathrm{NC}^0$ to bitwise-optimal PRGs and UOWHFs. As an additional contribution, we prove a projection-based compression theorem for locally samplable sources.

Tenure-track faculty at University of Colorado Boulder (apply by November 15, 2026)

from CCI: jobs
U. Colorado Boulder CS seeks applications for a TT Asst. Prof. position in Quantum Computation. We invite applications from all areas of QC; priority consideration will be given to: – Quantum computation theory: quantum algorithms, quantum complexity, quantum information, & quantum error correcting codes – Applications of QC to important or emerging areas such as […]

U. Colorado Boulder CS seeks applications for a TT Asst. Prof. position in Quantum Computation. We invite applications from all areas of QC; priority consideration will be given to:

– Quantum computation theory: quantum algorithms, quantum complexity, quantum information, & quantum error correcting codes
– Applications of QC to important or emerging areas such as quantum optimization & quantum ML

Website: https://jobs.colorado.edu/jobs/JobDetail/Tenure-Track-Faculty-in-Quantum-Computation/74741
Email: jgrochow@colorado.edu

By shacharlovett

Quantum Computers and Applied Mathematics at Brown University

from Gil Kalai
My host at Brown University was the fascinating Basilis Gidas whom I first met in Rio in 2018. Basilis has had a remarkable career, taking him from electrical and mechanical engineering through quantum field theory to many areas of applied … Continue reading →

My host at Brown University was the fascinating Basilis Gidas whom I first met in Rio in 2018. Basilis has had a remarkable career, taking him from electrical and mechanical engineering through quantum field theory to many areas of applied mathematics. In the context of AI, Basilis told me about a well-known passage in Plato—new to me—expressing concern that reliance on writing (new technology at the time) would weaken human memory. Overall, Basilis thinks that the most creative aspects of mathematics will remain human.

I had a wonderful and very intense time in Providence, where I gave an applied mathematics colloquium on my work on quantum computing. During my visit, I met and talked with quite a few mathematicians, chemists, physicists, computer scientists, and an economist Oded Galor.

Regarding quantum computation, I heard many excellent new questions about my point of view (I have collected earlier questions in this post), and learned about topics in chemistry and physics closely related to my work. I still have quite a bit to digest from these conversations. I also had a lovely dinner discussion about the notorious “measurement problem.”

Here are the slides of my talk: The Quantum Computer – A Miracle or Mirage.

The talk has five parts, and the audience and I concentrated on the second, devoted to my conjectures on correlated errors.

A few days ago I uploaded to the arXiv my paper The Fully Depolarizing Noise Conjecture for Entangled Physical States: A Twenty-Year Perspective, that is going to appear in: Fields of Logic and Computation IV: Essays dedicated to Yuri Gurevich, editted by: Guillermo Badia, Manfred Droste, Andreas Blass, and Nachum Dershowitz. (I noticed two even newer papers on the arXiv, on quantum cryptography by Yael Kalai!)

Of course, I also learned about advances in computer science and mathematics close to my other interests, heard new perspectives on the AI revolution in mathematics and the emotions it evokes, reconnected with old friends, and made some new ones. Let me mention Eli Upfal, now a distinguished computer scientist. Eli and I were graduate students at HUJI at the same time (he studied with Eli Shamir), and I had not seen him for several decades.

Greetings from Boston! For those who attended my talk here on “algebraic shifting,” here are the handouts. (There were not enough copies for everybody.) I plan to return to my Boston visit and to algebraic shifting a little later.

A bit of nostalgia: Providence was the first US city I set foot in, in 1978. It was my first trip outside Israel: I spent two weeks at a summer school in Montreal, then got a lift to Providence, where I took a bus to Boston. My hosts in Boston, Israeli sailing champions, took me sailing on the (then polluted) Charles River. They invited me to take the helm, and shortly afterward we all found ourselves in the water.

By Gil Kalai

TR26-227 | Interactive Proof of Proximity for Bipartiteness without Mixing | Guy Weissenberg

from ECCC Papers
Testing whether a bounded-degree $n$-vertex graph is bipartite or far from bipartite requires $\widetilde\Theta(\sqrt n)$ queries (Goldreich--Ron, Combinatorica'99, Algorithmica'02). An interactive proof of proximity (IPP) is a hybrid model of property testing and interactive proofs: the verifier faces the same task with the same query access, but it now interacts with an untrusted prover that sees the whole graph. Rothblum, Vadhan, and Wigderson (STOC'13) introduced this model and gave a one-round private-coin IPP for bipartiteness on well-mixing graphs with query and communication costs both $O(\log n)$ for fixed parameters, an exponential improvement over plain testing. The mixing assumption was used in their soundness analysis, and they asked whether it could be removed. We give a slightly modified version of the original RVW protocol and prove its soundness without the mixing promise. In the RVW protocol, soundness follows by bounding the probability of correctly guessing the parity of a hidden random walk from its start and endpoint. The best guess is wrong with probability equal to the overlap between the endpoint probabilities for even and odd numbers of edge traversals, and mixing gives one way to bound this overlap. We prove an overlap theorem that requires no mixing, building on the recent semidefinite programming (SDP) analysis of the Goldreich--Ron tester by Fei and Rubinfeld (2026).
Testing whether a bounded-degree $n$-vertex graph is bipartite or far from bipartite requires $\widetilde\Theta(\sqrt n)$ queries (Goldreich--Ron, Combinatorica'99, Algorithmica'02). An interactive proof of proximity (IPP) is a hybrid model of property testing and interactive proofs: the verifier faces the same task with the same query access, but it now interacts with an untrusted prover that sees the whole graph. Rothblum, Vadhan, and Wigderson (STOC'13) introduced this model and gave a one-round private-coin IPP for bipartiteness on well-mixing graphs with query and communication costs both $O(\log n)$ for fixed parameters, an exponential improvement over plain testing. The mixing assumption was used in their soundness analysis, and they asked whether it could be removed. We give a slightly modified version of the original RVW protocol and prove its soundness without the mixing promise. In the RVW protocol, soundness follows by bounding the probability of correctly guessing the parity of a hidden random walk from its start and endpoint. The best guess is wrong with probability equal to the overlap between the endpoint probabilities for even and odd numbers of edge traversals, and mixing gives one way to bound this overlap. We prove an overlap theorem that requires no mixing, building on the recent semidefinite programming (SDP) analysis of the Goldreich--Ron tester by Fei and Rubinfeld (2026).

Group Leader and Postdoc at Max Planck Institute for Informatics (apply by December 15, 2026)

from CCI: jobs
Max Planck Institute for Informatics (Saarbrücken) seeks Postdocs and Group Leaders in Algorithms & Complexity and related areas. Flexible start dates; strong research environment and travel support. Apply by 15 Dec 2026 with CV, publications, research plan, and 3 references. Website: www.mpi-inf.mpg.de/d1/offers/postdoc Email: join-d1@mpi-inf.mpg.de

Max Planck Institute for Informatics (Saarbrücken) seeks Postdocs and Group Leaders in Algorithms & Complexity and related areas. Flexible start dates; strong research environment and travel support. Apply by 15 Dec 2026 with CV, publications, research plan, and 3 references.

Website: https://www.mpi-inf.mpg.de/d1/offers/postdoc
Email: join-d1@mpi-inf.mpg.de

By shacharlovett

Saturday, October 03

TR26-226 | Multi-Access Randomness Saves Space, Even for Halting Algorithms | William Hoza

from ECCC Papers
We prove that every language in $\mathrm{P}^{\# \mathrm{P}}$ can be decided by a bounded-error randomized algorithm that uses only $O(\log n)$ bits of work space. The algorithm is guaranteed to halt for every input and every setting of the random tape. However, there is a catch: the algorithm uses a *multi-access* random tape, i.e., the algorithm scans both forward and backward across a read-only tape filled with an unlimited number of random bits. Prior work on log-space algorithms with multi-access randomness either focuses on polynomial-time algorithms, or else permits algorithms that sometimes run forever. Our algorithm always halts, but it uses exponential time. Thus, our work identifies a natural model of computation in which randomness is intrinsically useful, assuming $\mathrm{L} \neq \mathrm{P}^{\# \mathrm{P}}$.
We prove that every language in $\mathrm{P}^{\# \mathrm{P}}$ can be decided by a bounded-error randomized algorithm that uses only $O(\log n)$ bits of work space. The algorithm is guaranteed to halt for every input and every setting of the random tape. However, there is a catch: the algorithm uses a *multi-access* random tape, i.e., the algorithm scans both forward and backward across a read-only tape filled with an unlimited number of random bits. Prior work on log-space algorithms with multi-access randomness either focuses on polynomial-time algorithms, or else permits algorithms that sometimes run forever. Our algorithm always halts, but it uses exponential time. Thus, our work identifies a natural model of computation in which randomness is intrinsically useful, assuming $\mathrm{L} \neq \mathrm{P}^{\# \mathrm{P}}$.

TR26-225 | Beyond Width-3: Better Hitting Set Generators for Width-4 Read-Once Branching Programs | Gonen Krak

from ECCC Papers
We construct an explicit hitting set generator (HSG) for ordered read-once branching programs of width 4. For every length $n$ and every $\varepsilon \in (0,1)$, every width-4 length-$n$ program that accepts more than an $\varepsilon$ fraction of its inputs accepts at least one output of the generator, and the seed length is $$O\big((\log n)^{3/2} \cdot \sqrt{\log\log n} \cdot (1 + \log(1/\varepsilon))\big).$$ For constant $\varepsilon$ this is $O(\log^{3/2} n \cdot \sqrt{\log\log n})$. More generally, it is $o(\log^2 n)$ whenever $\varepsilon \ge 2^{-o(\sqrt{\log n/\log\log n})}$. This is the first explicit generator for width 4 that improves on the $O(\log^2 n)$ seed length of Nisan's generator (Combinatorica 1992). Our construction first reduces the hitting problem for width-4 programs to the hitting problem for programs with three live states and one rejecting state, following Doron and Hoza (RANDOM 2025). It then simplifies these programs by rounds of pseudorandom restrictions, and hits the simplified programs with a small-bias string that is modified in a small number of positions.
We construct an explicit hitting set generator (HSG) for ordered read-once branching programs of width 4. For every length $n$ and every $\varepsilon \in (0,1)$, every width-4 length-$n$ program that accepts more than an $\varepsilon$ fraction of its inputs accepts at least one output of the generator, and the seed length is $$O\big((\log n)^{3/2} \cdot \sqrt{\log\log n} \cdot (1 + \log(1/\varepsilon))\big).$$ For constant $\varepsilon$ this is $O(\log^{3/2} n \cdot \sqrt{\log\log n})$. More generally, it is $o(\log^2 n)$ whenever $\varepsilon \ge 2^{-o(\sqrt{\log n/\log\log n})}$. This is the first explicit generator for width 4 that improves on the $O(\log^2 n)$ seed length of Nisan's generator (Combinatorica 1992). Our construction first reduces the hitting problem for width-4 programs to the hitting problem for programs with three live states and one rejecting state, following Doron and Hoza (RANDOM 2025). It then simplifies these programs by rounds of pseudorandom restrictions, and hits the simplified programs with a small-bias string that is modified in a small number of positions.

Friday, October 02

TR26-224 | Explicit Nonlinear Functions beyond the Fourier bound | Swastik Kopparty, Rishabh Kothary, Shanthanu Rai

from ECCC Papers
We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb{F}_2^n \to \mathbb{F}_2^m$. Concretely, we want an $F$ and an $A = A(m,n)>0$ as small as possible, so that for every affine map $L: \mathbb{F}_2^n \to \mathbb{F}_2^m$ (of the form $L(x) = Mx + b$) we have: $$ agree(F,L) := |\{x \in \mathbb{F}_2^n \mid F(x) = L(x)\}| \leq A. $$ Such questions have been studied by Nyberg [1991, 1993], Carlet and Ding [2004, 2007], Liu, Mesnager and Chen [2017], Nagy [2025], and Biryukov, Turecek, and Udovenko [2026]. There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = \Theta(2^{n-m} + 2^{n/2}), $$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $\gamma>0$, a function $F: \mathbb{F}_2^n \to \mathbb{F}_2^m$ with $m = \mathcal{O}_{\gamma}(n)$, achieving $$ A(m,n) \leq (1+\gamma)^n. $$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb{F}_2^n \to \mathbb{F}_2^m$, with $m = \mathcal{O}_{\gamma,d}(n)$. Here the previously best bounds were of the form $A(m,n) = \mathcal{O}(2^{-\frac{n}{2^{d+1}}} \cdot 2^n)$ of Ben-Sasson and Kopparty [Kopparty's thesis, 2010], based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb{F}_q$ in place of $\mathbb{F}_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.
We study the problem of constructing highly nonlinear vectorial maps $F: \mathbb{F}_2^n \to \mathbb{F}_2^m$. Concretely, we want an $F$ and an $A = A(m,n)>0$ as small as possible, so that for every affine map $L: \mathbb{F}_2^n \to \mathbb{F}_2^m$ (of the form $L(x) = Mx + b$) we have: $$ agree(F,L) := |\{x \in \mathbb{F}_2^n \mid F(x) = L(x)\}| \leq A. $$ Such questions have been studied by Nyberg [1991, 1993], Carlet and Ding [2004, 2007], Liu, Mesnager and Chen [2017], Nagy [2025], and Biryukov, Turecek, and Udovenko [2026]. There is a classical method of constructing such functions from bent-functions and Fourier analytic ideas; the best bound achievable by this method is: $$ A(m,n) = \Theta(2^{n-m} + 2^{n/2}), $$ and in particular, is never smaller than $2^{n/2}$. In this work, we show how to construct highly nonlinear functions beyond this Fourier bound. Concretely, we show how to construct for every $\gamma>0$, a function $F: \mathbb{F}_2^n \to \mathbb{F}_2^m$ with $m = \mathcal{O}_{\gamma}(n)$, achieving $$ A(m,n) \leq (1+\gamma)^n. $$ Surprisingly, we even achieve the same quantitative behavior for the much harder question of having low agreement with $m$-tuples of degree $d$ polynomials $Q: \mathbb{F}_2^n \to \mathbb{F}_2^m$, with $m = \mathcal{O}_{\gamma,d}(n)$. Here the previously best bounds were of the form $A(m,n) = \mathcal{O}(2^{-\frac{n}{2^{d+1}}} \cdot 2^n)$ of Ben-Sasson and Kopparty [Kopparty's thesis, 2010], based on Gowers-norm-type arguments. All our results generalize to all finite fields $\mathbb{F}_q$ in place of $\mathbb{F}_2$. Our methods are based on a new connection to classical results on counting solutions to systems of polynomial equations via algebraic methods. This connection brings us to basic questions in combinatorics, about graphs and hypergraphs with simultaneously a small number of edges and independent sets.

TR26-223 | Space-Efficient Simulations Beyond Multitape Turing Machines | Danil Sibgatullin, Ryan Williams

from ECCC Papers
We explore new directions in simulating complex computations with low space, building on the work of Williams [STOC'25] and Cook-Mertz [STOC'24, SICOMP'25]. We define and study a strengthened version of the parallel external memory model, in which there are $P$ processors each with private internal memory $M$, all of which have shared access to an external memory. Like the standard external memory model, a processor is charged one step when it swaps up to $M$ bits of its internal memory with the external memory; unlike the usual external memory model, each processor also has free access to the input at no cost, and may read any subset of M bits of external memory (not just a contiguous block) in a single I/O step. We extend the reduction of Williams for multitape Turing machines to this vastly more general model, giving three main applications. 1. We show that the $P$-complete problem {\sc Lex First 2sat}, where even the best-known RAM algorithm runs in $O(n^{\omega})\leq O(n^{2.372})$ time on instances with $\Theta(n^2)$ clauses, nevertheless has an $\tilde{O}(\sqrt{n})$-space algorithm for $n$-variable instances. For dense instances with $\Theta(n^2)$ clauses, our space usage is less than the \emph{fourth root} of the best-known RAM running time. 2. We show that every \emph{unbounded} fan-in circuit of $n$ gates over the basis $\{$NOT, AND, OR, XOR$\}$ can be evaluated on any given input in only $O(\sqrt{n \log n})$ space. That is, the space usage is nearly a fourth root of the running time for dense circuits with $\Theta(n^2)$ wires. This result generalizes a space-efficient simulation of Shalunov for bounded fan-in circuits. 3. We consider the problem of simulating a $d$-dimensional cellular automaton, where the value of each cell is determined by its immediate neighbors. We show how to simulate $t$ time steps of a cellular automaton in only $O((t\log t)^{1-1/(d+1)})$ space. By an old theorem of Cook [1966], our space-efficient simulation of one-dimensional cellular automata yields another generalization of Williams' simulation of multitape time.
We explore new directions in simulating complex computations with low space, building on the work of Williams [STOC'25] and Cook-Mertz [STOC'24, SICOMP'25]. We define and study a strengthened version of the parallel external memory model, in which there are $P$ processors each with private internal memory $M$, all of which have shared access to an external memory. Like the standard external memory model, a processor is charged one step when it swaps up to $M$ bits of its internal memory with the external memory; unlike the usual external memory model, each processor also has free access to the input at no cost, and may read any subset of M bits of external memory (not just a contiguous block) in a single I/O step. We extend the reduction of Williams for multitape Turing machines to this vastly more general model, giving three main applications. 1. We show that the $P$-complete problem {\sc Lex First 2sat}, where even the best-known RAM algorithm runs in $O(n^{\omega})\leq O(n^{2.372})$ time on instances with $\Theta(n^2)$ clauses, nevertheless has an $\tilde{O}(\sqrt{n})$-space algorithm for $n$-variable instances. For dense instances with $\Theta(n^2)$ clauses, our space usage is less than the \emph{fourth root} of the best-known RAM running time. 2. We show that every \emph{unbounded} fan-in circuit of $n$ gates over the basis $\{$NOT, AND, OR, XOR$\}$ can be evaluated on any given input in only $O(\sqrt{n \log n})$ space. That is, the space usage is nearly a fourth root of the running time for dense circuits with $\Theta(n^2)$ wires. This result generalizes a space-efficient simulation of Shalunov for bounded fan-in circuits. 3. We consider the problem of simulating a $d$-dimensional cellular automaton, where the value of each cell is determined by its immediate neighbors. We show how to simulate $t$ time steps of a cellular automaton in only $O((t\log t)^{1-1/(d+1)})$ space. By an old theorem of Cook [1966], our space-efficient simulation of one-dimensional cellular automata yields another generalization of Williams' simulation of multitape time.

Beyond Light Cones: State Preparation Complexity in Quantum Spin Glasses

from arXiv: Computational Complexity

Authors: Omar Al-Ghattas, David Gamarnik, Bobak T Kiani

We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $Ω(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

Authors: Omar Al-Ghattas, David Gamarnik, Bobak T Kiani

We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $Ω(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

The Robustness of QAC0

from arXiv: Computational Complexity

Authors: Daniel Grier, Jackson Morris, Kewen Wu

In this work we study the robustness of $\mathsf{QAC}^0$ with respect to error tolerance and modifications to its gate-set. First, we investigate whether the non-zero error typically allowed for $\mathsf{QAC}^0$ circuits computing Boolean functions is truly necessary. We show that the error inherent in the parallel $W$-test of \cite{grier_morris_wu} can be eliminated entirely via a novel application of exact amplitude amplification in the many-copies context. Consequently, we find that $\mathsf{QAC}^0$ can \textit{exactly} simulate $\mathsf{TC}^0$ with polynomially many copies of the classical input and that for every fixed prime $p$ exact $\mathsf{QAC}^0$, $\mathsf{EQAC}^0$, can compute total Boolean functions outside of $\mathsf{AC}^0[p]$. Second, we ask to what extent the computational power of $\mathsf{QAC}^0$ follows from the fact that arbitrary single-qubit gates may be used at any point in the circuit. We find that $\mathsf{QAC}^0$ is in fact robust to restrictions on which single-qubit gates are permitted: every $\mathsf{QAC}^0$ circuit can be approximately implemented by a $\mathsf{QAC}^0$ circuit consisting of just generalized Toffoli, $S$, and Hadamard gates. Moreover, this approximating circuit can be constructed efficiently from a classical description of the original circuit.

Authors: Daniel Grier, Jackson Morris, Kewen Wu

In this work we study the robustness of $\mathsf{QAC}^0$ with respect to error tolerance and modifications to its gate-set. First, we investigate whether the non-zero error typically allowed for $\mathsf{QAC}^0$ circuits computing Boolean functions is truly necessary. We show that the error inherent in the parallel $W$-test of \cite{grier_morris_wu} can be eliminated entirely via a novel application of exact amplitude amplification in the many-copies context. Consequently, we find that $\mathsf{QAC}^0$ can \textit{exactly} simulate $\mathsf{TC}^0$ with polynomially many copies of the classical input and that for every fixed prime $p$ exact $\mathsf{QAC}^0$, $\mathsf{EQAC}^0$, can compute total Boolean functions outside of $\mathsf{AC}^0[p]$. Second, we ask to what extent the computational power of $\mathsf{QAC}^0$ follows from the fact that arbitrary single-qubit gates may be used at any point in the circuit. We find that $\mathsf{QAC}^0$ is in fact robust to restrictions on which single-qubit gates are permitted: every $\mathsf{QAC}^0$ circuit can be approximately implemented by a $\mathsf{QAC}^0$ circuit consisting of just generalized Toffoli, $S$, and Hadamard gates. Moreover, this approximating circuit can be constructed efficiently from a classical description of the original circuit.

Optimal transducers using symmetries

from arXiv: Computational Complexity

Authors: Benoît Dubus, Julien Ladeuze, Jérémie Roland

Transducers (Belovs, Jeffery and Yolcu, 2024) are a quantum computing framework describing a quantum algorithm as a unitary converting an input state into a target state using a catalyst, an auxiliary vector that is left unchanged. They are a powerful tool in quantum algorithm design, especially in the context of quantum query complexity: feasible points of the (dual) adversary semidefinite program directly translate into transducers and the optimal transduction complexity is equal to the adversary bound, i.e. the Las Vegas complexity, which is known to characterize bounded-error quantum query complexity. Moreover, contrary to bounded-error algorithms, transducers compose exactly, which limits overheads due to controlling errors in algorithms constructed by composition. Constructing efficient, let alone optimal, transducers in terms of quantum query complexity nevertheless remains a hard task since it still requires solving the adversary SDP and constructing the unitary to obtain an explicit algorithm. In this paper, we show how using the symmetry group of state-conversion problems simplifies both steps. First, using a symmetrization argument, we prove an optimal catalyst can always be chosen covariant under a representation of the symmetry group. Second, we prove that the transducer intertwines two different representations of the group and can thus be chosen block diagonal in the isotypic decomposition of the Hilbert space. Using those methods, we then derive optimal transducers, with optimal constants, for different widely used quantum algorithmic primitives, such as unstructured search, amplitude amplification and amplitude estimation. Our approach extends previous work on the use of representation theory to compute adversary lower bounds (Høyer, Lee, and {\v S}palek, 2007; Ambainis, Magnin, Roetteler and Roland, 2011) to the systematic construction of optimal algorithms.

Authors: Benoît Dubus, Julien Ladeuze, Jérémie Roland

Transducers (Belovs, Jeffery and Yolcu, 2024) are a quantum computing framework describing a quantum algorithm as a unitary converting an input state into a target state using a catalyst, an auxiliary vector that is left unchanged. They are a powerful tool in quantum algorithm design, especially in the context of quantum query complexity: feasible points of the (dual) adversary semidefinite program directly translate into transducers and the optimal transduction complexity is equal to the adversary bound, i.e. the Las Vegas complexity, which is known to characterize bounded-error quantum query complexity. Moreover, contrary to bounded-error algorithms, transducers compose exactly, which limits overheads due to controlling errors in algorithms constructed by composition. Constructing efficient, let alone optimal, transducers in terms of quantum query complexity nevertheless remains a hard task since it still requires solving the adversary SDP and constructing the unitary to obtain an explicit algorithm. In this paper, we show how using the symmetry group of state-conversion problems simplifies both steps. First, using a symmetrization argument, we prove an optimal catalyst can always be chosen covariant under a representation of the symmetry group. Second, we prove that the transducer intertwines two different representations of the group and can thus be chosen block diagonal in the isotypic decomposition of the Hilbert space. Using those methods, we then derive optimal transducers, with optimal constants, for different widely used quantum algorithmic primitives, such as unstructured search, amplitude amplification and amplitude estimation. Our approach extends previous work on the use of representation theory to compute adversary lower bounds (Høyer, Lee, and {\v S}palek, 2007; Ambainis, Magnin, Roetteler and Roland, 2011) to the systematic construction of optimal algorithms.

Short Resolution Refutations for CNFs with Bounded Weighted Incidence Treewidth

from arXiv: Computational Complexity

Authors: Shaowei Cai, Ziqun Li

It is an open problem in proof complexity whether every unsatisfiable CNF formula has an FPT-sized resolution refutation parameterized by incidence treewidth. In this paper, we establish several upper bounds on resolution refutation length related to this problem. Consider an unsatisfiable CNF formula $F$ with $n$ variables, $m$ clauses, maximum clause width $k$, and incidence treewidth $\mathrm{tw}^*(F)$. In this paper, we introduce two variants of incidence treewidth. Their definitions can be stated informally as follows. The first is log-weighted incidence treewidth $\mathrm{tw}_{\log}^*(F)$, which is the treewidth of the weighted incidence graph, in which variables have weight one and each clause has weight equal to the logarithm of its width. The second is partially log-weighted incidence treewidth $\mathrm{tw}^*_{\mathrm{plog}}(F)$, which is a refinement of log-weighted incidence treewidth. In this variant, for a nice tree decomposition of the incidence graph, each clause has weight one along a path selected for that clause and elsewhere has weight equal to the logarithm of one plus the number of its literals whose variables do not appear in any bag on that path, and variables have weight one. For every unsatisfiable CNF formula $F$, we prove the existence of (i) an FPT-sized resolution refutation parameterized by log-weighted incidence treewidth, with width at most $\mathrm{tw}_{\log}^*(F)+k$; (ii) a resolution refutation of length $(n+m)k^{O(\mathrm{tw}^*(F))}$ and width at most $\mathrm{tw}^*(F)+k$; (iii) an FPT-sized resolution refutation parameterized by partially log-weighted incidence treewidth; and (iv) an FPT-sized regular resolution refutation parameterized by log-weighted incidence treewidth. Our main idea is to construct FPT-sized $k$-DNF resolution refutations parameterized by incidence treewidth, and then convert them into resolution refutations.

Authors: Shaowei Cai, Ziqun Li

It is an open problem in proof complexity whether every unsatisfiable CNF formula has an FPT-sized resolution refutation parameterized by incidence treewidth. In this paper, we establish several upper bounds on resolution refutation length related to this problem. Consider an unsatisfiable CNF formula $F$ with $n$ variables, $m$ clauses, maximum clause width $k$, and incidence treewidth $\mathrm{tw}^*(F)$. In this paper, we introduce two variants of incidence treewidth. Their definitions can be stated informally as follows. The first is log-weighted incidence treewidth $\mathrm{tw}_{\log}^*(F)$, which is the treewidth of the weighted incidence graph, in which variables have weight one and each clause has weight equal to the logarithm of its width. The second is partially log-weighted incidence treewidth $\mathrm{tw}^*_{\mathrm{plog}}(F)$, which is a refinement of log-weighted incidence treewidth. In this variant, for a nice tree decomposition of the incidence graph, each clause has weight one along a path selected for that clause and elsewhere has weight equal to the logarithm of one plus the number of its literals whose variables do not appear in any bag on that path, and variables have weight one. For every unsatisfiable CNF formula $F$, we prove the existence of (i) an FPT-sized resolution refutation parameterized by log-weighted incidence treewidth, with width at most $\mathrm{tw}_{\log}^*(F)+k$; (ii) a resolution refutation of length $(n+m)k^{O(\mathrm{tw}^*(F))}$ and width at most $\mathrm{tw}^*(F)+k$; (iii) an FPT-sized resolution refutation parameterized by partially log-weighted incidence treewidth; and (iv) an FPT-sized regular resolution refutation parameterized by log-weighted incidence treewidth. Our main idea is to construct FPT-sized $k$-DNF resolution refutations parameterized by incidence treewidth, and then convert them into resolution refutations.

Can AI Oversight Be Zero Knowledge?

from arXiv: Computational Complexity

Authors: Alessandro Chiesa, Ziyi Guan, Burcu Yildiz

AI systems increasingly produce outputs from confidential data, such as a fitness-for-duty assessment from medical records or the predicted properties of a drug candidate from its secret structure. It is important to verify that such outputs are correct without revealing the underlying data. A recent line of work studies verification of AI outputs via interactive proofs and debate for oracle-aided computation, where correctness may depend on an oracle such as human judgment, a physical experiment, or the web. These works focus on verification by a verifier that runs much faster than the computation. However, such efficient verification is impossible for general oracle-aided computation, and these works therefore rely on additional assumptions. We focus instead on privacy: allowing the verifier to run in time polynomial in the computation, we ask whether interactive arguments for oracle-aided computation can be zero knowledge, so that the verifier learns nothing about the confidential data beyond the correctness of the output. We prove that, in general, they cannot. In the random oracle model, there are no zero-knowledge proofs for all oracle-aided computations, even if both the prover and the verifier are allowed to run much longer than the computation itself. The impossibility extends to debate, a canonical model for scalable oversight. On the positive side, we show that if the oracle attaches a cryptographic signature to each of its answers, then every oracle-aided computation can be verified in zero knowledge with an efficient prover and verifier, assuming only collision-resistant hash functions. Beyond privacy, this also gives an alternative approach to scalable oversight that relies neither on an honest opponent, as in debate, nor on the robustness of the computation, as in prior single-prover protocols.

Authors: Alessandro Chiesa, Ziyi Guan, Burcu Yildiz

AI systems increasingly produce outputs from confidential data, such as a fitness-for-duty assessment from medical records or the predicted properties of a drug candidate from its secret structure. It is important to verify that such outputs are correct without revealing the underlying data. A recent line of work studies verification of AI outputs via interactive proofs and debate for oracle-aided computation, where correctness may depend on an oracle such as human judgment, a physical experiment, or the web. These works focus on verification by a verifier that runs much faster than the computation. However, such efficient verification is impossible for general oracle-aided computation, and these works therefore rely on additional assumptions. We focus instead on privacy: allowing the verifier to run in time polynomial in the computation, we ask whether interactive arguments for oracle-aided computation can be zero knowledge, so that the verifier learns nothing about the confidential data beyond the correctness of the output. We prove that, in general, they cannot. In the random oracle model, there are no zero-knowledge proofs for all oracle-aided computations, even if both the prover and the verifier are allowed to run much longer than the computation itself. The impossibility extends to debate, a canonical model for scalable oversight. On the positive side, we show that if the oracle attaches a cryptographic signature to each of its answers, then every oracle-aided computation can be verified in zero knowledge with an efficient prover and verifier, assuming only collision-resistant hash functions. Beyond privacy, this also gives an alternative approach to scalable oversight that relies neither on an honest opponent, as in debate, nor on the robustness of the computation, as in prior single-prover protocols.

Trapdoored Clifford Operators and Applications

from arXiv: Computational Complexity

Authors: Minki Hhan, Hojune Lee

Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random $n$-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.

Authors: Minki Hhan, Hojune Lee

Random Clifford operators have numerous applications in quantum computing, including randomized benchmarking, classical shadows, and quantum authentication. However, sampling and implementing uniformly random $n$-qubit Clifford incur near-quadratic complexity due to the size of Clifford group. We introduce a cryptographic way to overcome these barriers: trapdoored Clifford operator distributions whose samples are computationally indistinguishable from uniformly random Cliffords, yet implementing them can be much faster given the trapdoor. We construct a distribution of trapdoored Clifford operators whose elements can be sampled and implemented in near-linear time under a variant of the learning parity with noise assumption. Our constructions allow fast tableau action on Pauli labels for classical simulation, and also can be optimized to admit polylogarithmic-depth implementation. Along the way, we construct trapdoored matrices over finite fields that support efficient multiplication by both a matrix and its inverse, resolving an open question left by Vaikuntanathan and Zamir [SODA'26]. We use these constructions to obtain faster protocols based on random Cliffords. We also explore their applications to the worst-case to average-case reductions for matrix and Clifford problems including the iterated matrix multiplication and Clifford circuit synthesis. In particular, we show the hardness of batching Clifford circuits: synthesizing circuits that apply the same Clifford to multiple registers is at least as hard as worst-case matrix multiplication, even when synthesis succeeds on a small constant fraction of random Cliffords. This extends to approximate implementations by general quantum circuits.

Lower Bound of 22 for 3x3 Matrix Multiplication over the Integers

from arXiv: Computational Complexity

Authors: Isaac Rudich, Louis-Martin Rousseau

Strassen showed that two 2x2 matrices can be multiplied with 7 multiplications instead of 8. Applied recursively, his algorithm multiplies two nxn matrices with O(n^2.807) multiplications, beating the naive O(n^3). The best known 3x3 recursive matrix multiplication algorithm uses 23 multiplications O(n^2.854). The best published lower bound of 21 (on algorithms with integer constants) leaves room for an algorithm with O(n^2.771) multiplications, and thus does not rule out the possibility of an algorithm that would beat Strassen's. We prove a lower bound of 22 multiplications for any 3x3 recursive algorithm with integer constants, proving that no such algorithm can do better than O(n^2.814) multiplications, and eliminating the possibility of a 3x3 algorithm that beats Strassen's 2x2 method. The proof builds on a recent decomposition method from Wang, who approached the problem by turning it into 496 subproblems. We provide exact solutions for 359 of them. The proof is in Lean; verification requires auditing only a few short files. The Lean formalization directly encodes statements about the limitations of recursive algorithms for matrix multiplication, as opposed to just a statement about the rank of the problem.

Authors: Isaac Rudich, Louis-Martin Rousseau

Strassen showed that two 2x2 matrices can be multiplied with 7 multiplications instead of 8. Applied recursively, his algorithm multiplies two nxn matrices with O(n^2.807) multiplications, beating the naive O(n^3). The best known 3x3 recursive matrix multiplication algorithm uses 23 multiplications O(n^2.854). The best published lower bound of 21 (on algorithms with integer constants) leaves room for an algorithm with O(n^2.771) multiplications, and thus does not rule out the possibility of an algorithm that would beat Strassen's. We prove a lower bound of 22 multiplications for any 3x3 recursive algorithm with integer constants, proving that no such algorithm can do better than O(n^2.814) multiplications, and eliminating the possibility of a 3x3 algorithm that beats Strassen's 2x2 method. The proof builds on a recent decomposition method from Wang, who approached the problem by turning it into 496 subproblems. We provide exact solutions for 359 of them. The proof is in Lean; verification requires auditing only a few short files. The Lean formalization directly encodes statements about the limitations of recursive algorithms for matrix multiplication, as opposed to just a statement about the rank of the problem.

Integer reachability in VASS with transfers: a refined complexity analysis

from arXiv: Computational Complexity

Authors: Tymoteusz Kucharek, Piotr Hofman

Integer reachability is NP-complete for vector addition systems with states (VASS), but becomes PSPACE-complete in the presence of transfer operations. We refine this complexity gap for single-transfer VASS by identifying structural features of transfers responsible for the increase in complexity. Each system induces a transfer graph whose vertices are counters and whose edges represent possible transfers. We classify its vertices as good or bad, according to the branching and cyclic structure of their reachable subgraphs. Let $b$ be the number of bad vertices. We show that every positive instance admits a polynomially verifiable certificate of size $|I|^{O(b+1)}$, where $|I|$ is the input size. Consequently, integer reachability for single-transfer VASS can be decided in nondeterministic time $|I|^{O(b+1)}$; in particular, it belongs to NP for every class with a bounded number of bad counters. Conversely, we show that bad counters provide sufficient structural power to encode space-bounded computation. For every transfer graph with $b$ bad vertices, we construct a single-transfer VASS that encodes the acceptance of a Turing machine using $b^{O(1)}$ tape cells. This yields PSPACE-hardness for every polynomial-time constructible family of transfer graphs containing linearly many bad vertices. Our results isolate the transfer patterns responsible for the complexity of integer reachability.

Authors: Tymoteusz Kucharek, Piotr Hofman

Integer reachability is NP-complete for vector addition systems with states (VASS), but becomes PSPACE-complete in the presence of transfer operations. We refine this complexity gap for single-transfer VASS by identifying structural features of transfers responsible for the increase in complexity. Each system induces a transfer graph whose vertices are counters and whose edges represent possible transfers. We classify its vertices as good or bad, according to the branching and cyclic structure of their reachable subgraphs. Let $b$ be the number of bad vertices. We show that every positive instance admits a polynomially verifiable certificate of size $|I|^{O(b+1)}$, where $|I|$ is the input size. Consequently, integer reachability for single-transfer VASS can be decided in nondeterministic time $|I|^{O(b+1)}$; in particular, it belongs to NP for every class with a bounded number of bad counters. Conversely, we show that bad counters provide sufficient structural power to encode space-bounded computation. For every transfer graph with $b$ bad vertices, we construct a single-transfer VASS that encodes the acceptance of a Turing machine using $b^{O(1)}$ tape cells. This yields PSPACE-hardness for every polynomial-time constructible family of transfer graphs containing linearly many bad vertices. Our results isolate the transfer patterns responsible for the complexity of integer reachability.

Beyond IP = PSPACE and QIP = PSPACE: Interactive Proofs in Arbitrary Physical Theories

from arXiv: Computational Complexity

Authors: Kishor Bharti

The equalities IP = PSPACE and QIP = PSPACE, the latter achievable with three messages, raise a basic question: how much of an interactive proof's power comes from the underlying physical theory? We study interactive proofs in general probabilistic theories, which include classical and quantum theory. The answer depends on what the prover and verifier exchange and how the theory specifies efficient operations. When they exchange only classical messages, protocols in every theory satisfying our standard assumptions decide exactly PSPACE. For protocols with a quantum verifier and quantum messages, allowing a prover to use any theory containing quantum theory does not increase the maximum acceptance probability. Thus, the three-message PSPACE result remains valid against such provers. When messages may be arbitrary systems, the interactive-proof class can strictly exceed PSPACE.

Authors: Kishor Bharti

The equalities IP = PSPACE and QIP = PSPACE, the latter achievable with three messages, raise a basic question: how much of an interactive proof's power comes from the underlying physical theory? We study interactive proofs in general probabilistic theories, which include classical and quantum theory. The answer depends on what the prover and verifier exchange and how the theory specifies efficient operations. When they exchange only classical messages, protocols in every theory satisfying our standard assumptions decide exactly PSPACE. For protocols with a quantum verifier and quantum messages, allowing a prover to use any theory containing quantum theory does not increase the maximum acceptance probability. Thus, the three-message PSPACE result remains valid against such provers. When messages may be arbitrary systems, the interactive-proof class can strictly exceed PSPACE.

A Degree--Size Relation for Resolution over Polynomials

from arXiv: Computational Complexity

Authors: Shuo Pang

For every constant-width CNF, we show that linear degree in polynomial calculus (PC) implies exponential size in resolution over constant-degree polynomials, over the same prime field. Applications include exponential lower bounds for CNFs in $\operatorname{Res}(\operatorname{PC}_r/\mathbb{F}_p)$ and hence in $\operatorname{Res}(\oplus_p)$, separations between different moduli, improved lower bounds for $\operatorname{Res}(k)$ up to $k=\varepsilon\log n$, proof-search consequences, and an implication of super-polynomial $AC^0[p]$-Frege bounds from very strong PC degree lower bounds. The proof uses the common-multiplier idea isolated from Braun [arXiv:2609.23015] to construct a Razborov--Smolensky approximation that preserves inferences, without introducing extension variables. The approximation errors are measured by ranks of the multiplication maps induced by the error-witness polynomials, modulo bounded-degree PC consequences.

Authors: Shuo Pang

For every constant-width CNF, we show that linear degree in polynomial calculus (PC) implies exponential size in resolution over constant-degree polynomials, over the same prime field. Applications include exponential lower bounds for CNFs in $\operatorname{Res}(\operatorname{PC}_r/\mathbb{F}_p)$ and hence in $\operatorname{Res}(\oplus_p)$, separations between different moduli, improved lower bounds for $\operatorname{Res}(k)$ up to $k=\varepsilon\log n$, proof-search consequences, and an implication of super-polynomial $AC^0[p]$-Frege bounds from very strong PC degree lower bounds. The proof uses the common-multiplier idea isolated from Braun [arXiv:2609.23015] to construct a Razborov--Smolensky approximation that preserves inferences, without introducing extension variables. The approximation errors are measured by ranks of the multiplication maps induced by the error-witness polynomials, modulo bounded-degree PC consequences.

Entrywise Logarithmic Matrix Algebra and Dichotomy of Planar Graph Homomorphisms (Part I)

from arXiv: Computational Complexity

Authors: Jin-Yi Cai, Zhuxiao Tang

We prove a complexity classification of counting planar graph homomorphisms with non-negative weights. For a real symmetric matrix $M$ with non-negative entries, the problem $\PlGH(M)$ is either (1) P-time computable over all graphs, or (2) \#P-hard in general but P-time computable over planar graphs, or (3) \#P-hard over planar graphs. Furthermore, $\PlGH(M)$ in (2) consists of precisely those that involve the P-time FKT algorithm to count planar perfect matchings with a holographic transformation. The dichotomy is achieved by forming a (centered) logarithmic matrix algebra (a vector space with bilinear multiplication) by taking entrywise logarithms of all realizable matrices from $M$ using planar edge gadgets and polynomial interpolation. The current version is part I, which contains the proof for the dichotomy of entrywise positive and positive definite matrices, which is at the core of the dichotomy for non-negative matrices. Part II contains the extension from entrywise positive and positive definite matrices to non-negative matrices.

Authors: Jin-Yi Cai, Zhuxiao Tang

We prove a complexity classification of counting planar graph homomorphisms with non-negative weights. For a real symmetric matrix $M$ with non-negative entries, the problem $\PlGH(M)$ is either (1) P-time computable over all graphs, or (2) \#P-hard in general but P-time computable over planar graphs, or (3) \#P-hard over planar graphs. Furthermore, $\PlGH(M)$ in (2) consists of precisely those that involve the P-time FKT algorithm to count planar perfect matchings with a holographic transformation. The dichotomy is achieved by forming a (centered) logarithmic matrix algebra (a vector space with bilinear multiplication) by taking entrywise logarithms of all realizable matrices from $M$ using planar edge gadgets and polynomial interpolation. The current version is part I, which contains the proof for the dichotomy of entrywise positive and positive definite matrices, which is at the core of the dichotomy for non-negative matrices. Part II contains the extension from entrywise positive and positive definite matrices to non-negative matrices.

Approximate Polynomial Satisfiability is in the Counting Hierarchy

from arXiv: Computational Complexity

Authors: Nikhil Balaji, Mahsa Shirmohammadi, Sébastien Tavenas, James Worrell

The Approximate polynomial satisfiability problem (APS), introduced by Guo, Saxena, and Sinhababu (CCC 2018), asks whether the zero vector lies in the Zariski closure of the image of a given polynomial map. Specifically, for a field $k$ with algebraic closure~$K$, the problem asks whether $\boldsymbol 0 \in\overline{\boldsymbol f(K^n)}$ for a polynomial map $\boldsymbol f=(f_1,\ldots,f_m)$ with $f_i\in k[X_1,\ldots,X_n]$. APS is a natural topological analogue of Hilbert's Nullstellensatz, namely the question of whether a given system of polynomial equations has a common zero. APS captures several problems in algebraic complexity, including border rank, hitting sets for border classes, and null-cone membership; it is known to be NP-hard and in PSPACE. We show that APS lies in the Counting Hierarchy (CH) over both the rationals and finite fields, substantially improving the known PSPACE upper bound. Our proof builds on a recent breakthrough due to Andrews, Garg, and Schost (FOCS 2026) on deciding Hilbert's Nullstellensatz in CH. As a corollary, our result improves the complexity of certifying hitting sets for border classes from PSPACE to CH. We also give a polynomial-time reduction of Hilbert's Nullstellensatz to APS, valid in any characteristic. In characteristic zero, we give a reduction of APS to the decision problem for the existential theory of real closed fields. Overall, our results place approximate polynomial satisfiability closer in complexity to exact polynomial feasibility and as a byproduct give improved complexity bounds for several problems arising in approximative complexity.

Authors: Nikhil Balaji, Mahsa Shirmohammadi, Sébastien Tavenas, James Worrell

The Approximate polynomial satisfiability problem (APS), introduced by Guo, Saxena, and Sinhababu (CCC 2018), asks whether the zero vector lies in the Zariski closure of the image of a given polynomial map. Specifically, for a field $k$ with algebraic closure~$K$, the problem asks whether $\boldsymbol 0 \in\overline{\boldsymbol f(K^n)}$ for a polynomial map $\boldsymbol f=(f_1,\ldots,f_m)$ with $f_i\in k[X_1,\ldots,X_n]$. APS is a natural topological analogue of Hilbert's Nullstellensatz, namely the question of whether a given system of polynomial equations has a common zero. APS captures several problems in algebraic complexity, including border rank, hitting sets for border classes, and null-cone membership; it is known to be NP-hard and in PSPACE. We show that APS lies in the Counting Hierarchy (CH) over both the rationals and finite fields, substantially improving the known PSPACE upper bound. Our proof builds on a recent breakthrough due to Andrews, Garg, and Schost (FOCS 2026) on deciding Hilbert's Nullstellensatz in CH. As a corollary, our result improves the complexity of certifying hitting sets for border classes from PSPACE to CH. We also give a polynomial-time reduction of Hilbert's Nullstellensatz to APS, valid in any characteristic. In characteristic zero, we give a reduction of APS to the decision problem for the existential theory of real closed fields. Overall, our results place approximate polynomial satisfiability closer in complexity to exact polynomial feasibility and as a byproduct give improved complexity bounds for several problems arising in approximative complexity.

Good Quantum Locally Testable Codes from Lossless Cubical Complexes

from arXiv: Computational Complexity

Authors: Itay Cohen, Itai Leigh, Assaf Reiner, Amnon Ta-Shma, Elad Tzalik

Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.

Authors: Itay Cohen, Itai Leigh, Assaf Reiner, Amnon Ta-Shma, Elad Tzalik

Sipser and Spielman constructed LDPC codes from either bipartite \emph{spectral} expanders or one-sided \emph{lossless} expanders. In higher dimensions, \emph{spectral} expansion similarly played a central role in the constructions of asymptotically good classical LTCs and qLDPC codes by Dinur, Evra, Livne, Lubotzky, and Mozes and by Panteleev and Kalachev. Alternatively, Lin and Hsieh constructed classical LTCs and qLDPC codes from two-dimensional \emph{lossless} cubical complexes. In this work we develop the higher-dimensional \emph{lossless} approach. We do not construct the required high-dimensional lossless cubical complexes; rather, we investigate what their existence would imply. We associate with a high-dimensional cubical complex a \emph{level chain complex}, whose chain groups are supported on the level sets of the Boolean cube rather than on its cells. Our main technical contribution is a clean local-to-global theorem for this structure: suitable one-dimensional lossless expansion in the directional graphs implies small-set coboundary expansion of the global level complex. As a consequence, sufficiently imbalanced, two-sided lossless four-dimensional cubical complexes give rise to asymptotically good quantum locally testable codes. We expect the local-to-global principle developed here to have further applications.

Hyperbolic Sphericity

from arXiv: Computational Geometry

Authors: Thomas Bläsius, Lennart Großkreutz, Jean-Pierre von der Heydt

The sphericity of a graph is the minimum dimension d such that the graph has an intersection representation of d-dimensional balls of equal radius. While sphericity has been studied in Euclidean space, we initiate the study of hyperbolic sphericity. The hyperbolic sphericity of a graph can be significantly smaller than its Euclidean counterpart, but, contrary to the Euclidean setting, depends strongly on the radius of the balls. We show that, if the radius of the balls can be chosen depending on the graph, the hyperbolic sphericity is upper bounded by the Euclidean sphericity. This extends a previous result for 2-dimensional hyperbolic space, i.e., uniform disk graphs, to arbitrary dimensions. Moreover, our proof is significantly simpler. If we fix the radius, i.e., do not make it dependent on the graph, we show that hyperbolic sphericity can be larger than Euclidean sphericity, but by at most 1. Additionally, we study how hyperbolic sphericity changes with the ball radius. We show that choosing a larger radius can substantially decrease the sphericity while increasing it by at most 1. We also provide a construction of a graph where the sphericity oscillates between different values as the radius increases. Besides being theoretically interesting, we note that these results are relevant for graph embeddings in machine learning, where one is interested in low-dimensional numeric representations of symbolic data like graphs.

Authors: Thomas Bläsius, Lennart Großkreutz, Jean-Pierre von der Heydt

The sphericity of a graph is the minimum dimension d such that the graph has an intersection representation of d-dimensional balls of equal radius. While sphericity has been studied in Euclidean space, we initiate the study of hyperbolic sphericity. The hyperbolic sphericity of a graph can be significantly smaller than its Euclidean counterpart, but, contrary to the Euclidean setting, depends strongly on the radius of the balls. We show that, if the radius of the balls can be chosen depending on the graph, the hyperbolic sphericity is upper bounded by the Euclidean sphericity. This extends a previous result for 2-dimensional hyperbolic space, i.e., uniform disk graphs, to arbitrary dimensions. Moreover, our proof is significantly simpler. If we fix the radius, i.e., do not make it dependent on the graph, we show that hyperbolic sphericity can be larger than Euclidean sphericity, but by at most 1. Additionally, we study how hyperbolic sphericity changes with the ball radius. We show that choosing a larger radius can substantially decrease the sphericity while increasing it by at most 1. We also provide a construction of a graph where the sphericity oscillates between different values as the radius increases. Besides being theoretically interesting, we note that these results are relevant for graph embeddings in machine learning, where one is interested in low-dimensional numeric representations of symbolic data like graphs.

Optimal Coresets for Hyperbolic Farthest-Point Queries via Ideal-Boundary Envelopes

from arXiv: Computational Geometry

Authors: Eunku Park

We study coresets for farthest-point queries in hyperbolic space. Given a nonempty finite set $P \subset \mathbb{H}^D$ and $0<\varepsilon \le 1$, we seek a coreset $P_{\varepsilon} \subseteq P$ whose farthest distance from every query point underestimates that of $P$ by at most an additive $\varepsilon$ and retains at least a $1-\varepsilon$ fraction of it. For every fixed $D \ge 2$, we prove that the optimal worst-case coreset size is $Θ\bigl(\varepsilon^{-(D-1)/2}\bigr)$. Our main geometric ingredient is an exact reduction from hyperbolic queries to an upper envelope on the ideal boundary. In the hyperboloid model, each input point induces a positive boundary-score function whose logarithm gives its asymptotic distance offset along geodesic rays. We define the \emph{ideal-boundary envelope} as the pointwise maximum of these functions and prove that the supremum additive loss over all queries equals the maximum logarithmic gap between the input and coreset envelopes. For the upper bound, we move the minimum-enclosing-ball center to the origin and normalize the spatial coordinates, obtaining a bounded Euclidean point set whose boundary envelope is bounded away from zero. A standard Euclidean kernel then approximates all directional score maxima simultaneously, and the structure theorem yields both guarantees. For the lower bound, a spherical packing on a fixed-radius hyperbolic sphere, together with antipodal queries and the hyperbolic cosine law, makes every input point indispensable, matching the upper bound even for either guarantee separately.

Authors: Eunku Park

We study coresets for farthest-point queries in hyperbolic space. Given a nonempty finite set $P \subset \mathbb{H}^D$ and $0<\varepsilon \le 1$, we seek a coreset $P_{\varepsilon} \subseteq P$ whose farthest distance from every query point underestimates that of $P$ by at most an additive $\varepsilon$ and retains at least a $1-\varepsilon$ fraction of it. For every fixed $D \ge 2$, we prove that the optimal worst-case coreset size is $Θ\bigl(\varepsilon^{-(D-1)/2}\bigr)$. Our main geometric ingredient is an exact reduction from hyperbolic queries to an upper envelope on the ideal boundary. In the hyperboloid model, each input point induces a positive boundary-score function whose logarithm gives its asymptotic distance offset along geodesic rays. We define the \emph{ideal-boundary envelope} as the pointwise maximum of these functions and prove that the supremum additive loss over all queries equals the maximum logarithmic gap between the input and coreset envelopes. For the upper bound, we move the minimum-enclosing-ball center to the origin and normalize the spatial coordinates, obtaining a bounded Euclidean point set whose boundary envelope is bounded away from zero. A standard Euclidean kernel then approximates all directional score maxima simultaneously, and the structure theorem yields both guarantees. For the lower bound, a spherical packing on a fixed-radius hyperbolic sphere, together with antipodal queries and the hyperbolic cosine law, makes every input point indispensable, matching the upper bound even for either guarantee separately.

Towards Strongly Aperiodic Monotiles in Higher Dimensions

from arXiv: Computational Geometry

Authors: Dmitry Kamenetsky

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism---the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings---to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: github.com/dimkadimon/Monotile-RN

Authors: Dmitry Kamenetsky

The discovery of Chair44 (Tsiokos, 2026) settled the three-dimensional einstein problem with a strongly aperiodic polyhedral monotile in $\mathbb{R}^3$. This note extends the underlying mechanism---the rep-$2^N$ chair $C_N = [0,2]^N \setminus (1,2]^N$ with corner/socket markings---to $\mathbb{R}^N$. Besides expository material (the rep-$2^N$ dissection and a conditional strong-aperiodicity theorem under lattice registration and hierarchical enforcement), the note makes a new computational contribution. We introduce a frame-marking formalism in which the marking of a tile is its full orientation frame and the matching rule is the contact language generated by the substitution itself; this makes the search for matching rules finite in every dimension. We give a finite certificate (coarsening closure, tightness, and a two-shell enclosure analysis) whose validity implies that every lattice-registered tiling by the marked tile is uniquely hierarchical, hence strongly aperiodic. For $N=3$ the certificate passes: it yields explicit facet matching rules on the 24 panels of $C_3$ (135 admissible facet-contact triples) and reproduces, from first principles and independently of published constructions, the Chair44 statistics 2388 $\to$ 44 admissible contacts (30 occurring), 33 one-shell clusters, 15 extendable, each forcing a unique supertile. Among the 2187 homochiral frame assignments of the 3D substitution with a translated central child, the certified one is unique up to conjugation. For $N=4$ the same pipeline is run on several structured families of frame assignments (canonical, $D_4$-, $Z_2\times Z_2$- and $Z_4$-symmetric, and a lift of the 3D solution); none is coarsening-closed, and we report the failure data. A self-similar marking of $C_4$ thus remains an explicitly finite, open computational problem, which we state precisely. Code: https://github.com/dimkadimon/Monotile-RN

Minimum Spanning Trees for Square Crop Plots

from arXiv: Computational Geometry

Authors: Mingyang Gong, Adiesha Liyanage, Braeden Sopp, Muzhou Chen, Binhai Zhu

Motivated by accessing crop plots in a field, where each crop plot can only be visited by a given pair of entry/exit points (we have three different types, each having four cases), we study the corresponding Minimum Spanning Tree (MST) problem of such a planar set of axis-aligned unit squares. It turns out that this crop plot distance does not satisfy triangle inequality, even though the whole setup of the problem is geometric. Hence additional care must be taken. The main results of this paper are as follows: (1) we prove that computing the MST of a set $P$ of unit squares under the crop plot distance is NP-hard, (2) the MST of $P$ can be approximated with a factor-4 approximation.

Authors: Mingyang Gong, Adiesha Liyanage, Braeden Sopp, Muzhou Chen, Binhai Zhu

Motivated by accessing crop plots in a field, where each crop plot can only be visited by a given pair of entry/exit points (we have three different types, each having four cases), we study the corresponding Minimum Spanning Tree (MST) problem of such a planar set of axis-aligned unit squares. It turns out that this crop plot distance does not satisfy triangle inequality, even though the whole setup of the problem is geometric. Hence additional care must be taken. The main results of this paper are as follows: (1) we prove that computing the MST of a set $P$ of unit squares under the crop plot distance is NP-hard, (2) the MST of $P$ can be approximated with a factor-4 approximation.

$k$-arrangements of pseudolines and pseudocircles

from arXiv: Computational Geometry

Authors: Jan Kynčl, Carolina Medina, Gelasio Salazar

A $k$-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly $k$ points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed $m\ge 1$, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size $m$. We show that, for every $m\ge 3$, the number of unavoidable simple $k$-arrangements of pseudolines of size $m$ grows exponentially with $k$, independently of $m$. For even $k$, we prove an analogous result for $k$-arrangements of pseudocircles.

Authors: Jan Kynčl, Carolina Medina, Gelasio Salazar

A $k$-arrangement of pseudolines is a set of bi-infinite curves in the plane such that any two of them intersect each other in exactly $k$ points, at which they cross, and it is simple if no three curves meet at a common point. Cyclic arrangements are the only simple $1$-arrangements of pseudolines that are unavoidable, in the Ramsey spirit: for each fixed $m\ge 1$, every sufficiently large simple $1$-arrangement of pseudolines has a cyclic subarrangement of size $m$. We show that, for every $m\ge 3$, the number of unavoidable simple $k$-arrangements of pseudolines of size $m$ grows exponentially with $k$, independently of $m$. For even $k$, we prove an analogous result for $k$-arrangements of pseudocircles.

Polynomial-time additive-error estimation of output probabilities for shallow quantum circuits

from arXiv: Data Structures and Algorithms

Authors: Matthew Coudron, Michael J. Gullans, Jon Nelson, Joel Rajakumar, Shi Jie Samuel Tan

We give a deterministic classical algorithm that estimates $|\langle x|U|0^n\rangle|^2$ to additive error $\varepsilon$ in $\mathrm{poly}(n, 1/\varepsilon)$ time, where $U$ is a constant-depth quantum circuit comprised of gates with bounded fan-in and arbitrary connectivity, and $x$ is an arbitrary $n$-bit output string. This improves over prior state-of-the-art algorithms that takes $n^{O(log(n))}$ time for the same task, $n^{O(log(log(n))}$ when $U$ is geometrically local, and $n^{O(1)}$ for 2D geometrically-local circuits.

Authors: Matthew Coudron, Michael J. Gullans, Jon Nelson, Joel Rajakumar, Shi Jie Samuel Tan

We give a deterministic classical algorithm that estimates $|\langle x|U|0^n\rangle|^2$ to additive error $\varepsilon$ in $\mathrm{poly}(n, 1/\varepsilon)$ time, where $U$ is a constant-depth quantum circuit comprised of gates with bounded fan-in and arbitrary connectivity, and $x$ is an arbitrary $n$-bit output string. This improves over prior state-of-the-art algorithms that takes $n^{O(log(n))}$ time for the same task, $n^{O(log(log(n))}$ when $U$ is geometrically local, and $n^{O(1)}$ for 2D geometrically-local circuits.

Stable and Online Algorithms for Random Matrix Discrepancy

from arXiv: Data Structures and Algorithms

Authors: Eren C. Kızıldağ, Shuangping Li

We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $κ>0$, find signs $σ_1,\dots,σ_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N σ_i A_i$ is at most $κ\sqrt{N}$. Focusing on the proportional regime $N/d^2\to τ\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $κ\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time \emph{recenter-and-round} algorithm that is noise-stable and succeeds whenever $τ=Ω(\frac{1}{κ^2\log(1/κ)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the \emph{Frobenius-greedy} algorithm, establishing that it succeeds when $τ>τ_{\rm FG}(κ)\sim \fracπ{4κ^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $Θ(\frac{1}{κ^2\log(1/κ)})$ for stable offline algorithms and $Θ(\frac{1}{κ^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $Θ(\log(1/κ))$, as shown by Maillard~\cite{maillard2025}.

Authors: Eren C. Kızıldağ, Shuangping Li

We study the average-case matrix discrepancy problem: given independent normalized $d\times d$ Gaussian orthogonal ensemble matrices $A_1,\dots,A_N$ and a fixed margin $κ>0$, find signs $σ_1,\dots,σ_N\in\{-1,1\}$ such that the operator norm of $\sum_{i=1}^N σ_i A_i$ is at most $κ\sqrt{N}$. Focusing on the proportional regime $N/d^2\to τ\in(0,\infty)$ as $d\to\infty$ followed by the small-margin limit $κ\downarrow 0$, we characterize the density required by stable offline algorithms and by online algorithms. In the offline setting, we construct a polynomial-time \emph{recenter-and-round} algorithm that is noise-stable and succeeds whenever $τ=Ω(\frac{1}{κ^2\log(1/κ)})$, along with a matching lower bound for all stable algorithms. In the online setting where each sign must be chosen irrevocably upon observing the corresponding matrix, we determine the exact limiting performance of the \emph{Frobenius-greedy} algorithm, establishing that it succeeds when $τ>τ_{\rm FG}(κ)\sim \fracπ{4κ^2}$, as well as a matching lower bound for all online algorithms by conditioning on a revealed prefix. At the core of our algorithms lies rotational symmetry, which enables us to transfer Frobenius norm control into operator norm guarantees. Together, our results identify the algorithmic phase transition points for random matrix discrepancy: $Θ(\frac{1}{κ^2\log(1/κ)})$ for stable offline algorithms and $Θ(\frac{1}{κ^2})$ for online algorithms. Both thresholds lie far above the satisfiability scale $Θ(\log(1/κ))$, as shown by Maillard~\cite{maillard2025}.

Coloring 3-colorable graphs with $O(n^{4/23})$ colors via a Gaussian-cover recursion

from arXiv: Data Structures and Algorithms

Authors: Emile Anand

We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\sqrt{97})/18+ε})=O(n^{0.17506\dots + ε})$ colors for every fixed $\smash{ε>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $μ$, and variance $v$, we prove $v\leq r^2-μ^2/(2\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.

Authors: Emile Anand

We give a randomized polynomial-time algorithm that colors any promised $3$-colorable graph on $n$ vertices with $\smash{O(n^{4/23}) = O(n^{0.17391\ldots})}$ colors, improving on the recent bounds of $O(n^{0.19539})$ by Bansal, Huang, and Lee and Narang and Tang who obtained $O(n^{(13-\sqrt{97})/18+ε})=O(n^{0.17506\dots + ε})$ colors for every fixed $\smash{ε>0}$. To prove our result, we start from a fixed-level semidefinite relaxation, where we use a finite-depth recursion on Gaussian covers. Fixing a root vertex, we group vertices by correlation with the root vector. Here, each step extends a cover of directions by one edge and transfers it to a successor group. Our key analytic ingredient is a variance bound for Gaussian maxima: for a maximum of $m\geq 2$ centered linear forms with coefficient norms at most $r$, mean $μ$, and variance $v$, we prove $v\leq r^2-μ^2/(2\log m)$ using Chen's Gaussian convexity theorem. Together with a variance-scale lower-tail estimate, this controls the threshold loss at each extension, which shows that root-conditioned vector colorings can either extract a large independent set from a group or bound its size, forcing a contradiction after constantly many steps. The resulting sparse-case guarantee combines with the dense progress bound of Kawarabayashi, Thorup, and Yoneda, and the recursion's numerical inequalities are verified via rational interval arithmetic.

Linear Programming Representations and Strongly Polynomial Algorithms for Robust Markov Decision Processes

from arXiv: Data Structures and Algorithms

Authors: Han Zhong, Yinyu Ye

We study linear programming (LP) representations and strongly polynomial algorithms for robust Markov decision processes (RMDPs) with rational polyhedral state-action rectangular uncertainty in rewards and transitions. By encoding a finite sequence of robust policy-iteration steps, we construct a single LP whose optimal solutions recover the robust optimal value and all optimal stationary randomized policies. At fixed discount, the LP has polynomial dimension and encoding length and can be constructed in strongly polynomial time. We also develop a general complexity analysis of robust policy iteration that combines the cost of minimizing over uncertainty sets with the number of iterations needed to evaluate a policy. For a fixed discount factor, we use this analysis to improve the known complexity bounds for $\ell_1$ and $\ell_\infty$ RMDPs and establish new strongly polynomial bounds for general interval, weighted $\ell_1$, and Wasserstein RMDPs, as well as turn-based stochastic games with these uncertainty sets.

Authors: Han Zhong, Yinyu Ye

We study linear programming (LP) representations and strongly polynomial algorithms for robust Markov decision processes (RMDPs) with rational polyhedral state-action rectangular uncertainty in rewards and transitions. By encoding a finite sequence of robust policy-iteration steps, we construct a single LP whose optimal solutions recover the robust optimal value and all optimal stationary randomized policies. At fixed discount, the LP has polynomial dimension and encoding length and can be constructed in strongly polynomial time. We also develop a general complexity analysis of robust policy iteration that combines the cost of minimizing over uncertainty sets with the number of iterations needed to evaluate a policy. For a fixed discount factor, we use this analysis to improve the known complexity bounds for $\ell_1$ and $\ell_\infty$ RMDPs and establish new strongly polynomial bounds for general interval, weighted $\ell_1$, and Wasserstein RMDPs, as well as turn-based stochastic games with these uncertainty sets.

Randomized Matvec Lower Bounds for Simplex-Based Matrix Games

from arXiv: Data Structures and Algorithms

Authors: Wendao Wu, Cong Fang

We prove randomized matrix-vector query lower bounds for two normalized matrix-game geometries: a Euclidean unit ball against a probability simplex, with row norms at most one, and two probability simplices, with entries of absolute value at most one. Each query returns $(Ax,A^\top y)$ for arbitrary real vectors. The algorithm must return a feasible pair with full saddle-point gap at most $\varepsilon$, with probability at least $2/3$ on every admissible matrix. For sufficiently small $\varepsilon$, the worst-case query complexities are $Ω(\varepsilon^{-2/3}/(\log^2(1/\varepsilon)\log\log(1/\varepsilon)))$ for ball-simplex games and $Ω(\varepsilon^{-2/3}/(\log^{7/3}(1/\varepsilon)\log\log(1/\varepsilon)))$ for simplex-simplex games. The hard instances have dimensions of order $\varepsilon^{-2/3}$ and $\varepsilon^{-2/3}/\log^{1/3}(1/\varepsilon)$, respectively, and the bounds extend to larger dimensions. These lower bounds match the deterministic upper bounds of Karmarkar, O'Carroll, and Sidford up to logarithmic factors. The proof extracts a fresh Gaussian core after adaptive two-sided queries and uses uncertainty in its smallest singular value to establish linear-system solve hardness. Two reductions transfer this hardness to matrix games by converting a small full gap into a small residual, with an additional logarithmic normalization loss only for simplex-simplex games.

Authors: Wendao Wu, Cong Fang

We prove randomized matrix-vector query lower bounds for two normalized matrix-game geometries: a Euclidean unit ball against a probability simplex, with row norms at most one, and two probability simplices, with entries of absolute value at most one. Each query returns $(Ax,A^\top y)$ for arbitrary real vectors. The algorithm must return a feasible pair with full saddle-point gap at most $\varepsilon$, with probability at least $2/3$ on every admissible matrix. For sufficiently small $\varepsilon$, the worst-case query complexities are $Ω(\varepsilon^{-2/3}/(\log^2(1/\varepsilon)\log\log(1/\varepsilon)))$ for ball-simplex games and $Ω(\varepsilon^{-2/3}/(\log^{7/3}(1/\varepsilon)\log\log(1/\varepsilon)))$ for simplex-simplex games. The hard instances have dimensions of order $\varepsilon^{-2/3}$ and $\varepsilon^{-2/3}/\log^{1/3}(1/\varepsilon)$, respectively, and the bounds extend to larger dimensions. These lower bounds match the deterministic upper bounds of Karmarkar, O'Carroll, and Sidford up to logarithmic factors. The proof extracts a fresh Gaussian core after adaptive two-sided queries and uses uncertainty in its smallest singular value to establish linear-system solve hardness. Two reductions transfer this hardness to matrix games by converting a small full gap into a small residual, with an additional logarithmic normalization loss only for simplex-simplex games.

Quantum state preparation for weighted d-DNNF

from arXiv: Data Structures and Algorithms

Authors: Steef Hegeman, Joon Hyung Lee, Alfons Laarman

The quantum state preparation problem is to, given a description of a quantum state, efficiently generate a quantum circuit computing the state. We show that for quantum states described by weighted d-DNNF (deterministic, decomposable pseudo-Boolean circuits) a quantum circuit computing the state can be obtained in linear time up to complex arithmetic.

Authors: Steef Hegeman, Joon Hyung Lee, Alfons Laarman

The quantum state preparation problem is to, given a description of a quantum state, efficiently generate a quantum circuit computing the state. We show that for quantum states described by weighted d-DNNF (deterministic, decomposable pseudo-Boolean circuits) a quantum circuit computing the state can be obtained in linear time up to complex arithmetic.

A computational phase diagram for the transverse field Ising model

from arXiv: Data Structures and Algorithms

Authors: Thuy-Duong Vuong

We study the transverse field Ising model, defined by the Hamiltonian $H =\frac{1}{2}\sum_{i, j\in [n]} J_{ij} Z_i Z_j +\sum_{i=1}^n h_i^z Z_i + η\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $η$ is the transverse field strength. Let $Δ(J)=λ_{\max}(J)-λ_{\min}(J)$ be the spectral width of $J.$ When the inverse temperature $β\geq0$ satisfies $Δ(J)\cdot\frac{\tanh(βη)}η\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(β)=\operatorname{Tr}(e^{-βH})$ to a given relative error $ε\in(0,1)$ in time polynomial in $n$, $β$, the model parameters, and $ε^{-1}$. When $ Δ(J) \cdot \frac{\tanh(βη)}η > 1 ,$ we show that approximating $ Z(β)$ within an $\exp(o(n))$-multiplicative factor is $\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $Δ(J)\cdot \frac{\tanh(βη)}η\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ ρ_β= \frac{e^{-βH}}{\operatorname{Tr}(e^{-βH})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \in \{I, X\}^{\otimes n}$, the algorithm further achieves arbitrarily small relative error.

Authors: Thuy-Duong Vuong

We study the transverse field Ising model, defined by the Hamiltonian $H =\frac{1}{2}\sum_{i, j\in [n]} J_{ij} Z_i Z_j +\sum_{i=1}^n h_i^z Z_i + η\sum_{i} X_i$ where $J $ is the symmetric interaction matrix, and $η$ is the transverse field strength. Let $Δ(J)=λ_{\max}(J)-λ_{\min}(J)$ be the spectral width of $J.$ When the inverse temperature $β\geq0$ satisfies $Δ(J)\cdot\frac{\tanh(βη)}η\leq1$, we give a randomized classical algorithm that approximates the partition function $Z(β)=\operatorname{Tr}(e^{-βH})$ to a given relative error $ε\in(0,1)$ in time polynomial in $n$, $β$, the model parameters, and $ε^{-1}$. When $ Δ(J) \cdot \frac{\tanh(βη)}η > 1 ,$ we show that approximating $ Z(β)$ within an $\exp(o(n))$-multiplicative factor is $\textbf{NP}$-hard, and thus unlikely to admit an efficient classical or quantum algorithms under standard complexity theoretic assumptions. Furthermore, in the regime $Δ(J)\cdot \frac{\tanh(βη)}η\leq 1,$ we provide an efficient randomized classical algorithm that approximates Pauli string observables of the Gibbs state $ ρ_β= \frac{e^{-βH}}{\operatorname{Tr}(e^{-βH})}$ within an arbitrarily small additive error. In the special case when the observable is also diagonal in the $X$-basis, i.e. $P \in \{I, X\}^{\otimes n}$, the algorithm further achieves arbitrarily small relative error.

Vertex-Failure Distance Oracles and Labeling Schemes: Compact and Constant-Approximate

from arXiv: Data Structures and Algorithms

Authors: Yaowei Long

We present new algorithms for the vertex-failure distance oracles and labeling schemes problems in undirected weighted graphs. A vertex-failure distance oracle is a data structure that, given two vertices $x$ and $y$ and a failed vertex set $F$ of size at most $f$, returns an approximation to the distance between $x$ and $y$ in $G \setminus F$. In the labeling-scheme setting, the data structure needs to be stored distributively as labels on the vertices, and each query $(x,y,F)$ must be answered by accessing only the labels of the vertices in $F \cup \{x,y\}$. For any $f\geq 1$ and $k \ge 1$, we obtain a vertex-failure distance oracle with $O(k^{6})$ approximation, space $\tilde{O}(f^{2}n^{1+1/k})$, query time $\tilde{O}(f^{5}n^{1/k})$, and polynomial preprocessing time. In particular, this is the first time-efficient oracle for multiple vertex failures with space close to linear, as well as the first constant-approximation oracle with polynomial space when tolerating $Ω(\log n)$ vertex failures. The previous results, due to [Duan-Gu-Ren, SODA'21], gave two alternatives: for any constant $c \ge 1$ and $ε>0$, one oracle has $\mathrm{poly}(\log n,f)$ approximation, space $n^{2+1/c}\mathrm{poly}(\log n,f)$, and query time $\mathrm{poly}(\log n,f^{c})$, while the other has $(1+ε)$ approximation, space $n^{2+1/c}(\log n/ε)^{O(f)}$, and query time $\mathrm{poly}(\log n,f^{c},1/ε)$. We also obtain a vertex-failure distance labeling scheme with $O(k^{6})$ approximation and label size $f^{3}n^{1/k}\log^{O(k)} n$. This is the first nontrivial distance labeling scheme for vertex failures. Our techniques build on recent tools related to length-constrained vertex expanders and also introduce a new expander-based shortcut sparsification. The latter also leads to a deterministic vertex-failure connectivity labeling scheme of size $\tilde{O}(f^{2})$.

Authors: Yaowei Long

We present new algorithms for the vertex-failure distance oracles and labeling schemes problems in undirected weighted graphs. A vertex-failure distance oracle is a data structure that, given two vertices $x$ and $y$ and a failed vertex set $F$ of size at most $f$, returns an approximation to the distance between $x$ and $y$ in $G \setminus F$. In the labeling-scheme setting, the data structure needs to be stored distributively as labels on the vertices, and each query $(x,y,F)$ must be answered by accessing only the labels of the vertices in $F \cup \{x,y\}$. For any $f\geq 1$ and $k \ge 1$, we obtain a vertex-failure distance oracle with $O(k^{6})$ approximation, space $\tilde{O}(f^{2}n^{1+1/k})$, query time $\tilde{O}(f^{5}n^{1/k})$, and polynomial preprocessing time. In particular, this is the first time-efficient oracle for multiple vertex failures with space close to linear, as well as the first constant-approximation oracle with polynomial space when tolerating $Ω(\log n)$ vertex failures. The previous results, due to [Duan-Gu-Ren, SODA'21], gave two alternatives: for any constant $c \ge 1$ and $ε>0$, one oracle has $\mathrm{poly}(\log n,f)$ approximation, space $n^{2+1/c}\mathrm{poly}(\log n,f)$, and query time $\mathrm{poly}(\log n,f^{c})$, while the other has $(1+ε)$ approximation, space $n^{2+1/c}(\log n/ε)^{O(f)}$, and query time $\mathrm{poly}(\log n,f^{c},1/ε)$. We also obtain a vertex-failure distance labeling scheme with $O(k^{6})$ approximation and label size $f^{3}n^{1/k}\log^{O(k)} n$. This is the first nontrivial distance labeling scheme for vertex failures. Our techniques build on recent tools related to length-constrained vertex expanders and also introduce a new expander-based shortcut sparsification. The latter also leads to a deterministic vertex-failure connectivity labeling scheme of size $\tilde{O}(f^{2})$.

Convergence of Kikuchi matrices to $Γ$-independent and $q$-Gaussian limits

from arXiv: Data Structures and Algorithms

Authors: Afonso S. Bandeira, Dmitriy Kunisky, Petar Nizić-Nikolac, Lucas Pesenti, Robert Wang

Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $Γ$-independent semicircular elements. This characterizes their limiting spectral distribution and yields improved bounds on their spectral norm, a key quantity in the analysis of algorithms for Tensor PCA. Finally, we show that, in an appropriate double limit, independent Kikuchi matrices converge to the $q$-Gaussian system, another central object in noncommutative probability.

Authors: Afonso S. Bandeira, Dmitriy Kunisky, Petar Nizić-Nikolac, Lucas Pesenti, Robert Wang

Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $Γ$-independent semicircular elements. This characterizes their limiting spectral distribution and yields improved bounds on their spectral norm, a key quantity in the analysis of algorithms for Tensor PCA. Finally, we show that, in an appropriate double limit, independent Kikuchi matrices converge to the $q$-Gaussian system, another central object in noncommutative probability.

Beating One Half for Online Bipartite Matching with Reusable Resources

from arXiv: Data Structures and Algorithms

Authors: Xiaohui Bei, Zhihao Gavin Tang, Wenhao Wu

We study online bipartite matching with unit-inventory reusable resources, where requests arrive in an adversarially fixed order, and each use of a resource makes it unavailable for an independent duration drawn from a resource-dependent distribution. The benchmark knows all requests in advance but cannot observe a duration before choosing the corresponding use. The classical Ranking algorithm of Karp, Vazirani, and Vazirani (STOC 1990) fixes a uniformly random priority order of the resources and matches each arriving request to its highest-priority available neighbor. It achieves the optimal competitive ratio $1-1/e$ for unweighted nonreusable resources, but whether it beats $1/2$ for reusable resources has remained open. We prove that, for unweighted resources with resource-dependent stochastic durations, Ranking achieves a competitive ratio of $(5-2\sqrt3)/3\approx0.511966$. We also give a black-box reduction from unweighted Ranking to resource-weighted matching: any unweighted competitive ratio $α>1/2$ yields a weighted ratio strictly above $1/2$. With independent sampling access to the duration distributions, the reduction gives a weighted ratio of $0.500034$. These results resolve two questions left open by Delong et al. (MOR 2024): whether Ranking beats $1/2$, and whether one can beat $1/2$ under stochastic durations. We analyze Ranking resource by resource, rather than request by request. For deterministic durations, this gives a reduction to random-order greedy for a coverage function. We then extend the analysis to stochastic durations by comparing the residual schedules of Ranking and a greedy algorithm, and apply a finer analysis of the random ranks to obtain the stated $0.511$ bound. For the weighted reduction, we apply Ranking within groups of similar weights and uses weighted greedy to control the loss between groups.

Authors: Xiaohui Bei, Zhihao Gavin Tang, Wenhao Wu

We study online bipartite matching with unit-inventory reusable resources, where requests arrive in an adversarially fixed order, and each use of a resource makes it unavailable for an independent duration drawn from a resource-dependent distribution. The benchmark knows all requests in advance but cannot observe a duration before choosing the corresponding use. The classical Ranking algorithm of Karp, Vazirani, and Vazirani (STOC 1990) fixes a uniformly random priority order of the resources and matches each arriving request to its highest-priority available neighbor. It achieves the optimal competitive ratio $1-1/e$ for unweighted nonreusable resources, but whether it beats $1/2$ for reusable resources has remained open. We prove that, for unweighted resources with resource-dependent stochastic durations, Ranking achieves a competitive ratio of $(5-2\sqrt3)/3\approx0.511966$. We also give a black-box reduction from unweighted Ranking to resource-weighted matching: any unweighted competitive ratio $α>1/2$ yields a weighted ratio strictly above $1/2$. With independent sampling access to the duration distributions, the reduction gives a weighted ratio of $0.500034$. These results resolve two questions left open by Delong et al. (MOR 2024): whether Ranking beats $1/2$, and whether one can beat $1/2$ under stochastic durations. We analyze Ranking resource by resource, rather than request by request. For deterministic durations, this gives a reduction to random-order greedy for a coverage function. We then extend the analysis to stochastic durations by comparing the residual schedules of Ranking and a greedy algorithm, and apply a finer analysis of the random ranks to obtain the stated $0.511$ bound. For the weighted reduction, we apply Ranking within groups of similar weights and uses weighted greedy to control the loss between groups.

Achieving Optimal Redundancy for Small Dynamic Rank/Select Dictionaries

from arXiv: Data Structures and Algorithms

Authors: Gabriel Marques Domingues

In this paper, we study the number of bits required to construct a dynamic dictionary with optimal time for $\texttt{rank}/\texttt{select}$ operations. Using the standard (multiplication) Word-RAM model with $w$-bit words, we construct a data-structure for a dynamic $\texttt{rank}/\texttt{select}$ dictionary for a set $S\subseteq\{0,1,\cdots,u-1\}$ of $n$ elements that, given a parameter $1\leq k\leq \log^*w$, uses $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log^{(k)}w)\text{ bits}$$ taking optimal $\mathcal{O}(k+\log_w n)$ time (worst-case) for all operations. We show optimality for $n=w^{\mathcal{O}(1)}$ by extending the lower bound of Li, Liang, Yu, and Zhou [FOCS 2023] to super-polynomial universes: any dynamic dictionary for $n\leq \sqrt{u}$ elements that uses $\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log^{(k)}n)$ bits requires $Ω(k)$ time for operations. Lastly, we extend the data-structure to a dynamic fully indexable dictionary (that also supports $\texttt{rank}/\texttt{select}$ on the complement of $S$).

Authors: Gabriel Marques Domingues

In this paper, we study the number of bits required to construct a dynamic dictionary with optimal time for $\texttt{rank}/\texttt{select}$ operations. Using the standard (multiplication) Word-RAM model with $w$-bit words, we construct a data-structure for a dynamic $\texttt{rank}/\texttt{select}$ dictionary for a set $S\subseteq\{0,1,\cdots,u-1\}$ of $n$ elements that, given a parameter $1\leq k\leq \log^*w$, uses $$\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log^{(k)}w)\text{ bits}$$ taking optimal $\mathcal{O}(k+\log_w n)$ time (worst-case) for all operations. We show optimality for $n=w^{\mathcal{O}(1)}$ by extending the lower bound of Li, Liang, Yu, and Zhou [FOCS 2023] to super-polynomial universes: any dynamic dictionary for $n\leq \sqrt{u}$ elements that uses $\operatorname{lg}\binom{u}{n}+\mathcal{O}(n\log^{(k)}n)$ bits requires $Ω(k)$ time for operations. Lastly, we extend the data-structure to a dynamic fully indexable dictionary (that also supports $\texttt{rank}/\texttt{select}$ on the complement of $S$).

Near-optimal quantum query lower bounds on bipartiteness and expansion testing in the bounded-degree graph model

from arXiv: Data Structures and Algorithms

Authors: Chandrima Kayal, Sayantan Sen, Dániel Szabó

In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that $\widetildeΘ(\sqrt{N})$ queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where $N$ denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform $\widetilde{O}(N^{1/3})$ queries, showing a polynomial speedup. They also proved that $\widetildeΩ(N^{1/4})$ queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight $\widetildeΩ(N^{1/3})$ quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.

Authors: Chandrima Kayal, Sayantan Sen, Dániel Szabó

In this work, we study bipartiteness and expansion testing, two canonical problems in graph property testing in the bounded-degree model through the lens of quantum query complexity. In the classical setting, it is known that $\widetildeΘ(\sqrt{N})$ queries are necessary and sufficient for both these testing problems (Goldreich and Ron, 1999, 2000 & 2002), where $N$ denotes the number of vertices of the input graph. Due to their significance, (Ambainis, Childs, and Liu, 2011) initiated the study of these problems in the quantum setting and designed quantum algorithms for bipartiteness and expansion testing that perform $\widetilde{O}(N^{1/3})$ queries, showing a polynomial speedup. They also proved that $\widetildeΩ(N^{1/4})$ queries are necessary for expansion testing, but the possibility of an exponential quantum advantage for bipartiteness testing remained open. Despite significant effort, there has been no improvement in these results in the last decade and a half. In this work, we prove essentially tight $\widetildeΩ(N^{1/3})$ quantum query lower bounds for both bipartiteness and expansion testing, thereby completely characterizing the quantum query complexity of these problems up to polylogarithmic factors. While our proofs use the polynomial method similarly to Ambainis, Childs, and Liu, we use intermediate problems that we relate to the main problems via reductions, and perform a more precise analysis of the resulting polynomials, leading to the near-optimal lower bounds.

Safe Hypergraph Contraction via Capacity-Aware Repair Certificates

from arXiv: Data Structures and Algorithms

Authors: Yu Deng, Xinyi Yang, Keren Zhu

Multilevel partitioners shrink circuit hypergraphs through vertex contractions, yet a contraction that satisfies block capacity can still eliminate every optimal balanced bipartition. We develop certified safe coarsening (CSC) to identify contractions that preserve an optimum without computing that optimum. CSC certifies a repair for any feasible partition that splits a candidate group: the repair must respect the fixed block capacities and must not increase the cut-net objective. Its bounds exclude hyperedges that capacity constraints force to be cut. A pair certificate checks individual merges, while a directed minimum-cut test certifies groups whose savings emerge only when vertices move together. We prove that certified disjoint batches and successive rounds with recertification retain at least one globally optimal feasible partition for hypergraphs with positive integer vertex and net weights. Experiments on exactly solvable instances confirm optimum preservation for every tested configuration; integration with KaHyPar lowers the sum of per-instance best cuts on circuit benchmarks, with additional runtime.

Authors: Yu Deng, Xinyi Yang, Keren Zhu

Multilevel partitioners shrink circuit hypergraphs through vertex contractions, yet a contraction that satisfies block capacity can still eliminate every optimal balanced bipartition. We develop certified safe coarsening (CSC) to identify contractions that preserve an optimum without computing that optimum. CSC certifies a repair for any feasible partition that splits a candidate group: the repair must respect the fixed block capacities and must not increase the cut-net objective. Its bounds exclude hyperedges that capacity constraints force to be cut. A pair certificate checks individual merges, while a directed minimum-cut test certifies groups whose savings emerge only when vertices move together. We prove that certified disjoint batches and successive rounds with recertification retain at least one globally optimal feasible partition for hypergraphs with positive integer vertex and net weights. Experiments on exactly solvable instances confirm optimum preservation for every tested configuration; integration with KaHyPar lowers the sum of per-instance best cuts on circuit benchmarks, with additional runtime.

Exact Locality Gaps for Matchable Semi-Matchings

from arXiv: Data Structures and Algorithms

Authors: Marek Gałązka, Hanna Wdowicka

An assignment of tasks to servers can resist every small improvement and still make tasks wait longer than necessary. We determine exactly how inefficient such an assignment can be when each task requires one unit of service and the eligibility constraints permit all tasks to use distinct servers. For every move size $r$ and maximum current server load $K$, we give a closed formula for the worst ratio between locally optimal and globally optimal total completion time. Local optimality here allows every feasible reassignment changing at most $r$ tasks. Every finite-cap bound is attained on a tree where each task has at most two eligible servers. Thus the worst behavior already occurs under simple eligibility constraints. At load cap two, the exact ratio is $1+1/(r+2)$, attained on a path with $r+2$ tasks. Without a load cap, the worst-case supremum is $3/2$ for single-task moves and approximately $1.294503159$ for two-task moves; its excess above one is $1/(r+2)+O(2^{-r}/r)$ as $r$ grows. The proof uses an explicit rational potential on a comparison graph and matching extremal constructions. These results give sharp guarantees for bounded-size local search on matchable semi-matchings, including exact guarantees under degree bounds.

Authors: Marek Gałązka, Hanna Wdowicka

An assignment of tasks to servers can resist every small improvement and still make tasks wait longer than necessary. We determine exactly how inefficient such an assignment can be when each task requires one unit of service and the eligibility constraints permit all tasks to use distinct servers. For every move size $r$ and maximum current server load $K$, we give a closed formula for the worst ratio between locally optimal and globally optimal total completion time. Local optimality here allows every feasible reassignment changing at most $r$ tasks. Every finite-cap bound is attained on a tree where each task has at most two eligible servers. Thus the worst behavior already occurs under simple eligibility constraints. At load cap two, the exact ratio is $1+1/(r+2)$, attained on a path with $r+2$ tasks. Without a load cap, the worst-case supremum is $3/2$ for single-task moves and approximately $1.294503159$ for two-task moves; its excess above one is $1/(r+2)+O(2^{-r}/r)$ as $r$ grows. The proof uses an explicit rational potential on a comparison graph and matching extremal constructions. These results give sharp guarantees for bounded-size local search on matchable semi-matchings, including exact guarantees under degree bounds.

Robust Non-Clairvoyant Scheduling with Classification Models

from arXiv: Data Structures and Algorithms

Authors: Anthony Dugois, Vincent Fagnon, Giorgio Lucarelli

We study the classical single-machine scheduling problem of minimizing the sum of completion times of jobs in a non-clairvoyant setting, where the processing time of each job remains unknown until its completion. This is a hard problem for which no constant competitive algorithm is possible. Inspired by robust optimization and learning-augmented algorithms, we introduce a novel robustness framework that leverages structural information provided by a classification model to overcome this limitation. Specifically, we assume that jobs are partitioned into classes and we have access to the confusion matrix of the classifier, whose entry $(k,\ell)$ indicates the number of jobs predicted to belong to class~$k$ but that actually belong to class~$\ell$. In this manner, we are able to characterize uncertainty as a set of permutations within each predicted class, rather than as a collection of discrete numerical scenarios, avoiding the computational difficulty of classical robust metrics, such as Min-Max and Min-Max Regret. In addition to these worst-case metrics, we also consider the expected objective over all scenarios. We first propose an optimal non-adaptive strategy that is oblivious with respect to all three robust criteria. We then investigate adaptive and randomized algorithms, showing that they can outperform the optimal non-adaptive strategy when the matrix exhibits particular structural properties.

Authors: Anthony Dugois, Vincent Fagnon, Giorgio Lucarelli

We study the classical single-machine scheduling problem of minimizing the sum of completion times of jobs in a non-clairvoyant setting, where the processing time of each job remains unknown until its completion. This is a hard problem for which no constant competitive algorithm is possible. Inspired by robust optimization and learning-augmented algorithms, we introduce a novel robustness framework that leverages structural information provided by a classification model to overcome this limitation. Specifically, we assume that jobs are partitioned into classes and we have access to the confusion matrix of the classifier, whose entry $(k,\ell)$ indicates the number of jobs predicted to belong to class~$k$ but that actually belong to class~$\ell$. In this manner, we are able to characterize uncertainty as a set of permutations within each predicted class, rather than as a collection of discrete numerical scenarios, avoiding the computational difficulty of classical robust metrics, such as Min-Max and Min-Max Regret. In addition to these worst-case metrics, we also consider the expected objective over all scenarios. We first propose an optimal non-adaptive strategy that is oblivious with respect to all three robust criteria. We then investigate adaptive and randomized algorithms, showing that they can outperform the optimal non-adaptive strategy when the matrix exhibits particular structural properties.

Factor Three Approximation for Edit Distance

from arXiv: Data Structures and Algorithms

Authors: Egor Gorbachev

We give randomized algorithms for $3$-approximate edit distance in $\widetilde{\mathcal{O}}(N^{11/6})$ time for unweighted edit distance and in $\widetilde{\mathcal{O}}(N^{40/21})$ time for arbitrary metric edit weights, where $N$ is the total input length. For non-metric costs, we prove an unconditional $Ω(N^2)$ oracle-query lower bound for every approximation factor depending only on $N$, even for symmetric weights or weights satisfying the triangle inequality (but not both). Under the Orthogonal Vectors Hypothesis, we show a similar result for constant-size alphabets. This holds even for symmetric weights over a size-$3$ alphabet or triangle-inequality weights over a size-$2$ alphabet. In contrast, for symmetric weights over a binary alphabet we show an $\widetilde{\mathcal{O}}(N^{40/21})$-time $3$-approximation algorithm.

Authors: Egor Gorbachev

We give randomized algorithms for $3$-approximate edit distance in $\widetilde{\mathcal{O}}(N^{11/6})$ time for unweighted edit distance and in $\widetilde{\mathcal{O}}(N^{40/21})$ time for arbitrary metric edit weights, where $N$ is the total input length. For non-metric costs, we prove an unconditional $Ω(N^2)$ oracle-query lower bound for every approximation factor depending only on $N$, even for symmetric weights or weights satisfying the triangle inequality (but not both). Under the Orthogonal Vectors Hypothesis, we show a similar result for constant-size alphabets. This holds even for symmetric weights over a size-$3$ alphabet or triangle-inequality weights over a size-$2$ alphabet. In contrast, for symmetric weights over a binary alphabet we show an $\widetilde{\mathcal{O}}(N^{40/21})$-time $3$-approximation algorithm.

When Is Deletion Ordering Tractable? From Update Dynamics to Permutation Structure

from arXiv: Data Structures and Algorithms

Authors: Xinyu Wang, Ziyu Zhao, Yixuan He, Xiaowen Chang Alex Smola

Given a fixed set of pending deletion requests, retraining from scratch after each request is prohibitive, so a prescribed request-wise policy processes them sequentially. The resulting terminal model can depend on their order. Rather than prescribing an ordering rule, we study the permutation objective induced by the fixed policy and ask when it admits simpler structure. We identify two independent reductions: position additivity represents the objective by request--position costs, reducing optimization to assignment and, with a shared positional profile, sorting; suffix localization removes dependence on the distant prefix while retaining interactions among the surviving requests. Under shared affine updates, we characterize the quadratic interactions that obstruct additivity, prove the reductions' independence, and show that suffix-conditioned assignment improves the approximation rate from O(p^L) toO(p^(2L)). Experiments recover both structures in executed objectives. A controlled damped-Newton sweep shows that stronger contraction shifts the objective toward shorter, more suffix-specific dependence, while two full-network policies exhibit distinct positional and within-suffix structure. Structures identified from compact execution sets also predict unseen orders. These results frame deletion ordering as identifying the computational structure induced by the executed updates.

Authors: Xinyu Wang, Ziyu Zhao, Yixuan He, Xiaowen Chang Alex Smola

Given a fixed set of pending deletion requests, retraining from scratch after each request is prohibitive, so a prescribed request-wise policy processes them sequentially. The resulting terminal model can depend on their order. Rather than prescribing an ordering rule, we study the permutation objective induced by the fixed policy and ask when it admits simpler structure. We identify two independent reductions: position additivity represents the objective by request--position costs, reducing optimization to assignment and, with a shared positional profile, sorting; suffix localization removes dependence on the distant prefix while retaining interactions among the surviving requests. Under shared affine updates, we characterize the quadratic interactions that obstruct additivity, prove the reductions' independence, and show that suffix-conditioned assignment improves the approximation rate from O(p^L) toO(p^(2L)). Experiments recover both structures in executed objectives. A controlled damped-Newton sweep shows that stronger contraction shifts the objective toward shorter, more suffix-specific dependence, while two full-network policies exhibit distinct positional and within-suffix structure. Structures identified from compact execution sets also predict unseen orders. These results frame deletion ordering as identifying the computational structure induced by the executed updates.

Settling the Pass Complexity of Streaming Set Cover

from arXiv: Data Structures and Algorithms

Authors: Sepehr Assadi, Janani Sundaresan

In the streaming set cover problem, $m$ sets from a universe of size $n$ are arriving one by one in a stream, and the algorithm is allowed to process the stream using one or a few passes and a space of $o(mn)$, which is sublinear in the input size. The goal is to determine the minimal (or approximately minimal) number of sets that cover the universe at the end of the last pass. This problem has been studied extensively over the years with rapid progress that led to several $O(\log{n})$-approximation algorithms in $\tilde{O}(mn^{1/p})$ space and $O(p)$ passes. However, progress on this front has largely stagnated over the past decade, despite the absence of any lower bounds that rule out even an $O(\log{n})$-approximation in $O(m)$ space and just two passes. We provide a simple explanation for this lack of progress by establishing an optimal three-way space-pass-approximation tradeoff for this problem: any $α$-approximation algorithm for streaming set cover requires $$ \widetildeΩ\Big(\frac{m}α \cdot \big(\frac{n}α\big)^{1/p}\Big) $$ space in $p$ passes whenever $α\ll n^{1/(p+1)}$. In light of prior work, this result is optimal up to constant factors in $p$ and logarithmic factors in $n,m$ for any $α\geq p$. Our bound is optimal with respect to the range of $α$ also, and fully settles the complexity of this fundamental problem in the streaming model. The proof of this result is (surprisingly) simple and non-technical and relies on a randomized reduction from a variant of the standard pointer chasing problem in communication complexity, using elementary properties of random sets.

Authors: Sepehr Assadi, Janani Sundaresan

In the streaming set cover problem, $m$ sets from a universe of size $n$ are arriving one by one in a stream, and the algorithm is allowed to process the stream using one or a few passes and a space of $o(mn)$, which is sublinear in the input size. The goal is to determine the minimal (or approximately minimal) number of sets that cover the universe at the end of the last pass. This problem has been studied extensively over the years with rapid progress that led to several $O(\log{n})$-approximation algorithms in $\tilde{O}(mn^{1/p})$ space and $O(p)$ passes. However, progress on this front has largely stagnated over the past decade, despite the absence of any lower bounds that rule out even an $O(\log{n})$-approximation in $O(m)$ space and just two passes. We provide a simple explanation for this lack of progress by establishing an optimal three-way space-pass-approximation tradeoff for this problem: any $α$-approximation algorithm for streaming set cover requires $$ \widetildeΩ\Big(\frac{m}α \cdot \big(\frac{n}α\big)^{1/p}\Big) $$ space in $p$ passes whenever $α\ll n^{1/(p+1)}$. In light of prior work, this result is optimal up to constant factors in $p$ and logarithmic factors in $n,m$ for any $α\geq p$. Our bound is optimal with respect to the range of $α$ also, and fully settles the complexity of this fundamental problem in the streaming model. The proof of this result is (surprisingly) simple and non-technical and relies on a randomized reduction from a variant of the standard pointer chasing problem in communication complexity, using elementary properties of random sets.

Best of Two Worlds: Combining High and Low Resolution to Compute Viewsheds on terrains

from arXiv: Data Structures and Algorithms

Authors: Laura Toma

The viewshed of a point $v$ on a grid terrain $T$, viewshed$_T(v)$, is defined as the set of grid points in $T$ that are visible from $v$. We describe a novel algorithm for computing viewshed$_T(v)$ using a multi-resolution approach: Given a parameter $k >1$ that represents the block size, we create a grid $T'$ which is a lower-resolution version of $T$, such that each point in $T'$ corresponds to a block of $\lceil \sqrt k \rceil $-by-$\lceil \sqrt k \rceil$ points in $T$. The key of our approach is using $T'$ to speed up the computation of viewshed$_T(v)$ while not introducing approximation. We compute viewshed$_T(v)$ in two steps: First we compute the viewshed of $v$ on $T'$, while maintaining the invariant that any block in $T'$ that is labeled as invisible may not contain any visible points. Thus, the first step's role is to use $T'$ to filter out blocks in $T$ that are guaranteed to be invisible. The second step considers the blocks that were labeled as visible in $T'$ and computes the visibility of their points with full accuracy using the data in $T$. Overall the algorithm runs in $O(n + \frac nk \lg \frac nk + k \lg k + l \cdot \lg n)$, where $l$ is the total size of visible blocks in $T'$. When $k = Ω(1)$ and $l = o(n) $, the running time of our algorithm improves on the previous best bound of $O(n \lg n)$. Our experimental results show the performance of the new algorithm in practice and a speedup of more than an order of magnitude compared to previous algorithms.

Authors: Laura Toma

The viewshed of a point $v$ on a grid terrain $T$, viewshed$_T(v)$, is defined as the set of grid points in $T$ that are visible from $v$. We describe a novel algorithm for computing viewshed$_T(v)$ using a multi-resolution approach: Given a parameter $k >1$ that represents the block size, we create a grid $T'$ which is a lower-resolution version of $T$, such that each point in $T'$ corresponds to a block of $\lceil \sqrt k \rceil $-by-$\lceil \sqrt k \rceil$ points in $T$. The key of our approach is using $T'$ to speed up the computation of viewshed$_T(v)$ while not introducing approximation. We compute viewshed$_T(v)$ in two steps: First we compute the viewshed of $v$ on $T'$, while maintaining the invariant that any block in $T'$ that is labeled as invisible may not contain any visible points. Thus, the first step's role is to use $T'$ to filter out blocks in $T$ that are guaranteed to be invisible. The second step considers the blocks that were labeled as visible in $T'$ and computes the visibility of their points with full accuracy using the data in $T$. Overall the algorithm runs in $O(n + \frac nk \lg \frac nk + k \lg k + l \cdot \lg n)$, where $l$ is the total size of visible blocks in $T'$. When $k = Ω(1)$ and $l = o(n) $, the running time of our algorithm improves on the previous best bound of $O(n \lg n)$. Our experimental results show the performance of the new algorithm in practice and a speedup of more than an order of magnitude compared to previous algorithms.

Beyond odd characteristic: Faster isomorphism testing of 2-groups of Frattini class 2

from arXiv: Data Structures and Algorithms

Authors: Joshua A. Grochow, Gábor Ivanyos, Youming Qiao, Xiaorui Sun

The finite group isomorphism problem asks whether two finite groups of order $N$ are isomorphic. The first algorithm, attributed to Tarjan (see Miller, STOC '78), runs in time $N^{\log N + O(1)}$. Despite intensive study, the current best known algorithm has a running time of $N^{(1 / 4 + o(1))\log N}$ (Rosenbaum, '13). $p$-groups of class $2$ have been recognized as the major bottleneck for faster group isomorphism. Recent progress has led to $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ where $p$ is odd (Sun, STOC '23; Ivanyos--Mendoza--Qiao--Sun--Zhang, FOCS '24; Grochow--Qiao--Stange--Sun, STOC '25). However, the case of $p=2$, which represents the majority of $p$-groups of class 2 assuming a well-known conjecture in group enumeration, remained elusive, with essentially no progress until now. In this paper, we present an algorithm for testing the isomorphism of two 2-groups of Frattini class 2 of order $N$ in time $N^{O((\log N)^{1/2})}$. To our knowledge, this is the first $N^{o(\log N)}$-time isomorphism algorithm for a class of $2$-groups that constitutes logarithmically almost all $2$-groups, in the sense that $\lim_{N \to \infty} \frac{\log(\text{\# 2-groups of Frattini class 2 and order } \leq N)}{\log(\text{\# 2-groups of order} \leq N)} = 1$. As our main tool, we present the first non-trivial algorithms for the quadratic form space/tuple isometry problems over $\mathbb{F}_2$. These algorithms rely on combinations of combinatorial and algebraic ideas, including finite matrix group algorithms developed by Luks (FOCS '92). As far as we know, this is the first time that matrix group algorithms are used to make progress on the worst-case complexity of $p$-group isomorphism.

Authors: Joshua A. Grochow, Gábor Ivanyos, Youming Qiao, Xiaorui Sun

The finite group isomorphism problem asks whether two finite groups of order $N$ are isomorphic. The first algorithm, attributed to Tarjan (see Miller, STOC '78), runs in time $N^{\log N + O(1)}$. Despite intensive study, the current best known algorithm has a running time of $N^{(1 / 4 + o(1))\log N}$ (Rosenbaum, '13). $p$-groups of class $2$ have been recognized as the major bottleneck for faster group isomorphism. Recent progress has led to $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ where $p$ is odd (Sun, STOC '23; Ivanyos--Mendoza--Qiao--Sun--Zhang, FOCS '24; Grochow--Qiao--Stange--Sun, STOC '25). However, the case of $p=2$, which represents the majority of $p$-groups of class 2 assuming a well-known conjecture in group enumeration, remained elusive, with essentially no progress until now. In this paper, we present an algorithm for testing the isomorphism of two 2-groups of Frattini class 2 of order $N$ in time $N^{O((\log N)^{1/2})}$. To our knowledge, this is the first $N^{o(\log N)}$-time isomorphism algorithm for a class of $2$-groups that constitutes logarithmically almost all $2$-groups, in the sense that $\lim_{N \to \infty} \frac{\log(\text{\# 2-groups of Frattini class 2 and order } \leq N)}{\log(\text{\# 2-groups of order} \leq N)} = 1$. As our main tool, we present the first non-trivial algorithms for the quadratic form space/tuple isometry problems over $\mathbb{F}_2$. These algorithms rely on combinations of combinatorial and algebraic ideas, including finite matrix group algorithms developed by Luks (FOCS '92). As far as we know, this is the first time that matrix group algorithms are used to make progress on the worst-case complexity of $p$-group isomorphism.

Sparsification Framework for Directed Densest Subgraph

from arXiv: Data Structures and Algorithms

Authors: Slobodan Mitrović, Theodore Pan

We develop a new approach for computing approximate directed densest subgraphs (DDS). Our main result is a sparsification procedure that reduces a directed graph $G$ on $n$ vertices to a graph with $n \cdot \text{poly} \log n$ edges while preserving enough structure to recover an approximate DDS of $G$. Instantiating this framework in several memory-constrained settings, we obtain the following improvements over the state of the art: In semi-streaming, we obtain a single-pass algorithm that computes a $(1-\varepsilon)$-approximate DDS. Previously, the only semi-streaming algorithm that computed a constant approximation of DDS was by Bahmani, Kumar, and Vassilvitskii (2012), providing a $0.5-\varepsilon$ approximation in $O(\log n)$ passes. Hence, our work completely closes the approximation gap between undirected and directed DS in the semi-streaming setting, matching the $(1-\varepsilon)$-approximate undirected DS algorithm by Esfandiari, Hajiaghayi, and Woodruff (2016). In the near-linear-memory MPC regime, we obtain an $O(1)$-round algorithm for $(1-\varepsilon)$-approximate DDS, improving over the $O(\sqrt{\log n})$-round $(0.5-\varepsilon)$-approximation algorithm of Mitrović and Pan (2024). In the sublinear-time setting, we obtain an algorithm using $\tilde{O}(n)$ time, space, and oracle queries to compute a $(1-\varepsilon)$-approximate DDS, improving over the $\tilde{O}(n^{1.5})$ time, space, and query algorithm of Esfandiari, Hajiaghayi, and Woodruff (2016).

Authors: Slobodan Mitrović, Theodore Pan

We develop a new approach for computing approximate directed densest subgraphs (DDS). Our main result is a sparsification procedure that reduces a directed graph $G$ on $n$ vertices to a graph with $n \cdot \text{poly} \log n$ edges while preserving enough structure to recover an approximate DDS of $G$. Instantiating this framework in several memory-constrained settings, we obtain the following improvements over the state of the art: In semi-streaming, we obtain a single-pass algorithm that computes a $(1-\varepsilon)$-approximate DDS. Previously, the only semi-streaming algorithm that computed a constant approximation of DDS was by Bahmani, Kumar, and Vassilvitskii (2012), providing a $0.5-\varepsilon$ approximation in $O(\log n)$ passes. Hence, our work completely closes the approximation gap between undirected and directed DS in the semi-streaming setting, matching the $(1-\varepsilon)$-approximate undirected DS algorithm by Esfandiari, Hajiaghayi, and Woodruff (2016). In the near-linear-memory MPC regime, we obtain an $O(1)$-round algorithm for $(1-\varepsilon)$-approximate DDS, improving over the $O(\sqrt{\log n})$-round $(0.5-\varepsilon)$-approximation algorithm of Mitrović and Pan (2024). In the sublinear-time setting, we obtain an algorithm using $\tilde{O}(n)$ time, space, and oracle queries to compute a $(1-\varepsilon)$-approximate DDS, improving over the $\tilde{O}(n^{1.5})$ time, space, and query algorithm of Esfandiari, Hajiaghayi, and Woodruff (2016).

The Power of Two-Choice Linear Probing

from arXiv: Data Structures and Algorithms

Authors: Amir Azarmehr, Michael A. Bender, William Kuszmaul, Rose Silver

This paper considers the following basic question: If an (ordered) linear-probing hash table is allowed \emph{two} hash functions, instead of one, how does this change the expected insertion and query time, as a function of the load factor $1 - ε$? We prove that the \emph{greedy two-choice insertion strategy} achieves polynomially better bounds than the single choice algorithm, but that one can even do \emph{much better} by using more sophisticated non-greedy strategies. Specifically, we show that there is an insertion strategy that does not evict elements (once an element is inserted, its hash choice is fixed) and that achieves expected query time $O(\log ε^{-1})$ with expected insertion time $O(ε^{-1})$. We then further show that, if one is allowed to evict elements (i.e., to change over time which hash function a given element uses), then it is possible to achieve expected query time $O(1)$ with expected insertion time $O(ε^{-1/2})$. This final result achieves an expected query time of $O(1)$ even when the hash table is filled to $100\%$ full. Combined, the results reveal that there is a surprisingly strong ``power of two choices'' phenomenon for linear-probing hash tables, allowing for a two-choice hash table to achieve significantly better bounds than what might at first seem to be possible.

Authors: Amir Azarmehr, Michael A. Bender, William Kuszmaul, Rose Silver

This paper considers the following basic question: If an (ordered) linear-probing hash table is allowed \emph{two} hash functions, instead of one, how does this change the expected insertion and query time, as a function of the load factor $1 - ε$? We prove that the \emph{greedy two-choice insertion strategy} achieves polynomially better bounds than the single choice algorithm, but that one can even do \emph{much better} by using more sophisticated non-greedy strategies. Specifically, we show that there is an insertion strategy that does not evict elements (once an element is inserted, its hash choice is fixed) and that achieves expected query time $O(\log ε^{-1})$ with expected insertion time $O(ε^{-1})$. We then further show that, if one is allowed to evict elements (i.e., to change over time which hash function a given element uses), then it is possible to achieve expected query time $O(1)$ with expected insertion time $O(ε^{-1/2})$. This final result achieves an expected query time of $O(1)$ even when the hash table is filled to $100\%$ full. Combined, the results reveal that there is a surprisingly strong ``power of two choices'' phenomenon for linear-probing hash tables, allowing for a two-choice hash table to achieve significantly better bounds than what might at first seem to be possible.

Query-efficient winner prediction in district-based elections

from arXiv: Data Structures and Algorithms

Authors: Koustav De, Debajyoti Kar, Swagato Sanyal

In a district-based election, N voters are partitioned into k districts, and each voter votes for one of m candidates. Each district elects a winner using the plurality rule (i.e. the candidate getting the largest number of votes is declared the winner, breaking ties as per some fixed rule), and the overall winner is determined by applying plurality to the district winners; we assume that there is a unique winner amongst the district winners. The margin of victory of such an election is the minimum number of votes that must be altered so that the current winner ceases to be the unique district winner. We study the problem of predicting the winner of a district-based election in the query complexity model, where one has query access to individual votes. The objective is to minimise the number of queries. This setting captures exit polling, where queries correspond to interviewing voters, and is closely related to problems in query complexity and property testing. Assuming that the margin of victory of the election is at least eps N, Dey, Kar and Sanyal (AAMAS 2023) gave algorithms for the case of two candidates with error probability del and query complexity tilde{O}(1/eps^6 log^2 1/del), which improves to tilde{O}(1/eps^4 log^2 1/del) under the additional assumption that district populations are balanced. Our main result is an adaptive randomised algorithm that, for an arbitrary district-based election and any error parameter del, with probability at least 1-del, predicts the winner correctly using tilde{O}(1/eps^2 log m/del log 1/del) queries. In particular, we improve the bounds of Dey et al. for arbitrary district populations and extend their results to any number of candidates. Furthermore, for constantly many candidates, our algorithm nearly matches a lower bound of Omega(1/eps^2 log 1/del) on the query complexity that holds even for two candidates and a single district.

Authors: Koustav De, Debajyoti Kar, Swagato Sanyal

In a district-based election, N voters are partitioned into k districts, and each voter votes for one of m candidates. Each district elects a winner using the plurality rule (i.e. the candidate getting the largest number of votes is declared the winner, breaking ties as per some fixed rule), and the overall winner is determined by applying plurality to the district winners; we assume that there is a unique winner amongst the district winners. The margin of victory of such an election is the minimum number of votes that must be altered so that the current winner ceases to be the unique district winner. We study the problem of predicting the winner of a district-based election in the query complexity model, where one has query access to individual votes. The objective is to minimise the number of queries. This setting captures exit polling, where queries correspond to interviewing voters, and is closely related to problems in query complexity and property testing. Assuming that the margin of victory of the election is at least eps N, Dey, Kar and Sanyal (AAMAS 2023) gave algorithms for the case of two candidates with error probability del and query complexity tilde{O}(1/eps^6 log^2 1/del), which improves to tilde{O}(1/eps^4 log^2 1/del) under the additional assumption that district populations are balanced. Our main result is an adaptive randomised algorithm that, for an arbitrary district-based election and any error parameter del, with probability at least 1-del, predicts the winner correctly using tilde{O}(1/eps^2 log m/del log 1/del) queries. In particular, we improve the bounds of Dey et al. for arbitrary district populations and extend their results to any number of candidates. Furthermore, for constantly many candidates, our algorithm nearly matches a lower bound of Omega(1/eps^2 log 1/del) on the query complexity that holds even for two candidates and a single district.

Faster Algorithms for Finding Small Induced Patterns in Sparse Host Graphs

from arXiv: Data Structures and Algorithms

Authors: Priyanshi Agrawal, Balagopal Komarath

We study algorithms for detecting induced subgraphs corresponding to fixed pattern graphs in host graphs. We show that at least five of the 21 connected graphs on five vertices can be detected in time roughly the product of the number of vertices and the number of edges, and that at least 65 of the 112 connected graphs on six vertices can be detected in time nearly quadratic in the number of edges. We also give algorithms for detecting induced paths and cycles on seven vertices, running in time roughly the number of vertices times the square of the number of edges. Our main technical tool is a generalized notion of tree decomposition width, called (p, q)-width. It yields algorithms whose running times depend on both the number of vertices and the number of edges, and are never worse than existing bounds. Whenever the host graph has fewer than roughly quadratically many edges in its number of vertices, our bounds are strictly faster. For some patterns, including the seven-vertex cycle, our algorithms are optimal under standard complexity-theoretic assumptions. We further develop this approach using pattern-based polynomials that exploit the structure of tree decompositions, not just their width. This gives algorithms for detecting induced paths and cycles on an even number of vertices in bipartite graphs, running in time roughly the (k-1)-th power of the number of edges for paths on 2k vertices, and that same bound times the number of vertices for cycles on 2k vertices. These are faster than the best known algorithms for general graphs.

Authors: Priyanshi Agrawal, Balagopal Komarath

We study algorithms for detecting induced subgraphs corresponding to fixed pattern graphs in host graphs. We show that at least five of the 21 connected graphs on five vertices can be detected in time roughly the product of the number of vertices and the number of edges, and that at least 65 of the 112 connected graphs on six vertices can be detected in time nearly quadratic in the number of edges. We also give algorithms for detecting induced paths and cycles on seven vertices, running in time roughly the number of vertices times the square of the number of edges. Our main technical tool is a generalized notion of tree decomposition width, called (p, q)-width. It yields algorithms whose running times depend on both the number of vertices and the number of edges, and are never worse than existing bounds. Whenever the host graph has fewer than roughly quadratically many edges in its number of vertices, our bounds are strictly faster. For some patterns, including the seven-vertex cycle, our algorithms are optimal under standard complexity-theoretic assumptions. We further develop this approach using pattern-based polynomials that exploit the structure of tree decompositions, not just their width. This gives algorithms for detecting induced paths and cycles on an even number of vertices in bipartite graphs, running in time roughly the (k-1)-th power of the number of edges for paths on 2k vertices, and that same bound times the number of vertices for cycles on 2k vertices. These are faster than the best known algorithms for general graphs.

Unifying and Extending Strong Simulation of Quantum Circuits

from arXiv: Data Structures and Algorithms

Authors: Floris Geerts, Rihan Hai, Matthias Lanzinger, Reinhard Pichler, Emanuel Sallinger, Daniel Unterberger

We establish functional aggregate queries (FAQs) as a unifying language for exact classical simulation of quantum circuits. A circuit becomes a sum-product query: factors encode gates, internal wire variables are aggregated, and free boundary variables index transition amplitudes. The central insight is that distinct sources of simulation tractability can be exploited within the same InsideOut evaluation scheme. The query specifies what is computed; the evaluation plan, semiring, and representation of intermediate factors determine the cost. This view unifies structural and algebraic simulation guarantees. With explicit factor representations, FAQ evaluation recovers the treewidth bound for tensor-network contraction and yields finer sparsity-sensitive bounds via fractional covers. Over a formal phase semiring, compressed intermediate factors recover rank-width-based simulation for compatible quadratic phase representations. For Clifford circuits, affine-quadratic factors are closed under multiplication and marginalization and remain polynomial in size, yielding polynomial-time exact amplitude computation without any bounded-width assumption. Beyond these recoveries, the framework yields a new tractability criterion: tensor layout symmetry width. This parameter combines local cut-rank with separator symmetry through exact tree-tensor representations. We give a constructive evaluation bound and exhibit a circuit family with bounded tensor layout symmetry width but unbounded phase-graph rank-width and circuit line-graph treewidth. These results establish representation-aware FAQ evaluation as a common algorithmic foundation for classical simulation and a systematic route to new tractable regimes.

Authors: Floris Geerts, Rihan Hai, Matthias Lanzinger, Reinhard Pichler, Emanuel Sallinger, Daniel Unterberger

We establish functional aggregate queries (FAQs) as a unifying language for exact classical simulation of quantum circuits. A circuit becomes a sum-product query: factors encode gates, internal wire variables are aggregated, and free boundary variables index transition amplitudes. The central insight is that distinct sources of simulation tractability can be exploited within the same InsideOut evaluation scheme. The query specifies what is computed; the evaluation plan, semiring, and representation of intermediate factors determine the cost. This view unifies structural and algebraic simulation guarantees. With explicit factor representations, FAQ evaluation recovers the treewidth bound for tensor-network contraction and yields finer sparsity-sensitive bounds via fractional covers. Over a formal phase semiring, compressed intermediate factors recover rank-width-based simulation for compatible quadratic phase representations. For Clifford circuits, affine-quadratic factors are closed under multiplication and marginalization and remain polynomial in size, yielding polynomial-time exact amplitude computation without any bounded-width assumption. Beyond these recoveries, the framework yields a new tractability criterion: tensor layout symmetry width. This parameter combines local cut-rank with separator symmetry through exact tree-tensor representations. We give a constructive evaluation bound and exhibit a circuit family with bounded tensor layout symmetry width but unbounded phase-graph rank-width and circuit line-graph treewidth. These results establish representation-aware FAQ evaluation as a common algorithmic foundation for classical simulation and a systematic route to new tractable regimes.

Dynamic Connectivity, Minimum Spanning Tree, and 2-Edge Connectivity with Polylogarithmic Worst-Case Update Time

from arXiv: Data Structures and Algorithms

Authors: Simon Meierhans, Maximilian Probst Gutenberg, Yu-Cheng Yeh

We give fully dynamic algorithms for maintaining connectivity, minimum spanning tree, and $2$-edge connectivity of a graph with worst-case polylogarithmic update time. Our algorithms are randomized and succeed with high probability against an adaptive adversary. For the minimum spanning tree and $2$-edge connectivity problems, this improves over the subpolynomial update time bounds obtained by Nanongkai, Saranurak, and Wulff-Nilsen [FOCS'17], Jin and Sun [FOCS'21], and Jin, Sun, and Thorup [SODA'24], respectively. The only randomized component of our algorithms is the computation of static expander decompositions, and a deterministic algorithm for said problem would directly imply deterministic algorithms for all three problems. This reduction is novel even for the connectivity problem.

Authors: Simon Meierhans, Maximilian Probst Gutenberg, Yu-Cheng Yeh

We give fully dynamic algorithms for maintaining connectivity, minimum spanning tree, and $2$-edge connectivity of a graph with worst-case polylogarithmic update time. Our algorithms are randomized and succeed with high probability against an adaptive adversary. For the minimum spanning tree and $2$-edge connectivity problems, this improves over the subpolynomial update time bounds obtained by Nanongkai, Saranurak, and Wulff-Nilsen [FOCS'17], Jin and Sun [FOCS'21], and Jin, Sun, and Thorup [SODA'24], respectively. The only randomized component of our algorithms is the computation of static expander decompositions, and a deterministic algorithm for said problem would directly imply deterministic algorithms for all three problems. This reduction is novel even for the connectivity problem.

Thursday, October 01

The Keynesian Subtweet

from Ben Recht
Utility maximization is unescapable. We should learn when it’s misapplied.

Hi there, argmin readers! Today’s post is a live blog of Class 9 of my graduate seminar “Forecasting: A Critical Retrospective.” The syllabus and list of past posts are here.

Every class I teach seems to require a lecture or two on expected utility maximization. It’s the basis of classification rules, so it’s integral to machine learning classes. It’s the simplest stochastic optimization problem, so it appears in my optimization classes. It’s a great motivator for probabilistic thinking, so it appears in probability classes. Perhaps a bit less well appreciated, it’s an indirect proper scoring rule for forecasts, so we have to talk about it in this class. And, I suppose, it’s a mathematical tool that scarily dominates the philosophy of many powerful people in government and industry alike. It’s worth understanding the nuts and bolts!

Part of what makes utility maximization so appealing is how simple the core idea is. We want to decide whether to act or not. We build a probabilistic model of the world under the action and compute the expected value of the benefit. We also build a model for what would happen if we don’t act and compute the expected value of the benefit of inaction. We choose to act if our calculations imply that action has more benefit than inaction.

What could be simpler? You compute probabilities. You compute costs. You multiply them together and add them up. Everything is beautifully quantified and calculable.

The only problem is these numbers are all made up. They are forecasts, and they are rarely justifiable. Outside of the casino, we rarely know how to calculate precise odds of outcomes. Worse, forecasting the prices out in the future is also inherently uncertain, often more uncertain than the odds calculations. Maximizing the utility of a rational actor or a general population is an appealing philosophical goal, but the precise rational calculations are always made about fantasy stories. Economists know this, and proceed with caution. In the comments of Tuesday’s post, Jordan Ellenberg flagged this quote from John Maynard Keynes.

“By ‘uncertain’ knowledge, let me explain, I do not mean merely to distinguish what is known for certain from what is only probable. The game of roulette is not subject, in this sense, to uncertainty… Even the weather is only moderately uncertain. The sense in which I am using the term is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealthowners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to be summed.”

Keynes here seems to be reluctantly defending utility maximization as the best of many bad options for decision making in the face of uncertainty. But I don’t think that’s his point at all! Keynes is defending his “General Theory of Employment” and arguing against this sort of utilitarian calculation.

As it is colloquially known, “Keynesian economics” — a term that sadly oversimplifies Keynes’ brilliant collected works — tells governments to stimulate economies during recessions. Keynes argues that we can’t accurately predict the future, and hence people tend to lean on status-quo bias. The safest bet is to assume nothing changes. But when the status quo is bad, people become risk-averse and hold their money. This hoarding prolongs recessions. Because the future is uncertain, people don’t act “rationally” when they fear they might not have money to spend later. Keynes thus argues that governments should step in to nudge citizens out of their undue pessimism by giving them extra money to spend.

Written in the pits of the Great Depression, Keynes’ argument in his 1937 article in the Quarterly Journal of Economics articulates how utility maximization goes wrong. The future is unknowable. Uncertainty makes people afraid. Fear makes hoarding more attractive. Mass hoarding perpetuates a vicious cycle of societal hardship. At this point, someone needs to step in to get the ball rolling, eating up some of the risk to inspire more confidence in people to spend their money.

In this class, we’re not going to debate the merits and challenges of recessionary stimulus spending. But Keynes’ article (which I have added to the reading list) articulates the nuance of forecasting under great uncertainty. Human psychology plays a key role. Today we’ll go through how this appears in the mathematics, with different utility functions expressing different models of risk aversion, and different algorithms for action. We’ll see that not only are the costs and benefits uncertain, but the algorithm itself can be shaped by different models of psychology. Expected utility maximization might be a reasonable way to engineer machines, but it’s not something that reasonable people do. Understanding the many hyperparameters of optimal decision making can help us better understand the psychology of people obsessed with forecasting.

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By Ben Recht

Computational Work Extraction: The Complexity of Catalysts

from arXiv: Computational Complexity

Authors: Atul Singh Arora, Shantanav Chakraborty, Alexandru Cojocaru, Sreyas Saminathan, Uttam Singh

We prove maximal separations: $n$-qubit systems can have $Θ(n)$ ergotropy, while every efficient process extracts negligible work, even for Hamiltonians consisting of single-qubit terms. We establish an unconditional existential separation and give an explicit construction in the random oracle model. Assuming the existence of quantum-secure pseudorandom functions, this separation extends to the plain model. This work uncovers an important connection between ergotropy and the complexity of catalytic computation---computation where auxiliary qubits must be finally restored to their initial state. Relative to a random oracle, we establish relational and decision problems that: (i) can be solved efficiently with $λ$ catalysts; but (ii) cannot be solved by any algorithm with $cλ$ catalysts, for any $c<1$. We show this by proving query lower bounds for quantum-space bounded algorithms. As a consequence, for computational ergotropy, catalysts prove to be surprisingly powerful---there is a family of Hamiltonians and states for which catalysts enable efficient extraction of the full $Θ(n)$ ergotropy, while every efficient non-catalytic process extracts negligible work. Furthermore, catalysts also allow us to introduce and instantiate the notion of pseudoergotropy---analogous to pseudorandomness. On the other hand, we show catalysts do not change (information-theoretic) ergotropy. Finally, our work also sheds light on the classical aspect of the problem. First, most of our constructions rely on classical states and Hamiltonians and therefore imply analogous results for classical ergotropy. Second, we show that certain proof of quantumness protocols can be used to generically separate classical and quantum catalytic ergotropy.

Authors: Atul Singh Arora, Shantanav Chakraborty, Alexandru Cojocaru, Sreyas Saminathan, Uttam Singh

We prove maximal separations: $n$-qubit systems can have $Θ(n)$ ergotropy, while every efficient process extracts negligible work, even for Hamiltonians consisting of single-qubit terms. We establish an unconditional existential separation and give an explicit construction in the random oracle model. Assuming the existence of quantum-secure pseudorandom functions, this separation extends to the plain model. This work uncovers an important connection between ergotropy and the complexity of catalytic computation---computation where auxiliary qubits must be finally restored to their initial state. Relative to a random oracle, we establish relational and decision problems that: (i) can be solved efficiently with $λ$ catalysts; but (ii) cannot be solved by any algorithm with $cλ$ catalysts, for any $c<1$. We show this by proving query lower bounds for quantum-space bounded algorithms. As a consequence, for computational ergotropy, catalysts prove to be surprisingly powerful---there is a family of Hamiltonians and states for which catalysts enable efficient extraction of the full $Θ(n)$ ergotropy, while every efficient non-catalytic process extracts negligible work. Furthermore, catalysts also allow us to introduce and instantiate the notion of pseudoergotropy---analogous to pseudorandomness. On the other hand, we show catalysts do not change (information-theoretic) ergotropy. Finally, our work also sheds light on the classical aspect of the problem. First, most of our constructions rely on classical states and Hamiltonians and therefore imply analogous results for classical ergotropy. Second, we show that certain proof of quantumness protocols can be used to generically separate classical and quantum catalytic ergotropy.

Computational Bounds for $f$-Routing

from arXiv: Computational Complexity

Authors: Oren Renard, Nicholas Spooner

The $f$-routing protocol is a leading candidate for quantum position verification (Kent, Munro, and Spiller, 2011), but security guarantees for explicit functions remain limited. We prove unconditional resource lower bounds for uniform attackers; our new techniques bypass communication-complexity bounds central to previous works, which are inherently at most linear in the input length. We show that, for input length $n$ and sufficiently small constant $ε>0$, a uniformly generated strategy using $q$ qubits and having description length $\mathrm{poly}(q)$, with success probability at least $1-ε$ on every input, implies the following computational bounds on $f$: 1. If the strategies are arbitrary quantum channels, then $f\in\mathrm{QSZK}(\mathrm{poly}(nq))$, where $\mathrm{QSZK}(T)$ is the class of languages having quantum statistical zero knowledge proofs in which the verifier runs in time $T$ (and the simulator in time $\mathrm{poly}(T)$). 2. If the strategies are explicit Pauli-sparse unitaries on $q$ qubits that have at most $s$ nonzero Pauli coefficients, then $f\in\mathrm{DTIME}(\mathrm{poly}(nqs))$. 3. If the strategies are Clifford+T circuits using at most $t$ magic gates, then $f\in\mathrm{DTIME}(\mathrm{poly}(nq2^t))$. Time and space hierarchies then yield explicit functions secure against polynomial and even quasipolynomial qubits $q$ under our computational restrictions. These bounds exceed the $q\le\log n$ bound of Bluhm, Christandl, and Speelman (2022) for inner product function $f=\mathrm{IP}$, at the cost of restricting adversarial computation and increasing honest evaluation complexity.

Authors: Oren Renard, Nicholas Spooner

The $f$-routing protocol is a leading candidate for quantum position verification (Kent, Munro, and Spiller, 2011), but security guarantees for explicit functions remain limited. We prove unconditional resource lower bounds for uniform attackers; our new techniques bypass communication-complexity bounds central to previous works, which are inherently at most linear in the input length. We show that, for input length $n$ and sufficiently small constant $ε>0$, a uniformly generated strategy using $q$ qubits and having description length $\mathrm{poly}(q)$, with success probability at least $1-ε$ on every input, implies the following computational bounds on $f$: 1. If the strategies are arbitrary quantum channels, then $f\in\mathrm{QSZK}(\mathrm{poly}(nq))$, where $\mathrm{QSZK}(T)$ is the class of languages having quantum statistical zero knowledge proofs in which the verifier runs in time $T$ (and the simulator in time $\mathrm{poly}(T)$). 2. If the strategies are explicit Pauli-sparse unitaries on $q$ qubits that have at most $s$ nonzero Pauli coefficients, then $f\in\mathrm{DTIME}(\mathrm{poly}(nqs))$. 3. If the strategies are Clifford+T circuits using at most $t$ magic gates, then $f\in\mathrm{DTIME}(\mathrm{poly}(nq2^t))$. Time and space hierarchies then yield explicit functions secure against polynomial and even quasipolynomial qubits $q$ under our computational restrictions. These bounds exceed the $q\le\log n$ bound of Bluhm, Christandl, and Speelman (2022) for inner product function $f=\mathrm{IP}$, at the cost of restricting adversarial computation and increasing honest evaluation complexity.

Quantum Algorithms for Minimum Generating Set

from arXiv: Computational Complexity

Authors: Bireswar Das, Udit Kumar, Kavita Samant, Dhara Thakkar

In this paper, we present a polynomial-time quantum algorithm for computing a minimum-sized generating set of solvable black-box groups. Next, we consider the class $Γ_d$ of black-box groups, where every non-abelian composition factor is isomorphic to a subgroup of the symmetric group $S_d$ for a fixed $d$. We design polynomial-time quantum algorithms to compute the direct product decomposition of abelian factor groups and solve the constructive membership problem for factor groups of groups from $Γ_d$. With the help of these algorithms, we design a quantum algorithm for computing a chief series of black-box groups from $Γ_d$. Using the chief series, we construct a polynomial-time quantum algorithm for computing minimum generating sets of black-box groups from $Γ_d$. Finally, we show that the minimum generating set problem for general black-box groups is in $\textrm{NP} \cap \textrm{coAM}$.

Authors: Bireswar Das, Udit Kumar, Kavita Samant, Dhara Thakkar

In this paper, we present a polynomial-time quantum algorithm for computing a minimum-sized generating set of solvable black-box groups. Next, we consider the class $Γ_d$ of black-box groups, where every non-abelian composition factor is isomorphic to a subgroup of the symmetric group $S_d$ for a fixed $d$. We design polynomial-time quantum algorithms to compute the direct product decomposition of abelian factor groups and solve the constructive membership problem for factor groups of groups from $Γ_d$. With the help of these algorithms, we design a quantum algorithm for computing a chief series of black-box groups from $Γ_d$. Using the chief series, we construct a polynomial-time quantum algorithm for computing minimum generating sets of black-box groups from $Γ_d$. Finally, we show that the minimum generating set problem for general black-box groups is in $\textrm{NP} \cap \textrm{coAM}$.

A physical and universal model of bosonic computations with Solovay-Kitaev theorem

from arXiv: Computational Complexity

Authors: Dorian Rudolph, Arsalan Motamedi, Dhruva Sambrani, Hamid Reza Naeij, Ulysse Chabaud, Sevag Gharibian, Saeed Mehraban

Bosonic quantum systems are among the leading architectures for quantum information processing, offering continuous-variable degrees of freedom with strong error-correction capabilities. However, standard bosonic quantum computation models such as the Lloyd-Braunstein [Lloyd and Braunstein, 1999] and hybrid oscillator-qubit models [Brenner, Dias, and Koenig, 2025; Liu et al., 2026] permit dramatic energy growth, leading to unphysical computational power and the breakdown of fundamental algorithmic tools such as universal and efficient compilation [Brenner et al., 2026; Rudolph et al., 2025]. To address this, we introduce a new model of bosonic quantum computation, Bosonic Energy-Preserving Quantum Computation (BEQC), in which energy is treated as a computational resource. Namely, energy is supplied solely through input coherent states, and all gates are generated by energy-preserving Hamiltonians. Thus, by construction, dramatic energy growth is impossible, making the model physically grounded. We next show that BEQC is a computationally robust and universal model in many respects, including: (1. Computational power) BEQC efficiently simulates all polynomial-energy computations in existing models, and exactly recovers BQP in the polynomial energy setting. It further admits several complexity-theoretic upper bounds when varying the energy, precision, and space parameters of the model. (2. Universal gate sets and state synthesis) BEQC has natural universal gate sets based on linear optics and Kerr interactions. In particular, we obtain a Solovay-Kitaev theorem which circumvents previous no-go results. We give various applications, including (a) a protocol for engineering GKP states with rigorous preparation guarantees, (b) Fock state preparation to exponential precision, and (c) native Fock space simulation of any qubit-based unitary.

Authors: Dorian Rudolph, Arsalan Motamedi, Dhruva Sambrani, Hamid Reza Naeij, Ulysse Chabaud, Sevag Gharibian, Saeed Mehraban

Bosonic quantum systems are among the leading architectures for quantum information processing, offering continuous-variable degrees of freedom with strong error-correction capabilities. However, standard bosonic quantum computation models such as the Lloyd-Braunstein [Lloyd and Braunstein, 1999] and hybrid oscillator-qubit models [Brenner, Dias, and Koenig, 2025; Liu et al., 2026] permit dramatic energy growth, leading to unphysical computational power and the breakdown of fundamental algorithmic tools such as universal and efficient compilation [Brenner et al., 2026; Rudolph et al., 2025]. To address this, we introduce a new model of bosonic quantum computation, Bosonic Energy-Preserving Quantum Computation (BEQC), in which energy is treated as a computational resource. Namely, energy is supplied solely through input coherent states, and all gates are generated by energy-preserving Hamiltonians. Thus, by construction, dramatic energy growth is impossible, making the model physically grounded. We next show that BEQC is a computationally robust and universal model in many respects, including: (1. Computational power) BEQC efficiently simulates all polynomial-energy computations in existing models, and exactly recovers BQP in the polynomial energy setting. It further admits several complexity-theoretic upper bounds when varying the energy, precision, and space parameters of the model. (2. Universal gate sets and state synthesis) BEQC has natural universal gate sets based on linear optics and Kerr interactions. In particular, we obtain a Solovay-Kitaev theorem which circumvents previous no-go results. We give various applications, including (a) a protocol for engineering GKP states with rigorous preparation guarantees, (b) Fock state preparation to exponential precision, and (c) native Fock space simulation of any qubit-based unitary.

A mixing time method for estimating the sample complexity of quantum state discrimination

from arXiv: Computational Complexity

Authors: Juntai Zhou, Felix Leditzky

We develop a mixing time method for estimating the sample complexity of quantum state discrimination. We start with considering the minimum-error discrimination of geometrically uniform pure state ensembles, and prove that its sample complexity has a tight estimate given by a quantum homogeneous mixing time [George et al., 2026] and a quantum version of the generalized Dobrushin coefficient [Wolfer, 2020]. This quantum mixing time further reduces to a classical one when the generating group $G$ forms a Gelfand pair with the stabilizer subgroup $H$ of the generator state. In this case the generalized Dobrushin coefficient can be fully expressed by representation-theoretic quantities of the commutative Hecke algebra $\operatorname{End}_G(\mathbb C[G/H])$. In particular, this method reduces the sample complexity estimation of learning quantum coupon collector states [Arunachalam et al., 2020] and learning phase states to classical mixing time problems. We apply this framework to answer the open problems of learning degree-$d$ phase states over $\mathbb F_q$ in [Alrabiah et al., 2026] and generalized Boolean phase states over $\mathbb Z_q$ [Arunachalam et al., 2023]. The framework also applies to hypergraph state ensembles, giving estimates expressed fully in terms of hypergraph data and recovering estimates for graph state ensembles in [Montanaro and Shao, 2022]. Finally, we extend the discussion to arbitrary mixed state ensembles with uniform priors, prove a sandwiched bound for minimum-error discrimination sample complexity by a quantum weakly mixing time, and provide a tight estimate for the minimax discrimination sample complexity from [D'Ariano et al., 2005] by a Dobrushin-type coefficient. We also discuss the method of strengthened data processing inequality [Gao and Rouz{é}, 2022] and give an upper bound in terms of a strengthened data processing inequality constant.

Authors: Juntai Zhou, Felix Leditzky

We develop a mixing time method for estimating the sample complexity of quantum state discrimination. We start with considering the minimum-error discrimination of geometrically uniform pure state ensembles, and prove that its sample complexity has a tight estimate given by a quantum homogeneous mixing time [George et al., 2026] and a quantum version of the generalized Dobrushin coefficient [Wolfer, 2020]. This quantum mixing time further reduces to a classical one when the generating group $G$ forms a Gelfand pair with the stabilizer subgroup $H$ of the generator state. In this case the generalized Dobrushin coefficient can be fully expressed by representation-theoretic quantities of the commutative Hecke algebra $\operatorname{End}_G(\mathbb C[G/H])$. In particular, this method reduces the sample complexity estimation of learning quantum coupon collector states [Arunachalam et al., 2020] and learning phase states to classical mixing time problems. We apply this framework to answer the open problems of learning degree-$d$ phase states over $\mathbb F_q$ in [Alrabiah et al., 2026] and generalized Boolean phase states over $\mathbb Z_q$ [Arunachalam et al., 2023]. The framework also applies to hypergraph state ensembles, giving estimates expressed fully in terms of hypergraph data and recovering estimates for graph state ensembles in [Montanaro and Shao, 2022]. Finally, we extend the discussion to arbitrary mixed state ensembles with uniform priors, prove a sandwiched bound for minimum-error discrimination sample complexity by a quantum weakly mixing time, and provide a tight estimate for the minimax discrimination sample complexity from [D'Ariano et al., 2005] by a Dobrushin-type coefficient. We also discuss the method of strengthened data processing inequality [Gao and Rouz{é}, 2022] and give an upper bound in terms of a strengthened data processing inequality constant.

Improved Quantum Random Self-Reduction for Linear Problems

from arXiv: Computational Complexity

Authors: Vahid R. Asadi, Shuichi Hirahara, Nobutaka Shimizu

We study quantum random self-reductions for linear problems over finite fields. Let $M\in\mathbb{F}^{n\times n}$ be an arbitrary matrix, and let $\mathcal{O}$ be an oracle that agrees with the linear map $x\mapsto Mx$ on an $\varepsilon$-fraction of inputs $x\sim\mathbb{F}^n$. Given coherent access to $\mathcal{O}$ and coherent entry access to $M$, we give a uniform quantum reduction that computes $Mx$ on any prescribed input $x$ with probability at least $2/3$ in time $\widetilde{O}(nT^{1/3})$, for $n\le T\le n^{3/2}$ and constant field size and $\varepsilon$, where $T$ is the cost of one coherent query to $\mathcal{O}$. In particular, when $T=\widetilde{O}(n)$, the reduction runs in time $\widetilde{O}(n^{4/3})$, improving the $\widetilde{O}(n^{3/2}+T)$ reduction of Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA 2024). Our reduction uses the Bogolyubov--Ruzsa subspace guaranteed by additive combinatorics, but it avoids learning this subspace explicitly, which was computationally expensive for the previous reduction; in particular, it does not recover a basis for its orthogonal complement. The main technical step is to decompose the inputs into sparse pieces and find a vector that lies outside the Bogolyubov--Ruzsa subspace via a quantum search based on amplitude amplification. This yields a tunable tradeoff between the cost of querying the average-case oracle and the cost of verifying matrix-vector products.

Authors: Vahid R. Asadi, Shuichi Hirahara, Nobutaka Shimizu

We study quantum random self-reductions for linear problems over finite fields. Let $M\in\mathbb{F}^{n\times n}$ be an arbitrary matrix, and let $\mathcal{O}$ be an oracle that agrees with the linear map $x\mapsto Mx$ on an $\varepsilon$-fraction of inputs $x\sim\mathbb{F}^n$. Given coherent access to $\mathcal{O}$ and coherent entry access to $M$, we give a uniform quantum reduction that computes $Mx$ on any prescribed input $x$ with probability at least $2/3$ in time $\widetilde{O}(nT^{1/3})$, for $n\le T\le n^{3/2}$ and constant field size and $\varepsilon$, where $T$ is the cost of one coherent query to $\mathcal{O}$. In particular, when $T=\widetilde{O}(n)$, the reduction runs in time $\widetilde{O}(n^{4/3})$, improving the $\widetilde{O}(n^{3/2}+T)$ reduction of Asadi, Golovnev, Gur, Shinkar, and Subramanian (SODA 2024). Our reduction uses the Bogolyubov--Ruzsa subspace guaranteed by additive combinatorics, but it avoids learning this subspace explicitly, which was computationally expensive for the previous reduction; in particular, it does not recover a basis for its orthogonal complement. The main technical step is to decompose the inputs into sparse pieces and find a vector that lies outside the Bogolyubov--Ruzsa subspace via a quantum search based on amplitude amplification. This yields a tunable tradeoff between the cost of querying the average-case oracle and the cost of verifying matrix-vector products.

Security Properties of Neural Networks as Decision Problems

from arXiv: Computational Complexity

Authors: Adrian Wurm

Certifying a deployed neural network raises decision problems that the verification literature has not classified: whether the model carries a backdoor planted in its training data, whether a fault in its stored parameters can drive it into an unsafe state, whether its output leaks a private part of its input. We formalise eight such problems and classify what we can. The organising observation is a logical one. The function computed by a piecewise linear network, together with all its node values, is definable by a quantifier-free formula of real addition of size linear in the network, so a property of the network is a quantifier-alternation sentence, which Sontag's 1985 theorem places in the polynomial hierarchy at the level of its prefix. Membership results are thus corollaries, and the argument makes plain what they need: that the quantified objects are inputs rather than the network's own parameters. Non-interference, monotonicity and counterfactual fairness have exactly the complexity of network equivalence and of interval verification, all co-NP- complete over ReLU. Detection of backdoor triggers from a quantised alphabet is Sigma_2^P-complete, one level above robustness certification, so it does not reduce to polynomially many robustness queries unless the hierarchy collapses. Inversion resistance is co-NP-complete for every l_p metric, p a fixed positive integer. Quantifying over parameters instead of inputs - the fault model of bit-flip attacks, radiation upsets and analog accelerators - makes verification exists-R-complete already for networks of identity nodes, for which every previously studied problem is in P, and it stays so when each parameter is confined to a box of inverse-polynomial width; the corresponding safety question is forall-R-complete for ReLU.

Authors: Adrian Wurm

Certifying a deployed neural network raises decision problems that the verification literature has not classified: whether the model carries a backdoor planted in its training data, whether a fault in its stored parameters can drive it into an unsafe state, whether its output leaks a private part of its input. We formalise eight such problems and classify what we can. The organising observation is a logical one. The function computed by a piecewise linear network, together with all its node values, is definable by a quantifier-free formula of real addition of size linear in the network, so a property of the network is a quantifier-alternation sentence, which Sontag's 1985 theorem places in the polynomial hierarchy at the level of its prefix. Membership results are thus corollaries, and the argument makes plain what they need: that the quantified objects are inputs rather than the network's own parameters. Non-interference, monotonicity and counterfactual fairness have exactly the complexity of network equivalence and of interval verification, all co-NP- complete over ReLU. Detection of backdoor triggers from a quantised alphabet is Sigma_2^P-complete, one level above robustness certification, so it does not reduce to polynomially many robustness queries unless the hierarchy collapses. Inversion resistance is co-NP-complete for every l_p metric, p a fixed positive integer. Quantifying over parameters instead of inputs - the fault model of bit-flip attacks, radiation upsets and analog accelerators - makes verification exists-R-complete already for networks of identity nodes, for which every previously studied problem is in P, and it stays so when each parameter is confined to a box of inverse-polynomial width; the corresponding safety question is forall-R-complete for ReLU.

QMA(2) with Limited Shared Entanglement

from arXiv: Computational Complexity

Authors: Alex Della Schiava, Ranitha Mataraarachchi

A $\mathsf{QMA}(2)$ protocol involves two provers submitting unentangled witnesses to a polynomial-time quantum verifier. In The Power of Unentanglement (ToC, 2009), Aaronson et al. proposed $\mathsf{QMA}(2;h)$, a variant of $\mathsf{QMA}(2)$ in which the two provers may share $h$ EPR pairs. Our main result shows that the power of $\mathsf{QMA}(2)$ remains unchanged for up to logarithmically many shared EPR pairs: $\mathsf{QMA}(2;h)=\mathsf{QMA}(2)$ for $h=O(\log n)$, where $n$ is the input length. The result follows from a simulation using four unentangled witnesses, combined with the Harrow-Montanaro equality $\mathsf{QMA}(4)=\mathsf{QMA}(2)$ (FOCS, 2010). We also prove monotonicity in the EPR budget: $\mathsf{QMA}(2;h)\subseteq\mathsf{QMA}(2;H)$ for $h\le H$, preserving completeness and soundness. Combined with input padding, this shows that establishing $\mathsf{QMA}(2;n^\varepsilon)=\mathsf{QMA}(2)$ for any fixed $\varepsilon>0$ would imply equality for every polynomially bounded budget, resolving the open problem raised by Aaronson et al. Finally, we extend these results to a variant of the model in which the provers may use local operations and classical communication (LOCC) during witness preparation. For logarithmic-size witnesses and inverse-polynomial gaps, both models remain equivalent to their unentangled counterpart when $h=O(\log n)$. We show that extending this equivalence to any superlogarithmic EPR budget in the LOCC model would imply $\mathsf{NP}\subseteq\mathsf{BQP}$.

Authors: Alex Della Schiava, Ranitha Mataraarachchi

A $\mathsf{QMA}(2)$ protocol involves two provers submitting unentangled witnesses to a polynomial-time quantum verifier. In The Power of Unentanglement (ToC, 2009), Aaronson et al. proposed $\mathsf{QMA}(2;h)$, a variant of $\mathsf{QMA}(2)$ in which the two provers may share $h$ EPR pairs. Our main result shows that the power of $\mathsf{QMA}(2)$ remains unchanged for up to logarithmically many shared EPR pairs: $\mathsf{QMA}(2;h)=\mathsf{QMA}(2)$ for $h=O(\log n)$, where $n$ is the input length. The result follows from a simulation using four unentangled witnesses, combined with the Harrow-Montanaro equality $\mathsf{QMA}(4)=\mathsf{QMA}(2)$ (FOCS, 2010). We also prove monotonicity in the EPR budget: $\mathsf{QMA}(2;h)\subseteq\mathsf{QMA}(2;H)$ for $h\le H$, preserving completeness and soundness. Combined with input padding, this shows that establishing $\mathsf{QMA}(2;n^\varepsilon)=\mathsf{QMA}(2)$ for any fixed $\varepsilon>0$ would imply equality for every polynomially bounded budget, resolving the open problem raised by Aaronson et al. Finally, we extend these results to a variant of the model in which the provers may use local operations and classical communication (LOCC) during witness preparation. For logarithmic-size witnesses and inverse-polynomial gaps, both models remain equivalent to their unentangled counterpart when $h=O(\log n)$. We show that extending this equivalence to any superlogarithmic EPR budget in the LOCC model would imply $\mathsf{NP}\subseteq\mathsf{BQP}$.

The Commuting Local Hamiltonian Problem: Relativized Evidence Against BQP-Hardness

from arXiv: Computational Complexity

Authors: Itay Shalit, Mark Zhandry

The commuting local-Hamiltonian (CLH) problem is a restriction of the local-Hamiltonian problem, in which the terms of the Hamiltonian are required to pairwise commute. A long line of work has shown that the problem lies in $\mathsf{NP}$ for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is $\mathsf{QMA}$-complete. CLH is complete for the complexity class $\mathsf{QIMA}$, defined through quantum verifiers whose local gates commute. Therefore, CLH is $\mathsf{QMA}$-complete if and only if $\mathsf{QIMA}=\mathsf{QMA}$. In this work, we introduce a classical-oracle analogue $\mathsf{QIMA}^{\mathcal{O}}$ and construct a classical oracle $\mathcal{O}$ such that $\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}$. Since $\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}$ for any classical oracle $\mathcal{O}$, this implies $\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}$ for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is $\mathsf{BQP}$-hard, and hence also against the possibility that it is $\mathsf{QMA}$-complete.

Authors: Itay Shalit, Mark Zhandry

The commuting local-Hamiltonian (CLH) problem is a restriction of the local-Hamiltonian problem, in which the terms of the Hamiltonian are required to pairwise commute. A long line of work has shown that the problem lies in $\mathsf{NP}$ for certain families of commuting local Hamiltonians. Nevertheless, there has been no formal evidence against the possibility that the general CLH problem is $\mathsf{QMA}$-complete. CLH is complete for the complexity class $\mathsf{QIMA}$, defined through quantum verifiers whose local gates commute. Therefore, CLH is $\mathsf{QMA}$-complete if and only if $\mathsf{QIMA}=\mathsf{QMA}$. In this work, we introduce a classical-oracle analogue $\mathsf{QIMA}^{\mathcal{O}}$ and construct a classical oracle $\mathcal{O}$ such that $\mathsf{BQP}^{\mathcal{O}}\not\subseteq\mathsf{QIMA}^{\mathcal{O}}$. Since $\mathsf{BQP}^{\mathcal{O}}\subseteq\mathsf{QMA}^{\mathcal{O}}$ for any classical oracle $\mathcal{O}$, this implies $\mathsf{QIMA}^{\mathcal{O}}\neq\mathsf{QMA}^{\mathcal{O}}$ for our constructed oracle. Thus, our result provides relativized evidence against the possibility that the general commuting local-Hamiltonian problem is $\mathsf{BQP}$-hard, and hence also against the possibility that it is $\mathsf{QMA}$-complete.

Frustration Free Stoquastic Local Hamiltonian with Sub-Constant Gap is in NP

from arXiv: Computational Complexity

Authors: Eshan Chattopadhyay, Oren Renard, Nicholas Spooner

We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the $β$-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when $β= 1/\mathrm{poly}(n)$. Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap $β= Ω(1)$. We present an improved algorithm and analysis, establishing membership in NP even when $β= Ω(1/(\log\log n))$. We complement our result with an explicit example demonstrating why the analysis does not extend directly to $β= o(1/\log\log n)$.

Authors: Eshan Chattopadhyay, Oren Renard, Nicholas Spooner

We continue the study of the Stoquastic Local Hamiltonian problem, a physically motivated restriction of the QMA-complete Local Hamiltonian problem (Kitaev, Shen, and Vyalyi, 2002). For the $β$-gapped, frustration-free case, Bravyi, Bessen, and Terhal (2006) showed that the problem is MA-complete when $β= 1/\mathrm{poly}(n)$. Aharonov and Grilo (2019) derandomized this algorithm and proved membership in NP for constant gap $β= Ω(1)$. We present an improved algorithm and analysis, establishing membership in NP even when $β= Ω(1/(\log\log n))$. We complement our result with an explicit example demonstrating why the analysis does not extend directly to $β= o(1/\log\log n)$.

Experimentally Testable Quantum Advantage in Shallow Circuits

from arXiv: Computational Complexity

Authors: Kishor Bharti, Adán Cabello

Experimental tests of shallow-circuit quantum advantage require explicit classical bounds at finite circuit sizes. We refine the finite-size classical soundness bound of Aasnaess's graph-distributed construction to depend linearly on the number of players. Combined with standard disjoint-player repetition, this gives a two-round test on a single processor for any finite nonlocal game with a finite-dimensional perfect quantum strategy and classical winning probability $γ<1$, with arbitrarily small classical success. The method is based on playing $m$ copies with disjoint players and teleporting each player's register to a uniformly random one of $N$ sites before the questions are revealed. Each answer bit of a depth-$D$, fan-in-$K$ classical response with fixed wiring depends on at most $K^D$ question wires, making cross-player dependencies unlikely when $N$ is large. The quantum implementation has constant depth per round and wins with certainty. A classical device wins with probability at most $γ^m+O(mK^D/N)$, which vanishes as $O(\log N/N)$ for $m=\lceil\log_{1/γ}N\rceil$ at fixed $D$ and $K$. We present an explicit proposal for an experimentally testable quantum advantage with 99 qubits.

Authors: Kishor Bharti, Adán Cabello

Experimental tests of shallow-circuit quantum advantage require explicit classical bounds at finite circuit sizes. We refine the finite-size classical soundness bound of Aasnaess's graph-distributed construction to depend linearly on the number of players. Combined with standard disjoint-player repetition, this gives a two-round test on a single processor for any finite nonlocal game with a finite-dimensional perfect quantum strategy and classical winning probability $γ<1$, with arbitrarily small classical success. The method is based on playing $m$ copies with disjoint players and teleporting each player's register to a uniformly random one of $N$ sites before the questions are revealed. Each answer bit of a depth-$D$, fan-in-$K$ classical response with fixed wiring depends on at most $K^D$ question wires, making cross-player dependencies unlikely when $N$ is large. The quantum implementation has constant depth per round and wins with certainty. A classical device wins with probability at most $γ^m+O(mK^D/N)$, which vanishes as $O(\log N/N)$ for $m=\lceil\log_{1/γ}N\rceil$ at fixed $D$ and $K$. We present an explicit proposal for an experimentally testable quantum advantage with 99 qubits.

A Separation Between Types of Quantum Oracle Separations

from arXiv: Computational Complexity

Authors: Scott Aaronson, Adam Bouland, Jordan Docter, Barak Nehoran

Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B. We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes $\mathsf{PostBQP}$ and $\mathsf{PreciseBQP}$, the exponentially precise variant of $\mathsf{BQP}$. Both classes equal $\mathsf{PP}$ in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.

Authors: Scott Aaronson, Adam Bouland, Jordan Docter, Barak Nehoran

Recent works have demonstrated that quantum oracles have subtle behavior, as access to inverse, conjugate or controlled queries can exponentially change the query complexity of certain tasks. Inspired by these works, we introduce the notion of meta-complexity of quantum relativization. We ask: for any two quantum complexity classes, under which "types" of quantum oracles are they equal or separated? Different pantheons of oracles (or quantum oracle types, e.g. unitary vs state, poly- vs superpoly-dimensional, closed under inverse or not) form a partially ordered set based on their power in separating complexity classes. Moreover, two oracle pantheons A and B are separated if there exists a pair of complexity classes that are separated under an oracle from pantheon A but yet the complexity classes are equivalent under all oracles from pantheon B. We show that this meta-complexity can be nontrivial by giving a complete classification, within the family of oracle pantheons defined in this paper, of which models can separate the complexity classes $\mathsf{PostBQP}$ and $\mathsf{PreciseBQP}$, the exponentially precise variant of $\mathsf{BQP}$. Both classes equal $\mathsf{PP}$ in the unrelativized setting. Within our taxonomy, they remain equal relative to real or polynomial-dimensional unitary oracles and whenever inverse or conjugate access is supplied. In contrast, we give a separation relative to forward-only complex diagonal unitaries of superpolynomial dimension, as well as a separation relative to single-qubit state-preparation oracles. We view this as a test case for the meta-complexity of oracles which underscores the subtlety inherent to the relativization of quantum complexity classes.

Top-Down Lower Bounds for All Depths

from arXiv: Computational Complexity

Authors: Oliver Korten

We prove that Parity requires $2^{n^{Ω(1)}}$ size De Morgan circuits of constant depth using a new method which is completely "top-down" in the sense of [HJP95]. The proof relies crucially on the core ideas developed in a line of work [HJP95, PPZ99, MW19, GRSS24] which previously established top-down lower bounds for circuits of depth 3 and 4. We first present a proof of a lower bound $\exp(n^{3^{-d}})$. In this case, nearly all of the relevant combinatorial ideas necessary for the proof are already present in some form in [GRSS24]. We then present two extensions of this argument, the first achieving a lower bound $\exp(ε_d n^{1/(2d-2)})$ for some $ε_d>0$ depending only on $d$, and the second achieving the essentially tight lower bound $\exp(ε_d n^{1/(d-1)})$. These improved results each hinge on establishing a key lemma which quantifies the extent to which a high entropy random variable in $\{0,1\}^n$ will look close to uniform after projecting it onto a random small set of coordinates $R \subseteq [n]$.

Authors: Oliver Korten

We prove that Parity requires $2^{n^{Ω(1)}}$ size De Morgan circuits of constant depth using a new method which is completely "top-down" in the sense of [HJP95]. The proof relies crucially on the core ideas developed in a line of work [HJP95, PPZ99, MW19, GRSS24] which previously established top-down lower bounds for circuits of depth 3 and 4. We first present a proof of a lower bound $\exp(n^{3^{-d}})$. In this case, nearly all of the relevant combinatorial ideas necessary for the proof are already present in some form in [GRSS24]. We then present two extensions of this argument, the first achieving a lower bound $\exp(ε_d n^{1/(2d-2)})$ for some $ε_d>0$ depending only on $d$, and the second achieving the essentially tight lower bound $\exp(ε_d n^{1/(d-1)})$. These improved results each hinge on establishing a key lemma which quantifies the extent to which a high entropy random variable in $\{0,1\}^n$ will look close to uniform after projecting it onto a random small set of coordinates $R \subseteq [n]$.

Karp's NP-complete problems over first-order definable structures

from arXiv: Computational Complexity

Authors: Aidan Healy, Bartek Klin

We determine the decidability of Karp's NP-complete problems on structures which are first-order definable over the theory of equality, also known as orbit-finite sets with atoms or nominal sets.

Authors: Aidan Healy, Bartek Klin

We determine the decidability of Karp's NP-complete problems on structures which are first-order definable over the theory of equality, also known as orbit-finite sets with atoms or nominal sets.

Parameterized Hardness of Mixed 2-Sided Orthant Depth

from arXiv: Computational Geometry

Authors: Michelle Döring, Georg Tennigkeit

We consider the maximum-depth problem for mixed 2-sided orthants in R^d: each region imposes one lower bound and one upper bound on distinct coordinates, and the task is to find a point contained in as many regions as possible. We show that the corresponding decision problem is W[1]-hard when parameterized by the dimension. Our reduction from MultiColoredClique uses two coordinates per color class and only polynomially many orthants.

Authors: Michelle Döring, Georg Tennigkeit

We consider the maximum-depth problem for mixed 2-sided orthants in R^d: each region imposes one lower bound and one upper bound on distinct coordinates, and the task is to find a point contained in as many regions as possible. We show that the corresponding decision problem is W[1]-hard when parameterized by the dimension. Our reduction from MultiColoredClique uses two coordinates per color class and only polynomially many orthants.

Flexible discrete translational surfaces

from arXiv: Computational Geometry

Authors: Georg Nawratil

We give a full list of translational nets which flex within their class of discrete surfaces of translation, by reducing the classification problem to the one of flexible complete bipartite frameworks on the sphere, for which the solution is known. We also obtained two novel classes which correspond to Bottema's spherical 16-bar mechanisms and the constant diagonal angle frameworks. Based on an algorithm for the construction of all flexible translational nets, we also discuss flexible translational tubes and toroids. Furthermore, we present novel results for both topologies which are implied by Bottema's spherical 16-bar mechanisms.

Authors: Georg Nawratil

We give a full list of translational nets which flex within their class of discrete surfaces of translation, by reducing the classification problem to the one of flexible complete bipartite frameworks on the sphere, for which the solution is known. We also obtained two novel classes which correspond to Bottema's spherical 16-bar mechanisms and the constant diagonal angle frameworks. Based on an algorithm for the construction of all flexible translational nets, we also discuss flexible translational tubes and toroids. Furthermore, we present novel results for both topologies which are implied by Bottema's spherical 16-bar mechanisms.

A Unified Dual Method for Matching Problems

from arXiv: Computational Geometry

Authors: Guillaume Houry, Ferdinand Genans, Jean Feydy, François-Xavier Vialard

Matching problems are ubiquitous in data science as they enable the alignment of structured objects and distributions. While existing solvers are often tailored to specific matching formulations, we unify a broad class of such problems within a common mathematical and optimization framework based on duality theory. Theoretically, we demonstrate that matching objectives decomposable as a difference of convex (DC) functions can be recast as implicit registration problems. This connection links matching to another well-studied class of objectives and yields a dual formulation amenable to natural optimization strategies. We then apply these findings to quadratic matching (QM) problems, which admit DC decompositions and for which we provide extensive convergence guarantees. Our framework applies to Gromov-Wasserstein (GW), as well as its unbalanced formulation and several variants, which are increasingly popular QM problems. Numerically, we implement our algorithms at scale for various data modalities such as graphs, point clouds, meshes, and word embeddings. Finally, our modular approach allows us to explore new formulations such as fracture matching, broadening the scope of problems that can be addressed within this framework..

Authors: Guillaume Houry, Ferdinand Genans, Jean Feydy, François-Xavier Vialard

Matching problems are ubiquitous in data science as they enable the alignment of structured objects and distributions. While existing solvers are often tailored to specific matching formulations, we unify a broad class of such problems within a common mathematical and optimization framework based on duality theory. Theoretically, we demonstrate that matching objectives decomposable as a difference of convex (DC) functions can be recast as implicit registration problems. This connection links matching to another well-studied class of objectives and yields a dual formulation amenable to natural optimization strategies. We then apply these findings to quadratic matching (QM) problems, which admit DC decompositions and for which we provide extensive convergence guarantees. Our framework applies to Gromov-Wasserstein (GW), as well as its unbalanced formulation and several variants, which are increasingly popular QM problems. Numerically, we implement our algorithms at scale for various data modalities such as graphs, point clouds, meshes, and word embeddings. Finally, our modular approach allows us to explore new formulations such as fracture matching, broadening the scope of problems that can be addressed within this framework..

A differentiability framework for zigzag persistent homology via linear interpolation

from arXiv: Computational Geometry

Authors: Enrico Maria Ferrari, Clemens Bannwart, Matteo Biagetti

Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.

Authors: Enrico Maria Ferrari, Clemens Bannwart, Matteo Biagetti

Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.

Gibbs Sampling in the Shattered Phase by Decoded Quantum Interferometry

from arXiv: Data Structures and Algorithms

Authors: Leo Zhou, Noah Shutty, Mark Sellke, Stephen P. Jordan

We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $β_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=αk$ with fixed $α>1$, both Prange's algorithm and DQI can sample at any inverse temperature $β< \tanh^{-1}(1/α)$ for sufficiently large $k$, well beyond the dynamical threshold $β_{\rm dyn} \sim \sqrt{2\ln k / (αk)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.

Authors: Leo Zhou, Noah Shutty, Mark Sellke, Stephen P. Jordan

We apply Decoded Quantum Interferometry (DQI) to sample from the Gibbs measures of classical Ising spin Hamiltonians. We show that this Gibbs sampling problem reduces to a quantum decoding problem, and the temperature achievable by DQI is determined by the performance of decoding algorithms. We then focus on the task of Gibbs sampling for classical Ising $k$-spin glasses (or Max-$k$-XORSAT) on random Erdős-Rényi hypergraphs with average degree $D\ge k$. In a temperature range beginning asymptotically at the predicted dynamical phase transition, $β_{\rm dyn}(k,D) = \sqrt{(2\ln k)/D}\times [1+o_{k\to\infty}(1)]$, we show that shattering and disorder chaos form a topological barrier that obstructs many algorithms, including Glauber dynamics and any algorithm whose output distribution is "stable" under perturbations of the input. In contrast, we prove that this barrier can be broken both by a classical algorithm based on Prange's method, and by DQI equipped with a quantum decoder. For example, when $D=αk$ with fixed $α>1$, both Prange's algorithm and DQI can sample at any inverse temperature $β< \tanh^{-1}(1/α)$ for sufficiently large $k$, well beyond the dynamical threshold $β_{\rm dyn} \sim \sqrt{2\ln k / (αk)}$. Therefore, our results show that DQI can overcome topological barriers that obstruct stable algorithms.

Quantum Fine-Grained Lower Bounds for SetDisjointness via Sub-Linear Reductions from 3SUM

from arXiv: Data Structures and Algorithms

Authors: Jeremy Huang, Young Kun Ko, Chunhao Wang

In classical fine-grained complexity, the 3SUM Conjecture is used to prove a variety of conditional lower bounds on data structure and graph problems via an initial reduction to the SetDisjointness problem. However, there is an $\tilde{O}(n)$-time quantum algorithm for 3SUM and a direct application of Grover's algorithm to SetDisjointness queries beats the state-of-the-art classical conditional bound by Kopelowitz, Pettie, and Porat (SODA 2016); this shows that these classical bounds do not apply in the quantum setting. Thus establishing analogous conditional lower bounds in the quantum setting requires applying the quantum 3SUM Conjecture to a \emph{quantum} fine-grained reduction from 3SUM to SetDisjointness. We give the first sub-linear time quantum reductions from 3SUM to online SetDisjointness. Via our reduction, the quantum 3SUM conjecture implies a $p + 2q \geqslant 1$ tradeoff bound for quantum SetDisjointness algorithms with $O(N^p)$ preprocessing time and $O(N^q)$ query time. We also give an analogous reduction from 3XOR. These results are derived from a general framework for fine-grained reductions to SetDisjointness which applies to any Abelian 3-Orthogonal Array (3OA) problem with suitable almost-linear hash functions.

Authors: Jeremy Huang, Young Kun Ko, Chunhao Wang

In classical fine-grained complexity, the 3SUM Conjecture is used to prove a variety of conditional lower bounds on data structure and graph problems via an initial reduction to the SetDisjointness problem. However, there is an $\tilde{O}(n)$-time quantum algorithm for 3SUM and a direct application of Grover's algorithm to SetDisjointness queries beats the state-of-the-art classical conditional bound by Kopelowitz, Pettie, and Porat (SODA 2016); this shows that these classical bounds do not apply in the quantum setting. Thus establishing analogous conditional lower bounds in the quantum setting requires applying the quantum 3SUM Conjecture to a \emph{quantum} fine-grained reduction from 3SUM to SetDisjointness. We give the first sub-linear time quantum reductions from 3SUM to online SetDisjointness. Via our reduction, the quantum 3SUM conjecture implies a $p + 2q \geqslant 1$ tradeoff bound for quantum SetDisjointness algorithms with $O(N^p)$ preprocessing time and $O(N^q)$ query time. We also give an analogous reduction from 3XOR. These results are derived from a general framework for fine-grained reductions to SetDisjointness which applies to any Abelian 3-Orthogonal Array (3OA) problem with suitable almost-linear hash functions.

Verifiable quantum advantage based on polynomials with planted structures

from arXiv: Data Structures and Algorithms

Authors: Markus Bläser, Michael Gullans, Dominik Hangleiter, Yuxuan Liu, Youming Qiao

A central question in the theory of quantum advantage is whether there are quantum advantage protocols with similar resource requirements as random circuit sampling that are also verifiable just from the classical outputs of the quantum computation. Here, we develop the idea of simulation secrets for verifiable advantage. A verifier can use a simulation secret to evaluate a cross-entropy test faster than it would take a classical adversary to pass the test. We instantiate this idea using IQP circuits described by cubic polynomials with planted independent spaces. These correspond to the largest independent set in the orbit of a polynomial under the general linear group and yield a low-rank stabilizer decomposition of the corresponding state. We conjecture that large independent spaces are invisible to a computationally bounded adversary, and therefore they cannot exploit them to pass the protocol. A second conjecture regards the fine-grained complexity of producing samples that pass the cross-entropy test for uniformly random polynomials. Under these conjectures, our scheme results in a polynomial gap between the verification time and the time a classical adversary would need to pass the protocol---both are exponential. It has a potential application to generating classically certifiable randomness, since the output distributions have high min-entropy. We estimate that the planted polynomial scheme is implementable using 100 logical qubits at logical error rates around $10^{-6}$.

Authors: Markus Bläser, Michael Gullans, Dominik Hangleiter, Yuxuan Liu, Youming Qiao

A central question in the theory of quantum advantage is whether there are quantum advantage protocols with similar resource requirements as random circuit sampling that are also verifiable just from the classical outputs of the quantum computation. Here, we develop the idea of simulation secrets for verifiable advantage. A verifier can use a simulation secret to evaluate a cross-entropy test faster than it would take a classical adversary to pass the test. We instantiate this idea using IQP circuits described by cubic polynomials with planted independent spaces. These correspond to the largest independent set in the orbit of a polynomial under the general linear group and yield a low-rank stabilizer decomposition of the corresponding state. We conjecture that large independent spaces are invisible to a computationally bounded adversary, and therefore they cannot exploit them to pass the protocol. A second conjecture regards the fine-grained complexity of producing samples that pass the cross-entropy test for uniformly random polynomials. Under these conjectures, our scheme results in a polynomial gap between the verification time and the time a classical adversary would need to pass the protocol---both are exponential. It has a potential application to generating classically certifiable randomness, since the output distributions have high min-entropy. We estimate that the planted polynomial scheme is implementable using 100 logical qubits at logical error rates around $10^{-6}$.

Polynomial-time algorithm for exact $(1,2)$-center problem under continuous Fréchet distance

from arXiv: Data Structures and Algorithms

Authors: Soumya Bhattacharya, Serene Rasheed, Sasanka Roy

In this paper, we explore the $(1,2)$-center problem for polygonal curves under continuous Fréchet distance. The $(k,\ell)$-center problem, in general, is known to be NP-hard. Aronov, Filtser, Horton, Katz, and Sheikhan (WADS'19) gave a polynomial-time algorithm for the $(1,2)$-center of curves in the plane under the discrete Fréchet distance. To the best of our knowledge, the $(1,2)$-center under continuous Fréchet distance has not been studied yet. We present a polynomial time algorithm to solve the problem exactly under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm, running in $O\bigr((n^2r+nr^2)^{2+ε}\bigl)$ time for curves in the plane where $r$ is the number of input curves and $n$ is the maximum complexity of any curve. Further, for curves in any dimension $d$, the expected time to compute the center using the algorithm is $O\bigr((n^2r+nr^2)^{2(d-1)+ε}\bigl)$. We have also shown that an $(1+\widetildeε)-$factor approximation of $(1,2)$-center can be computed in $O(n^2r+nr^2+1/ε^s)$ time for any $ε>\widetildeε>0$ and some constant $s$ for curves in the plane. For curves in the plane, we have shown that, with the center restricted to be horizontal, we can compute the exact center in $O(n^2r+nr^2)$ time. An algorithm has been introduced to find a $3$-factor approximation of the $(1,2)$-center in time linear in the number of curves. A formulation was introduced by de Berg, Mehrabi, and Ophelders (CCCG'17) to measure Fréchet distance between a curve and a query segment under $\mathbb{L}_2$ norm for curves in the plane. We have shown the formulation is valid under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm for curves in $\mathbb{R}^d$.

Authors: Soumya Bhattacharya, Serene Rasheed, Sasanka Roy

In this paper, we explore the $(1,2)$-center problem for polygonal curves under continuous Fréchet distance. The $(k,\ell)$-center problem, in general, is known to be NP-hard. Aronov, Filtser, Horton, Katz, and Sheikhan (WADS'19) gave a polynomial-time algorithm for the $(1,2)$-center of curves in the plane under the discrete Fréchet distance. To the best of our knowledge, the $(1,2)$-center under continuous Fréchet distance has not been studied yet. We present a polynomial time algorithm to solve the problem exactly under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm, running in $O\bigr((n^2r+nr^2)^{2+ε}\bigl)$ time for curves in the plane where $r$ is the number of input curves and $n$ is the maximum complexity of any curve. Further, for curves in any dimension $d$, the expected time to compute the center using the algorithm is $O\bigr((n^2r+nr^2)^{2(d-1)+ε}\bigl)$. We have also shown that an $(1+\widetildeε)-$factor approximation of $(1,2)$-center can be computed in $O(n^2r+nr^2+1/ε^s)$ time for any $ε>\widetildeε>0$ and some constant $s$ for curves in the plane. For curves in the plane, we have shown that, with the center restricted to be horizontal, we can compute the exact center in $O(n^2r+nr^2)$ time. An algorithm has been introduced to find a $3$-factor approximation of the $(1,2)$-center in time linear in the number of curves. A formulation was introduced by de Berg, Mehrabi, and Ophelders (CCCG'17) to measure Fréchet distance between a curve and a query segment under $\mathbb{L}_2$ norm for curves in the plane. We have shown the formulation is valid under both $\mathbb{L}_2$ and $\mathbb{L}_\infty$ norm for curves in $\mathbb{R}^d$.

Exponential quantum speedup for $\mathbb{F}_3^n$-Subset-Sum? Or, rigorous classical algorithms for Binary-Error LWE

from arXiv: Data Structures and Algorithms

Authors: Robin Kothari, Tony Metger, Ryan O'Donnell, Noah Shutty, Kewen Wu

We study vector subset sum over $\mathbb{F}_3^n$: given $m$ random vectors from $\mathbb{F}_3^n$, find a nonempty subset that sums to zero; the smaller $m$, the more difficult it is to find such a subset. Chen, Liu, and Zhandry (EUROCRYPT'22) introduced an efficient quantum algorithm that solves this problem when $m\approx n^2/2$, where a naive classical algorithm would require exponential time. Subsequently, Kothari, O'Donnell, and Wu (STOC'2026) gave an efficient classical algorithm that only requires $m \approx n^2/3$ vectors, thus removing the hope for an exponential quantum advantage in this parameter regime. Using the framework of Chen, Liu, and Zhandry, we give quantum algorithms that require much fewer input vectors, renewing the possibility of an exponential quantum speedup: for any fixed $ε>0$, our quantum algorithm solves $\mathbb{F}_3$-subset sum in polynomial time with $m=ε\cdot n^2$ vectors. More generally, we establish a full sample--time tradeoff that interpolates between exponential and polynomial runtime. The main ingredient is a deterministic classical algorithm for the binary-error Learning-with-Errors problem, which is of independent cryptographic interest. For this, we rigorously establish a sample--time tradeoff that was predicted by earlier algebraic heuristics. For vector subset sums over larger fields, we also significantly improve classical algorithms in Kothari, O'Donnell, and Wu (STOC'2026).

Authors: Robin Kothari, Tony Metger, Ryan O'Donnell, Noah Shutty, Kewen Wu

We study vector subset sum over $\mathbb{F}_3^n$: given $m$ random vectors from $\mathbb{F}_3^n$, find a nonempty subset that sums to zero; the smaller $m$, the more difficult it is to find such a subset. Chen, Liu, and Zhandry (EUROCRYPT'22) introduced an efficient quantum algorithm that solves this problem when $m\approx n^2/2$, where a naive classical algorithm would require exponential time. Subsequently, Kothari, O'Donnell, and Wu (STOC'2026) gave an efficient classical algorithm that only requires $m \approx n^2/3$ vectors, thus removing the hope for an exponential quantum advantage in this parameter regime. Using the framework of Chen, Liu, and Zhandry, we give quantum algorithms that require much fewer input vectors, renewing the possibility of an exponential quantum speedup: for any fixed $ε>0$, our quantum algorithm solves $\mathbb{F}_3$-subset sum in polynomial time with $m=ε\cdot n^2$ vectors. More generally, we establish a full sample--time tradeoff that interpolates between exponential and polynomial runtime. The main ingredient is a deterministic classical algorithm for the binary-error Learning-with-Errors problem, which is of independent cryptographic interest. For this, we rigorously establish a sample--time tradeoff that was predicted by earlier algebraic heuristics. For vector subset sums over larger fields, we also significantly improve classical algorithms in Kothari, O'Donnell, and Wu (STOC'2026).

Solving Sparse SDPs in Sublinear Time: A Classical Algorithm Inspired by the Quantum OR Lemma

from arXiv: Data Structures and Algorithms

Authors: Fernando G. S. L. Brandão, Alexander M. Dalzell, András Gilyén, Francisca Vasconcelos

We give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved $\widetilde{O}(\sqrt{n}+\sqrt{m})$ dependence on matrix dimension $n$ and constraint number $m$. Compared with the $\widetilde{O}(mn)$ runtime of existing classical methods, this suggests a quartic quantum speedup when $m \approx n$. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs. Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let $s$ denote the the input matrix sparsity and $γ:=Rr/\varepsilon$ capture dependence on the primal $(R)$ and dual $(r)$ radii as well as target accuracy $(\varepsilon)$. When $γ^2\leq\min\{m,n/s\}$, our solver runs in time $\widetilde{O}\left(nsγ^{4.5}+msγ^2\right)$. For $γ=O(1)$, this is $\widetilde{O}\left((n+m)s\right)$ and sublinear in the $O(mns)$ input size. Similar to the quantum algorithms, this matches known lower bounds with respect to $m$ and $n$, up to logarithmic factors. This implies that, with respect to dimensions $m$ and $n$, there is no super-quadratic quantum advantage for generic sparse SDP solving.

Authors: Fernando G. S. L. Brandão, Alexander M. Dalzell, András Gilyén, Francisca Vasconcelos

We give the first sublinear-time classical solvers for sparse semidefinite programs in the bounded-radius regime, without low-rank assumptions or Frobenius norm dependence on the constraint matrices. For constant precision and bounded primal and dual radii, prior quantum algorithms of Brandão et al. (2019) and van Apeldoorn and Gilyén (2019) achieved $\widetilde{O}(\sqrt{n}+\sqrt{m})$ dependence on matrix dimension $n$ and constraint number $m$. Compared with the $\widetilde{O}(mn)$ runtime of existing classical methods, this suggests a quartic quantum speedup when $m \approx n$. Beyond a usual Grover speedup, this separation relies on the Quantum OR lemma, whose sample-reuse mechanism decouples the cost of Gibbs-state preparation from constraint search. We show that this reuse mechanism is classically realizable for sparse SDPs. Our main technical contribution is a classical procedure for simultaneously estimating many expectation values with respect to a sparse Hamiltonian's Gibbs state. This combines randomized Lánczos filtering with an efficient sampling-based estimator. We also introduce a stochastic online-learning framework for SDP solving, substantially improving accuracy-dependence over standard oracle-based MMWU approaches. Let $s$ denote the the input matrix sparsity and $γ:=Rr/\varepsilon$ capture dependence on the primal $(R)$ and dual $(r)$ radii as well as target accuracy $(\varepsilon)$. When $γ^2\leq\min\{m,n/s\}$, our solver runs in time $\widetilde{O}\left(nsγ^{4.5}+msγ^2\right)$. For $γ=O(1)$, this is $\widetilde{O}\left((n+m)s\right)$ and sublinear in the $O(mns)$ input size. Similar to the quantum algorithms, this matches known lower bounds with respect to $m$ and $n$, up to logarithmic factors. This implies that, with respect to dimensions $m$ and $n$, there is no super-quadratic quantum advantage for generic sparse SDP solving.

Mixing FM-indexes and CSAs: backward search over an order-1 rank encoding

from arXiv: Data Structures and Algorithms

Authors: Travis Gagie

FM-indexes and compressed suffix arrays (CSAs) are often treated as interchangeable, but they behave differently as the alphabet grows. An FM-index step costs about one cache miss per level of a wavelet tree, so it gets slower with the alphabet size. A CSA step is a binary search whose range shrinks as characters get rarer. We describe a simple hybrid. Each character of the text is replaced by the rank of its frequency among the characters that follow the previous character. We backward-search on this encoding, which is over a small, skewed alphabet, and recover the one piece of information the encoding loses (the first character of the pattern) with a single CSA-like step on an array we call $\PsiE$. Counting is exact, and locating works with standard suffix-array sampling. A prototype on synthetic repetitive data shows that the hybrid is the fastest of the indexes we tried at intermediate alphabet sizes with 1\% noise, but even its compact version is 1.7 to 3.9 times larger than a compressed run-length CSA or FM-index of the original text, because the encoding and $\PsiE$ together have more runs than the original Burrows--Wheeler transform. Whether that changes on real data, such as parses and minimizer digests, is the main open question.

Authors: Travis Gagie

FM-indexes and compressed suffix arrays (CSAs) are often treated as interchangeable, but they behave differently as the alphabet grows. An FM-index step costs about one cache miss per level of a wavelet tree, so it gets slower with the alphabet size. A CSA step is a binary search whose range shrinks as characters get rarer. We describe a simple hybrid. Each character of the text is replaced by the rank of its frequency among the characters that follow the previous character. We backward-search on this encoding, which is over a small, skewed alphabet, and recover the one piece of information the encoding loses (the first character of the pattern) with a single CSA-like step on an array we call $\PsiE$. Counting is exact, and locating works with standard suffix-array sampling. A prototype on synthetic repetitive data shows that the hybrid is the fastest of the indexes we tried at intermediate alphabet sizes with 1\% noise, but even its compact version is 1.7 to 3.9 times larger than a compressed run-length CSA or FM-index of the original text, because the encoding and $\PsiE$ together have more runs than the original Burrows--Wheeler transform. Whether that changes on real data, such as parses and minimizer digests, is the main open question.

Conditioning-Free Non-Uniform Quantum Fourier and Chebyshev Transforms

from arXiv: Data Structures and Algorithms

Authors: Chaowen Guan, Akshit Katiyar

We present an efficient quantum algorithm for the non-uniform Chebyshev transform. It is defined as the projection of a function onto Chebyshev polynomials sampled at given nodes that are uniform in $x\in[-1,1]$, and hence non-uniform in the angle $θ=\arccos x$, a setting that QFT-based quantum Chebyshev transforms cannot handle. Our construction is based on an improvement of an existing Non-uniform Quantum Fourier Transform (NUQFT) whereby we remove the conditioning from non-uniform node sampling. Hence, error bounds are independent of the geometry-dependent parameter $κ$ of prior work. We use the fact that Chebyshev transform matrix is the average of two Type-II non-uniform DFTs, which we implement with a single controlled NUQFT circuit. We provide explicit oracle constructions, including the row-access oracle previously left as an assumption. The resulting $\varepsilon$-accurate block encoding has $O(1)$ normalization and uses $O(L)$ qubits and $\widetilde O(L^2)$ gates, where $L=\log N+\log(1/\varepsilon)$. We give an end-to-end implementation with complexity analysis, including the success probability and output-state error.

Authors: Chaowen Guan, Akshit Katiyar

We present an efficient quantum algorithm for the non-uniform Chebyshev transform. It is defined as the projection of a function onto Chebyshev polynomials sampled at given nodes that are uniform in $x\in[-1,1]$, and hence non-uniform in the angle $θ=\arccos x$, a setting that QFT-based quantum Chebyshev transforms cannot handle. Our construction is based on an improvement of an existing Non-uniform Quantum Fourier Transform (NUQFT) whereby we remove the conditioning from non-uniform node sampling. Hence, error bounds are independent of the geometry-dependent parameter $κ$ of prior work. We use the fact that Chebyshev transform matrix is the average of two Type-II non-uniform DFTs, which we implement with a single controlled NUQFT circuit. We provide explicit oracle constructions, including the row-access oracle previously left as an assumption. The resulting $\varepsilon$-accurate block encoding has $O(1)$ normalization and uses $O(L)$ qubits and $\widetilde O(L^2)$ gates, where $L=\log N+\log(1/\varepsilon)$. We give an end-to-end implementation with complexity analysis, including the success probability and output-state error.

Quantum oblique eigenprojection

from arXiv: Data Structures and Algorithms

Authors: Alexander M. Dalzell, Yuan Su

Every square matrix decomposes its underlying Hilbert space into generalized, nonorthogonal eigensubspaces. We show that a quantum computer can perform such an oblique eigenprojection $Π$ given block encoding access to the input matrix. Our approach has a query complexity nearly linear in the inverse gap and a normalization factor close to $\lVertΠ\rVert$ under a spectral-set condition on the input. This covers common assumptions on the numerical range or diagonalizability and matches known results for orthogonal eigenprojections. We achieve this with a two-sided block preconditioning that uses a discrete Fourier transform of the matrix resolvent. We describe applications to: (i) preparing eigenstates of matrices with complex eigenvalues, extending the quantum eigenvalue transformation algorithm of Low and Su beyond real spectra; (ii) solving continuous-time algebraic Riccati equations, cubically speeding up a prior solver of Rodenas-Ruiz, Zhao, and Lee; and (iii) solving ordinary Sylvester equations, quadratically improving a direct augmented method of Wang and Liu. Our result suggests a promising route to applying nonanalytic matrix functions on quantum computers.

Authors: Alexander M. Dalzell, Yuan Su

Every square matrix decomposes its underlying Hilbert space into generalized, nonorthogonal eigensubspaces. We show that a quantum computer can perform such an oblique eigenprojection $Π$ given block encoding access to the input matrix. Our approach has a query complexity nearly linear in the inverse gap and a normalization factor close to $\lVertΠ\rVert$ under a spectral-set condition on the input. This covers common assumptions on the numerical range or diagonalizability and matches known results for orthogonal eigenprojections. We achieve this with a two-sided block preconditioning that uses a discrete Fourier transform of the matrix resolvent. We describe applications to: (i) preparing eigenstates of matrices with complex eigenvalues, extending the quantum eigenvalue transformation algorithm of Low and Su beyond real spectra; (ii) solving continuous-time algebraic Riccati equations, cubically speeding up a prior solver of Rodenas-Ruiz, Zhao, and Lee; and (iii) solving ordinary Sylvester equations, quadratically improving a direct augmented method of Wang and Liu. Our result suggests a promising route to applying nonanalytic matrix functions on quantum computers.

Dynamic Time Warping in the Low-Distance Regime

from arXiv: Data Structures and Algorithms

Authors: Itai Boneh, Shay Golan, Tomasz Kociumaka

Dynamic Time Warping (DTW) is a classical similarity measure for strings and time series that allows local stretching. Given non-empty strings $S,T$ over an alphabet $Σ$ and a cost function $δ:Σ^2\to\mathbb{R}_{\ge0}$, $DTW_δ(S,T)$ is the minimum total cost of equal-length expansions of $S$ and $T$ obtained by duplicating characters. For strings of length at most $n$, DTW is computable in $O(n^2)$ time, and this is conditionally optimal under the Orthogonal Vectors Hypothesis (OVH). We study the low-distance regime, where an integer $k$ upper-bounds $DTW_δ(S,T)$, assuming $δ(a,a)=0$ and $δ(a,b)\ge1$ for $a\ne b$. For several classical similarity measures, this regime admits $O(n+\operatorname{poly}(k))$ algorithms, whereas for DTW with metric costs the best known bound is $O(nk)$. We show that this dependence is essentially optimal: assuming OVH, computing DTW requires $n^{1-o(1)}k$ time even for the discrete mismatch-cost function, which assigns cost $1$ to every mismatch. The lower bound applies to the whole spectrum of thresholds $k$ between constant and linear in $n$. Our reduction from Orthogonal Vectors encodes vector coordinates in the lengths of equal-character runs. The resulting instances are very structured: collapsing runs to single characters reveals long substrings with short periods. We complement the lower bound with a $\tilde O(n+\operatorname{poly}(k))$-time algorithm whenever, after collapsing runs in the inputs, every substring with period $O(k)$ has length $\operatorname{poly}(k)$. Finally, we extend this lower bound to DTW pattern matching, which asks whether any non-empty substring of a length-$n$ text has DTW distance at most $k$ from a length-$m$ pattern. We prove that the classic $O(nm)$-time dynamic-programming algorithm is near-optimal under OVH, even when $k=O(\log n)$.

Authors: Itai Boneh, Shay Golan, Tomasz Kociumaka

Dynamic Time Warping (DTW) is a classical similarity measure for strings and time series that allows local stretching. Given non-empty strings $S,T$ over an alphabet $Σ$ and a cost function $δ:Σ^2\to\mathbb{R}_{\ge0}$, $DTW_δ(S,T)$ is the minimum total cost of equal-length expansions of $S$ and $T$ obtained by duplicating characters. For strings of length at most $n$, DTW is computable in $O(n^2)$ time, and this is conditionally optimal under the Orthogonal Vectors Hypothesis (OVH). We study the low-distance regime, where an integer $k$ upper-bounds $DTW_δ(S,T)$, assuming $δ(a,a)=0$ and $δ(a,b)\ge1$ for $a\ne b$. For several classical similarity measures, this regime admits $O(n+\operatorname{poly}(k))$ algorithms, whereas for DTW with metric costs the best known bound is $O(nk)$. We show that this dependence is essentially optimal: assuming OVH, computing DTW requires $n^{1-o(1)}k$ time even for the discrete mismatch-cost function, which assigns cost $1$ to every mismatch. The lower bound applies to the whole spectrum of thresholds $k$ between constant and linear in $n$. Our reduction from Orthogonal Vectors encodes vector coordinates in the lengths of equal-character runs. The resulting instances are very structured: collapsing runs to single characters reveals long substrings with short periods. We complement the lower bound with a $\tilde O(n+\operatorname{poly}(k))$-time algorithm whenever, after collapsing runs in the inputs, every substring with period $O(k)$ has length $\operatorname{poly}(k)$. Finally, we extend this lower bound to DTW pattern matching, which asks whether any non-empty substring of a length-$n$ text has DTW distance at most $k$ from a length-$m$ pattern. We prove that the classic $O(nm)$-time dynamic-programming algorithm is near-optimal under OVH, even when $k=O(\log n)$.

Shadow Quantum Singular Value Transformation with Shallow Quantum Circuits

from arXiv: Data Structures and Algorithms

Authors: Nai-Hui Chia, Hyunseong Kim, Chia-Ying Lin

We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state $|ψ\rangle$, a Hermitian matrix $H$, a polynomial $f$, and a set of observables $\{O_1,\dots,O_m\}$, the goal is to estimate $\langleψ|f(H)^{\dagger}O_j f(H)|ψ\rangle$ for all $j\in\{1,\dots,m\}$. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of $f(H)$. It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary. We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with $H$ and $|ψ\rangle$ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from $H$, $f$, and $|ψ\rangle$ without prior knowledge of the observables or explicit preparation of the target state proportional to $f(H)|ψ\rangle$. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.

Authors: Nai-Hui Chia, Hyunseong Kim, Chia-Ying Lin

We introduce shadow quantum singular value transformation (Shadow QSVT): given an initial state $|ψ\rangle$, a Hermitian matrix $H$, a polynomial $f$, and a set of observables $\{O_1,\dots,O_m\}$, the goal is to estimate $\langleψ|f(H)^{\dagger}O_j f(H)|ψ\rangle$ for all $j\in\{1,\dots,m\}$. Shadow QSVT provides a systematic route to reduce the quantum resources required by standard QSVT, which constructs a unitary block-encoding of $f(H)$. It uses structure in the input state and observables, together with the fact that many applications require only observable estimates rather than synthesizing the full unitary. We present three algorithms that exploit structure in the initial state and observables to reduce quantum circuit depth. First, we develop a state-aware QSVT algorithm that prepares the target state with low circuit depth when the Krylov subspace associated with $H$ and $|ψ\rangle$ is low-dimensional or admits an accurate low-dimensional approximation. Second, we introduce an observable-aware Shadow QSVT algorithm that combines a new observable-aware Krylov subspace with history states to further reduce circuit depth and gate complexity. Finally, we develop Classical Shadow QSVT, which constructs a classical representation from $H$, $f$, and $|ψ\rangle$ without prior knowledge of the observables or explicit preparation of the target state proportional to $f(H)|ψ\rangle$. This representation enables estimation of the target quantities for observables specified after the quantum computation. Together, these three algorithms provide tools for reducing the circuit depth of QSVT-based computations across a range of settings.

Policy Iteration Is Not Strongly Polynomial for Deterministic Markov Decision Processes: The Price of Algorithmic Anarchy

from arXiv: Data Structures and Algorithms

Authors: Han Zhong, Yinyu Ye

We establish an exponential iteration lower bound in the number of states for Howard's policy iteration on deterministic discounted Markov decision processes, with at most two actions per state. This rules out strong polynomiality of Howard's policy iteration when the discount factor is part of the input and yields an exponential separation from the simplex method with Dantzig's pivoting rule, which is proved to be strongly polynomial on this class. Even when each reward is restricted to logarithmic bit length, we obtain a stretched-exponential iteration lower bound. The gap between Howard's decentralized and simultaneous selfish improvements and Dantzig's coordinated selection of a single action with the largest gain across all states reveals a ``price'' of algorithmic anarchy.

Authors: Han Zhong, Yinyu Ye

We establish an exponential iteration lower bound in the number of states for Howard's policy iteration on deterministic discounted Markov decision processes, with at most two actions per state. This rules out strong polynomiality of Howard's policy iteration when the discount factor is part of the input and yields an exponential separation from the simplex method with Dantzig's pivoting rule, which is proved to be strongly polynomial on this class. Even when each reward is restricted to logarithmic bit length, we obtain a stretched-exponential iteration lower bound. The gap between Howard's decentralized and simultaneous selfish improvements and Dantzig's coordinated selection of a single action with the largest gain across all states reveals a ``price'' of algorithmic anarchy.

Sharper Gaussian Covers in the Bansal--Huang--Lee Coloring Framework

from arXiv: Data Structures and Algorithms

Authors: Konstantin Makarychev, Yury Makarychev

Bansal, Huang, and Lee recently gave a polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.19539})$ colors. We improve their bound to $O(n^{0.17794})$ colors. The improvement comes from a sharper rule for combining Gaussian covers. We prove the rule from Ehrhard's inequality by comparing the probability that a Gaussian vector misses a polyhedron at two thresholds. At the farther threshold a union bound records how many halfspaces define the polyhedron, and this information reduces the loss in the combination. We then organize the successive neighborhood steps of the Bansal-Huang-Lee argument into a single recursion. The smaller loss lets the recursion run one step farther than before, and at that step it would require more distinct vertices than the graph contains. This rules out the remaining case. Combining the resulting bounded-degree guarantee with the dense-graph algorithm of Kawarabayashi, Thorup, and Yoneda gives a randomized polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.17794})$ colors.

Authors: Konstantin Makarychev, Yury Makarychev

Bansal, Huang, and Lee recently gave a polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.19539})$ colors. We improve their bound to $O(n^{0.17794})$ colors. The improvement comes from a sharper rule for combining Gaussian covers. We prove the rule from Ehrhard's inequality by comparing the probability that a Gaussian vector misses a polyhedron at two thresholds. At the farther threshold a union bound records how many halfspaces define the polyhedron, and this information reduces the loss in the combination. We then organize the successive neighborhood steps of the Bansal-Huang-Lee argument into a single recursion. The smaller loss lets the recursion run one step farther than before, and at that step it would require more distinct vertices than the graph contains. This rules out the remaining case. Combining the resulting bounded-degree guarantee with the dense-graph algorithm of Kawarabayashi, Thorup, and Yoneda gives a randomized polynomial-time algorithm that colors every $3$-colorable graph on $n$ vertices with $O(n^{0.17794})$ colors.

Super-Quadratic Quantum Speedups for Combinatorial Optimization via Tilted Walks

from arXiv: Data Structures and Algorithms

Authors: Guneykan Ozgul, Shouvanik Chakrabarti

We introduce quantum tilted walks, a quantum algorithmic framework for solving exact combinatorial optimization problems. The framework applies an average of powers of a tilted Hamiltonian that biases the discriminant matrix of a base Markov chain (mixer) with the objective function. Our starting point is quantum short-path algorithms, which prepare the ground state of such a Hamiltonian and obtain super-quadratic speedups over exhaustive search for certain combinatorial optimization problems. Recently, Le Gall and Tamaki~(arXiv:2604.12131) developed a classical conditioning-and-search algorithm for weighted MAX-E$k$-LIN2 and weighted MAX-$k$-CSP. Under the same assumptions, their algorithm is only sub-quadratically slower than quantum short-path algorithms. Consequently, existing short-path algorithms do not establish a super-quadratic speedup over this stronger classical baseline. For maximization problems, we give conditions under which tilted walks increase the amplitude on the target state with high objective value when initialized from a starting state with lower objective value. This framework captures conditioning-and-search and yields super-quadratic speedups over it for the same problems. While our framework recovers quantum short-path algorithms as a special case, it neither requires ground-state preparation nor initialization in the ground state of the base mixer. We demonstrate these advantages on a synthetic optimization problem for which tilted walks achieve a super-quadratic speedup whereas the short-path algorithms do not.

Authors: Guneykan Ozgul, Shouvanik Chakrabarti

We introduce quantum tilted walks, a quantum algorithmic framework for solving exact combinatorial optimization problems. The framework applies an average of powers of a tilted Hamiltonian that biases the discriminant matrix of a base Markov chain (mixer) with the objective function. Our starting point is quantum short-path algorithms, which prepare the ground state of such a Hamiltonian and obtain super-quadratic speedups over exhaustive search for certain combinatorial optimization problems. Recently, Le Gall and Tamaki~(arXiv:2604.12131) developed a classical conditioning-and-search algorithm for weighted MAX-E$k$-LIN2 and weighted MAX-$k$-CSP. Under the same assumptions, their algorithm is only sub-quadratically slower than quantum short-path algorithms. Consequently, existing short-path algorithms do not establish a super-quadratic speedup over this stronger classical baseline. For maximization problems, we give conditions under which tilted walks increase the amplitude on the target state with high objective value when initialized from a starting state with lower objective value. This framework captures conditioning-and-search and yields super-quadratic speedups over it for the same problems. While our framework recovers quantum short-path algorithms as a special case, it neither requires ground-state preparation nor initialization in the ground state of the base mixer. We demonstrate these advantages on a synthetic optimization problem for which tilted walks achieve a super-quadratic speedup whereas the short-path algorithms do not.

Robust and Learned Online Matching in Growing Trees

from arXiv: Data Structures and Algorithms

Authors: Marek Gałązka, Hanna Wdowicka

We study irrevocable maximum-cardinality matching in trees revealed by successive leaf attachments, with a known horizon and an exogenous growth law that is misspecified or unknown. For deterministic affine attachment forecasts with nonnegative degree reinforcement, the optimal threshold policy loses at most twice the cumulative expected conditional total-variation error relative to an online oracle knowing the actual growth law. This follows from a unit-span property of the Bellman continuation score and has no additional horizon factor. A four-vertex example attains the coefficient two for the specified deterministic policy, and a two-model argument gives a lower bound linear in the model-error budget for arbitrary policies under general misspecification. For uniform-preferential attachment, the local error has an exact expression through the leaf count. When its constant mixture parameter is unknown, we estimate it from the same growing tree and update the threshold policy at geometric times. A parameter-sensitivity bound for individual Bellman prices and uniform degree-moment estimates yield expected regret $O(\sqrt{n}\log^2 n)$, using $O(n^2\log n)$ arithmetic operations and $O(n)$ stored entries. The exact minimax rate remains open.

Authors: Marek Gałązka, Hanna Wdowicka

We study irrevocable maximum-cardinality matching in trees revealed by successive leaf attachments, with a known horizon and an exogenous growth law that is misspecified or unknown. For deterministic affine attachment forecasts with nonnegative degree reinforcement, the optimal threshold policy loses at most twice the cumulative expected conditional total-variation error relative to an online oracle knowing the actual growth law. This follows from a unit-span property of the Bellman continuation score and has no additional horizon factor. A four-vertex example attains the coefficient two for the specified deterministic policy, and a two-model argument gives a lower bound linear in the model-error budget for arbitrary policies under general misspecification. For uniform-preferential attachment, the local error has an exact expression through the leaf count. When its constant mixture parameter is unknown, we estimate it from the same growing tree and update the threshold policy at geometric times. A parameter-sensitivity bound for individual Bellman prices and uniform degree-moment estimates yield expected regret $O(\sqrt{n}\log^2 n)$, using $O(n^2\log n)$ arithmetic operations and $O(n)$ stored entries. The exact minimax rate remains open.

Component-Weighted Centroid Search for Exact Incremental BPE

from arXiv: Data Structures and Algorithms

Authors: Harshit Verma, Rex Ying

Exact incremental BPE maintains the canonical tokenization state after every appended byte. The recent algorithm of Jiang and Gong (2026) does this in $O(\log^2 t)$ worst-case time, where $t$ is the maximum canonical token length. Its centroid search visits $O(\log t)$ components and can pay another $O(\log t)$ for ordered point location at each one. Within Jiang and Gong's normalized/proper merge-stage model, we change only that local search. Each interval is weighted by the size of the recursive component it selects, so a move from size $m$ to size $m'$ costs $O(1+\log(m/m'))$. These charges telescope, giving $O(\log t)$ time per append and $O(n\log t)$ over an $n$-byte stream, with the same BPE semantics and asymptotic space. We also construct a normalized proper BPE family over a fixed alphabet where count-balanced search uses $Θ(\log^2 t)$ probes on a reachable update, while the weighted search uses $Θ(\log t)$. A Rust implementation matches the predicted probe counts on every tested instance. On ordinary vocabularies the queried degrees are small, however, and the improvement is a worst-case guarantee rather than an average-speed result.

Authors: Harshit Verma, Rex Ying

Exact incremental BPE maintains the canonical tokenization state after every appended byte. The recent algorithm of Jiang and Gong (2026) does this in $O(\log^2 t)$ worst-case time, where $t$ is the maximum canonical token length. Its centroid search visits $O(\log t)$ components and can pay another $O(\log t)$ for ordered point location at each one. Within Jiang and Gong's normalized/proper merge-stage model, we change only that local search. Each interval is weighted by the size of the recursive component it selects, so a move from size $m$ to size $m'$ costs $O(1+\log(m/m'))$. These charges telescope, giving $O(\log t)$ time per append and $O(n\log t)$ over an $n$-byte stream, with the same BPE semantics and asymptotic space. We also construct a normalized proper BPE family over a fixed alphabet where count-balanced search uses $Θ(\log^2 t)$ probes on a reachable update, while the weighted search uses $Θ(\log t)$. A Rust implementation matches the predicted probe counts on every tested instance. On ordinary vocabularies the queried degrees are small, however, and the improvement is a worst-case guarantee rather than an average-speed result.

Learning Random Quantum Circuits and the Emergence of Pseudorandomness

from arXiv: Data Structures and Algorithms

Authors: Srinivasan Arunachalam, Qizhao Huang, Makrand Sinha

We give an efficient algorithm for learning $k$-dimensional brickwork random quantum circuits using only copies of the output state obtained by applying $U$ to the all-zero input. For a depth-$d$ circuit on $n$ sites with random $2\ell$-qubit gates, the algorithm learns the original circuit $U$ with high probability in $\text{poly}(n,2^{\ell d})$ time for every constant dimensional lattice. In particular, the algorithm is polynomial time as long as $\ell d = O(\log n)$. In one dimension, this reaches the natural boundary suggested by pseudorandomness: pseudorandom states require $\ell d=ω(\log n)$, and structured cryptographic constructions suggest that this scale may be achievable from above. In higher dimensions, it remains plausible that the same $\ell d=ω(\log n)$ scale remains the threshold for ancilla-free pseudorandomness, and our results help clarify the conditions under which pseudorandomness can arise in this setting. The main idea behind our algorithm is a local correlation criterion that identifies gates in the final layer without learning their entire backward light cones, avoiding a bottleneck in the previous approaches. A key technical ingredient is a Carbery-Wright type anticoncentration inequality for low-degree polynomials of Haar random unitaries whose small-ball exponent is independent of the matrix dimension. The dimension-independent exponent is crucial for handling gates of growing locality. This anticoncentration result may also be of independent interest.

Authors: Srinivasan Arunachalam, Qizhao Huang, Makrand Sinha

We give an efficient algorithm for learning $k$-dimensional brickwork random quantum circuits using only copies of the output state obtained by applying $U$ to the all-zero input. For a depth-$d$ circuit on $n$ sites with random $2\ell$-qubit gates, the algorithm learns the original circuit $U$ with high probability in $\text{poly}(n,2^{\ell d})$ time for every constant dimensional lattice. In particular, the algorithm is polynomial time as long as $\ell d = O(\log n)$. In one dimension, this reaches the natural boundary suggested by pseudorandomness: pseudorandom states require $\ell d=ω(\log n)$, and structured cryptographic constructions suggest that this scale may be achievable from above. In higher dimensions, it remains plausible that the same $\ell d=ω(\log n)$ scale remains the threshold for ancilla-free pseudorandomness, and our results help clarify the conditions under which pseudorandomness can arise in this setting. The main idea behind our algorithm is a local correlation criterion that identifies gates in the final layer without learning their entire backward light cones, avoiding a bottleneck in the previous approaches. A key technical ingredient is a Carbery-Wright type anticoncentration inequality for low-degree polynomials of Haar random unitaries whose small-ball exponent is independent of the matrix dimension. The dimension-independent exponent is crucial for handling gates of growing locality. This anticoncentration result may also be of independent interest.

Connected Dominating Set on Semi-Ladder-Free Graphs

from arXiv: Data Structures and Algorithms

Authors: Sobyasachi Chatterjee, Sushmita Gupta, Saket Saurabh, Sanjay Seetharaman, Anannya Upasana

We study \textsc{Connected Dominating Set} on graphs whose closed-neighborhood set systems are $d$-semi-ladder-free. This structural condition strictly generalizes the biclique-free setting and provides a natural regime for connectivity-constrained domination. We obtain both a fixed-parameter algorithm and an approximate kernelization framework for the problem on this class. Our algorithmic result is based on a new compact representation theorem for inclusion-wise minimal set covers in $d$-semi-ladder-free set systems. Although the number of minimal set covers of size at most $k$ may be as large as $n^{Ω(k)}$, we show that all such set covers can nevertheless be encoded by a family of at most $k^{kd+1}$ tuples, and that this family can be enumerated in time $\Oh(k^{kd+2}\cdot nm)$. Combining this representation with a \textsc{Group Steiner Tree} subroutine, we obtain an algorithm for \textsc{Connected Set Cover}, which in turn yields an algorithm for \textsc{Connected Dominating Set} running in time $k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$ and polynomial space. For the preprocessing result, we introduce grouped domination cores and dominator cores, and prove polynomial upper bounds on their sizes in $d$-semi-ladder-free graphs. Using these structures, we obtain, for every fixed $d$ and $\varepsilon>0$, a polynomial-time $(1+\varepsilon)$-lossy compression for \textsc{Connected Dominating Set} to an equivalent reduced instance of size $k^{\Oh(d^2/\varepsilon)}$. The reduced instance is a \textsc{Connected Dominating Set} instance on a $(d+2)$-semi-ladder-free graph.

Authors: Sobyasachi Chatterjee, Sushmita Gupta, Saket Saurabh, Sanjay Seetharaman, Anannya Upasana

We study \textsc{Connected Dominating Set} on graphs whose closed-neighborhood set systems are $d$-semi-ladder-free. This structural condition strictly generalizes the biclique-free setting and provides a natural regime for connectivity-constrained domination. We obtain both a fixed-parameter algorithm and an approximate kernelization framework for the problem on this class. Our algorithmic result is based on a new compact representation theorem for inclusion-wise minimal set covers in $d$-semi-ladder-free set systems. Although the number of minimal set covers of size at most $k$ may be as large as $n^{Ω(k)}$, we show that all such set covers can nevertheless be encoded by a family of at most $k^{kd+1}$ tuples, and that this family can be enumerated in time $\Oh(k^{kd+2}\cdot nm)$. Combining this representation with a \textsc{Group Steiner Tree} subroutine, we obtain an algorithm for \textsc{Connected Set Cover}, which in turn yields an algorithm for \textsc{Connected Dominating Set} running in time $k^{kd+2}\cdot 2^k \cdot n^{\Oh(1)}$ and polynomial space. For the preprocessing result, we introduce grouped domination cores and dominator cores, and prove polynomial upper bounds on their sizes in $d$-semi-ladder-free graphs. Using these structures, we obtain, for every fixed $d$ and $\varepsilon>0$, a polynomial-time $(1+\varepsilon)$-lossy compression for \textsc{Connected Dominating Set} to an equivalent reduced instance of size $k^{\Oh(d^2/\varepsilon)}$. The reduced instance is a \textsc{Connected Dominating Set} instance on a $(d+2)$-semi-ladder-free graph.

Testing Induced-Subgraph Freeness in Outerplanar Graphs under the Random-Neighbor Oracle

from arXiv: Data Structures and Algorithms

Authors: Pan Peng, Kefan Yu

We prove that, for every fixed nonempty graph $H$, induced-$H$-freeness is testable with $\varepsilon^{-O_H(1)}$ queries on outerplanar graphs with no maximum-degree bound in the $\textit{random-neighbor model}$, where each query at a vertex returns a uniformly random neighbor. Thus, the query complexity is polynomial in $1/\varepsilon$ and independent of the number $n$ of vertices. Previously, the best bound known for this problem was the $\operatorname{poly}(\log n)$-query guarantee that follows from the general outerplanar-graph tester of Babu, Khoury, and Newman (2016) in the stronger $\textit{adjacency-list model}$, which provides exact degree queries and indexed access to neighbors. Our tester has $\textit{two-sided error}$, which is necessary in general: induced-$P_3$-freeness has no one-sided constant-query tester in the random-neighbor model, even on outerplanar graphs of maximum degree two.

Authors: Pan Peng, Kefan Yu

We prove that, for every fixed nonempty graph $H$, induced-$H$-freeness is testable with $\varepsilon^{-O_H(1)}$ queries on outerplanar graphs with no maximum-degree bound in the $\textit{random-neighbor model}$, where each query at a vertex returns a uniformly random neighbor. Thus, the query complexity is polynomial in $1/\varepsilon$ and independent of the number $n$ of vertices. Previously, the best bound known for this problem was the $\operatorname{poly}(\log n)$-query guarantee that follows from the general outerplanar-graph tester of Babu, Khoury, and Newman (2016) in the stronger $\textit{adjacency-list model}$, which provides exact degree queries and indexed access to neighbors. Our tester has $\textit{two-sided error}$, which is necessary in general: induced-$P_3$-freeness has no one-sided constant-query tester in the random-neighbor model, even on outerplanar graphs of maximum degree two.

$(α, β)$ Spanners and Hybrid Spanners with Nearly Tight Bounds

from arXiv: Data Structures and Algorithms

Authors: Shiri Chechik, Gur Lifshitz

For an $n$-vertex undirected, unweighted graph $G=(V,E)$ and a positive integer $k$, we present new spanner constructions with $O_k(n^{1+1/k})$ edges that achieve nearly optimal guarantees for all distances $d\le k$. Specifically, we construct a spanner $H\subseteq G$ with $O(n^{1+1/k}+(k+d\log d)n)$ edges, ensuring that any pair at original distance at most $d$ satisfies $\mathrm{dist}_H(u,v)\le 2k+O(d\log d)$. Equivalently, the multiplicative stretch for pairs at distance $d$ is $2k/d+O(\log d)$. In particular, setting $d=k/\log k$ yields an $(O(\log k),O(k))$-spanner with $O(n^{1+1/k}+kn)$ edges. For comparison, Ben-Levy and Parter (SODA'20) obtained, for every fixed $\varepsilon>0$ and sufficiently large $k$, an $(O(k^\varepsilon),O_\varepsilon(k))$-spanner with $O_{\varepsilon,k}(n^{1+1/k})$ edges. Our result improves the multiplicative stretch from $O(k^\varepsilon)$ to $O(\log k)$ while keeping the additive term linear in $k$, bringing us closer to the goal of $(O(1),O(k))$-spanners. Furthermore, Ben-Levy and Parter obtained multiplicative stretch $O_\varepsilon(k/d)$ for distances $d\le k^{1-\varepsilon}$, for every fixed $\varepsilon>0$, and an explicit bound of $7k/d$ for $d\le\sqrt{k}/2$. We achieve $2k/d+O(\log d)$, which is $(2+o(1))k/d$ whenever $d=o(k/\log k)$. Our second result is an improved construction of $k$-hybrid spanners, which guarantee stretch $2k-1$ for adjacent pairs and $k$ for non-adjacent pairs. Parter's original construction uses $O(k^2 n^{1+1/k})$ edges; we achieve the same guarantees with $O(n^{1+1/k}+kn)$ edges, removing the $k^2$ factor from the $n^{1+1/k}$ term. For every fixed $k$, our edge bound is optimal up to a constant factor under Erdős' girth conjecture.

Authors: Shiri Chechik, Gur Lifshitz

For an $n$-vertex undirected, unweighted graph $G=(V,E)$ and a positive integer $k$, we present new spanner constructions with $O_k(n^{1+1/k})$ edges that achieve nearly optimal guarantees for all distances $d\le k$. Specifically, we construct a spanner $H\subseteq G$ with $O(n^{1+1/k}+(k+d\log d)n)$ edges, ensuring that any pair at original distance at most $d$ satisfies $\mathrm{dist}_H(u,v)\le 2k+O(d\log d)$. Equivalently, the multiplicative stretch for pairs at distance $d$ is $2k/d+O(\log d)$. In particular, setting $d=k/\log k$ yields an $(O(\log k),O(k))$-spanner with $O(n^{1+1/k}+kn)$ edges. For comparison, Ben-Levy and Parter (SODA'20) obtained, for every fixed $\varepsilon>0$ and sufficiently large $k$, an $(O(k^\varepsilon),O_\varepsilon(k))$-spanner with $O_{\varepsilon,k}(n^{1+1/k})$ edges. Our result improves the multiplicative stretch from $O(k^\varepsilon)$ to $O(\log k)$ while keeping the additive term linear in $k$, bringing us closer to the goal of $(O(1),O(k))$-spanners. Furthermore, Ben-Levy and Parter obtained multiplicative stretch $O_\varepsilon(k/d)$ for distances $d\le k^{1-\varepsilon}$, for every fixed $\varepsilon>0$, and an explicit bound of $7k/d$ for $d\le\sqrt{k}/2$. We achieve $2k/d+O(\log d)$, which is $(2+o(1))k/d$ whenever $d=o(k/\log k)$. Our second result is an improved construction of $k$-hybrid spanners, which guarantee stretch $2k-1$ for adjacent pairs and $k$ for non-adjacent pairs. Parter's original construction uses $O(k^2 n^{1+1/k})$ edges; we achieve the same guarantees with $O(n^{1+1/k}+kn)$ edges, removing the $k^2$ factor from the $n^{1+1/k}$ term. For every fixed $k$, our edge bound is optimal up to a constant factor under Erdős' girth conjecture.

Consensus for Compressed Static Functions

from arXiv: Data Structures and Algorithms

Authors: Dominik Rosch, Jonatan Ziegler

The Consensus technique marked a breakthrough in the construction of minimal perfect hash functions (MPHFs), reaching a linear tradeoff between construction time and space overhead relative to the optimum. Consensus provides a clever scheme to search for and encode seeds of tasks in random data structures. We apply Consensus to the related field of compressed static functions (CSFs). These data structures store a function $f: S \to Σ$ such that querying a key $x \in S$ returns $f(x)$ and querying $x \not \in S$ returns an arbitrary value. CSFs do not need to store the keys $S$ and only need space close to the zeroth-order empirical entropy of the multiset of values. Often, some values are much more common than others. In these cases, CSFs can use less space than their non-compressed counterparts. CSFs are a useful building block, for example in database design and bioinformatics. We introduce Consensus-CSF, which can reach arbitrarily close to the empirical entropy $n H_0$, with a construction time of $n \exp(\tilde{\cal{O}} (\sqrt{1 / δ}))$ for space usage of $n H_0 (1 + δ)$ when assuming some parameters of the value distribution to be constants. This tradeoff beats previously implemented approaches that can only reach some fixed threshold above the entropy lower bound. We enable Consensus in the setting of CSFs, which is less structured than MPHFs, with the introduction of task insertions. Our approach randomly distributes the keys into one-bit Consensus tasks and then strategically inserts additional tasks in places where the construction would get stuck otherwise. We provide an implemented version of our algorithm which reaches the same order of magnitude in space overhead as competitors but is not competitive in practice. Beyond these results, we present a new way to think and reason about Consensus, which may also be applied to other problems.

Authors: Dominik Rosch, Jonatan Ziegler

The Consensus technique marked a breakthrough in the construction of minimal perfect hash functions (MPHFs), reaching a linear tradeoff between construction time and space overhead relative to the optimum. Consensus provides a clever scheme to search for and encode seeds of tasks in random data structures. We apply Consensus to the related field of compressed static functions (CSFs). These data structures store a function $f: S \to Σ$ such that querying a key $x \in S$ returns $f(x)$ and querying $x \not \in S$ returns an arbitrary value. CSFs do not need to store the keys $S$ and only need space close to the zeroth-order empirical entropy of the multiset of values. Often, some values are much more common than others. In these cases, CSFs can use less space than their non-compressed counterparts. CSFs are a useful building block, for example in database design and bioinformatics. We introduce Consensus-CSF, which can reach arbitrarily close to the empirical entropy $n H_0$, with a construction time of $n \exp(\tilde{\cal{O}} (\sqrt{1 / δ}))$ for space usage of $n H_0 (1 + δ)$ when assuming some parameters of the value distribution to be constants. This tradeoff beats previously implemented approaches that can only reach some fixed threshold above the entropy lower bound. We enable Consensus in the setting of CSFs, which is less structured than MPHFs, with the introduction of task insertions. Our approach randomly distributes the keys into one-bit Consensus tasks and then strategically inserts additional tasks in places where the construction would get stuck otherwise. We provide an implemented version of our algorithm which reaches the same order of magnitude in space overhead as competitors but is not competitive in practice. Beyond these results, we present a new way to think and reason about Consensus, which may also be applied to other problems.

A Quantum Scaling Algorithm for Maximum-Weight Perfect Matching in General Graphs

from arXiv: Data Structures and Algorithms

Authors: Kourosh Mirsohi, Sandy Irani, Michael T. Goodrich

Quantum speed-ups have been obtained for many fundamental graph problems, including most variants of matching. A notable exception, however, is the maximum-weight perfect matching (MWPM) problem in general graphs with integer edge weights, which is arguably the most challenging variant of matching. We present a quantum algorithm for MWPM in general graphs that runs in \( \widetilde{O}(n m^{2/3}\log W) \) time, where $W$ is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of \( \widetilde{O}(m\sqrt n \log W) \) in the dense regime, where $m\ge n^{3/2}$. To the best of our knowledge, this is the first quantum algorithm to obtain an asymptotic improvement over the best classical combinatorial algorithm for the MWPM problem in general graphs. The running time of our method accounts for QRAM initialization and access overheads up to polylogarithmic factors, as well as all classical updates to the data structures. At a high level, our algorithm is based on a classical framework due to Duan, Pettie, and Su, but our algorithm requires replacing certain classical tasks with quantum methods, alternative analysis of classical procedures, and the use of alternative data structures that can be implemented effectively in the QRAM model.

Authors: Kourosh Mirsohi, Sandy Irani, Michael T. Goodrich

Quantum speed-ups have been obtained for many fundamental graph problems, including most variants of matching. A notable exception, however, is the maximum-weight perfect matching (MWPM) problem in general graphs with integer edge weights, which is arguably the most challenging variant of matching. We present a quantum algorithm for MWPM in general graphs that runs in \( \widetilde{O}(n m^{2/3}\log W) \) time, where $W$ is an upper bound on the magnitude of the edge weights. This is an improvement over the best known classical combinatorial bound of \( \widetilde{O}(m\sqrt n \log W) \) in the dense regime, where $m\ge n^{3/2}$. To the best of our knowledge, this is the first quantum algorithm to obtain an asymptotic improvement over the best classical combinatorial algorithm for the MWPM problem in general graphs. The running time of our method accounts for QRAM initialization and access overheads up to polylogarithmic factors, as well as all classical updates to the data structures. At a high level, our algorithm is based on a classical framework due to Duan, Pettie, and Su, but our algorithm requires replacing certain classical tasks with quantum methods, alternative analysis of classical procedures, and the use of alternative data structures that can be implemented effectively in the QRAM model.

Multidimensional Resource Scheduling with Small Demands

from arXiv: Data Structures and Algorithms

Authors: Yossi Azar, Rathish Das, Hao Sun

We study multidimensional resource scheduling. Each job $i$ has a $d$-dimensional resource-demand vector $v_i$ and a processing time $s_i$. The scheduler assigns a start time to each job, subject to the constraint that, at every time, the total demand of the jobs being processed does not exceed $1$ in any resource dimension. The objective is to minimize the makespan. We focus on the regime in which every individual resource demand is small. We ask whether the favorable \emph{small-vector phenomenon} known for multidimensional vector packing, which corresponds to the special case of unit processing times, extends to jobs with heterogeneous processing times. The key difficulty is that arbitrary processing times create temporal interactions across multiple duration scales: a long job may overlap many shorter jobs, while feasibility must be maintained throughout every job's execution interval. We prove that the small-vector phenomenon persists in this temporal setting. For any $0<ε<1/4$, after normalizing the maximum processing time to $1$, if every coordinate of every demand vector is at most $O(ε^2/\log(d/ε))$, we give a randomized offline algorithm that produces a schedule with expected makespan at most $ (1+6ε)\mathrm{OPT}+3$. Thus, sufficiently small resource demands admit asymptotically near-optimal schedules in arbitrary dimension, despite heterogeneous processing times. We also obtain constant competitive ratios in the online setting. Let $T$ denote the ratio between the maximum and minimum processing times. If every coordinate is at most $O(1/(\log d\log T))$, we give a randomized $O(1)$-competitive algorithm, with a competitive ratio independent of both $d$ and $T$. We further derandomize our approach, obtaining a deterministic $O(1)$-competitive algorithm under a comparable smallness assumption.

Authors: Yossi Azar, Rathish Das, Hao Sun

We study multidimensional resource scheduling. Each job $i$ has a $d$-dimensional resource-demand vector $v_i$ and a processing time $s_i$. The scheduler assigns a start time to each job, subject to the constraint that, at every time, the total demand of the jobs being processed does not exceed $1$ in any resource dimension. The objective is to minimize the makespan. We focus on the regime in which every individual resource demand is small. We ask whether the favorable \emph{small-vector phenomenon} known for multidimensional vector packing, which corresponds to the special case of unit processing times, extends to jobs with heterogeneous processing times. The key difficulty is that arbitrary processing times create temporal interactions across multiple duration scales: a long job may overlap many shorter jobs, while feasibility must be maintained throughout every job's execution interval. We prove that the small-vector phenomenon persists in this temporal setting. For any $0<ε<1/4$, after normalizing the maximum processing time to $1$, if every coordinate of every demand vector is at most $O(ε^2/\log(d/ε))$, we give a randomized offline algorithm that produces a schedule with expected makespan at most $ (1+6ε)\mathrm{OPT}+3$. Thus, sufficiently small resource demands admit asymptotically near-optimal schedules in arbitrary dimension, despite heterogeneous processing times. We also obtain constant competitive ratios in the online setting. Let $T$ denote the ratio between the maximum and minimum processing times. If every coordinate is at most $O(1/(\log d\log T))$, we give a randomized $O(1)$-competitive algorithm, with a competitive ratio independent of both $d$ and $T$. We further derandomize our approach, obtaining a deterministic $O(1)$-competitive algorithm under a comparable smallness assumption.

MultiTable: A Faster Hash Table at any Physical Load Factor up to and Including One

from arXiv: Data Structures and Algorithms

Authors: Maksym Petkus

We present \emph{multitable} and its Rust reference implementation: a stable hash table both materially faster at equal physical memory and more flexible than the SwissTable in its Rust's hashbrown implementation. As an arithmetic mean over 84 configurations it delivers $\mathbf{2.1\times}$ hashbrown's throughput when both hash the same raw bytes and $\mathbf{1.9\times}$ when hashbrown is keyed on native integers, its best case; on negative lookups alone, $3.2\times$ and $2.9\times$. Multitable reaches \textbf{any physical load factor} up to and \textbf{including one} ($0.9999$ demonstrated), exactly for the requested capacity, compared to hashbrown which doubles at $0.777$ for 4-byte keys and values. At $75\%$ saturation of hashbrown (assumed average case of its rigid ladder) and multitable sized to $0.97$ physical load factor, hashbrown takes $66\%$ more space. The lookup probe count has no cliff as the load factor approaches one. Bucket size, physical load factor, and failure budget are parameters, and the multitable can be grown without rehashing. We implement two variants of multitable: plain and filtered. At equal physical memory on an Apple M2 Pro the filtered multitable leads hashbrown in all $84$ insert, hit, and miss configurations. Multitable is more \textbf{memory-efficient}, at equal mixed-lookup throughput on the map of $4$-byte keys and values the filtered multitable needs up to $12\%$ fewer bytes than hashbrown, and the plain multitable is $18\%$ smaller, holding $\mathbf{22\%}$ more keys in the same memory.

Authors: Maksym Petkus

We present \emph{multitable} and its Rust reference implementation: a stable hash table both materially faster at equal physical memory and more flexible than the SwissTable in its Rust's hashbrown implementation. As an arithmetic mean over 84 configurations it delivers $\mathbf{2.1\times}$ hashbrown's throughput when both hash the same raw bytes and $\mathbf{1.9\times}$ when hashbrown is keyed on native integers, its best case; on negative lookups alone, $3.2\times$ and $2.9\times$. Multitable reaches \textbf{any physical load factor} up to and \textbf{including one} ($0.9999$ demonstrated), exactly for the requested capacity, compared to hashbrown which doubles at $0.777$ for 4-byte keys and values. At $75\%$ saturation of hashbrown (assumed average case of its rigid ladder) and multitable sized to $0.97$ physical load factor, hashbrown takes $66\%$ more space. The lookup probe count has no cliff as the load factor approaches one. Bucket size, physical load factor, and failure budget are parameters, and the multitable can be grown without rehashing. We implement two variants of multitable: plain and filtered. At equal physical memory on an Apple M2 Pro the filtered multitable leads hashbrown in all $84$ insert, hit, and miss configurations. Multitable is more \textbf{memory-efficient}, at equal mixed-lookup throughput on the map of $4$-byte keys and values the filtered multitable needs up to $12\%$ fewer bytes than hashbrown, and the plain multitable is $18\%$ smaller, holding $\mathbf{22\%}$ more keys in the same memory.

Breaking the $2^n$ barrier for directed hamiltonicity

from arXiv: Data Structures and Algorithms

Authors: Tomohiro Koana, Soh Kumabe

We give a randomized algorithm for Directed Hamiltonian Cycle on $n$-vertex directed graphs that runs in time $O^*((375/196)^n)=O^*(1.9133^n)$. For general directed graphs, this is the first improvement in the exponential base over the classical $O^*(2^n)$-time algorithms of Bellman and Held--Karp (1962). To obtain this improvement, we first give a $(2-2^{-d})^n \, \text{poly}(n,W)$-time algorithm for counting Hamiltonian paths modulo two at each total weight when at most $d$ distinct weights from $\{1,\ldots,W\}$ enter each vertex. The algorithm combines the Laplacian determinant method of Björklund, Kaski, and Koutis (ICALP 2017) with a random linearization also used by Arvind and Guruswami (IPEC 2021). To apply the isolation lemma while keeping $d$ small, we randomly delete and duplicate arcs, partitioning the incoming copies at each vertex into $d$ groups, where $d\ge2$. We show that, if the input graph has a Hamiltonian path from $s$ to $t$, then with probability at least $\left(1-\frac{1}{1+(2^d-1)^2}\right)^{n-1}$ one can select one group at each vertex other than $s$ so that the selected arcs contain an odd number of such paths.

Authors: Tomohiro Koana, Soh Kumabe

We give a randomized algorithm for Directed Hamiltonian Cycle on $n$-vertex directed graphs that runs in time $O^*((375/196)^n)=O^*(1.9133^n)$. For general directed graphs, this is the first improvement in the exponential base over the classical $O^*(2^n)$-time algorithms of Bellman and Held--Karp (1962). To obtain this improvement, we first give a $(2-2^{-d})^n \, \text{poly}(n,W)$-time algorithm for counting Hamiltonian paths modulo two at each total weight when at most $d$ distinct weights from $\{1,\ldots,W\}$ enter each vertex. The algorithm combines the Laplacian determinant method of Björklund, Kaski, and Koutis (ICALP 2017) with a random linearization also used by Arvind and Guruswami (IPEC 2021). To apply the isolation lemma while keeping $d$ small, we randomly delete and duplicate arcs, partitioning the incoming copies at each vertex into $d$ groups, where $d\ge2$. We show that, if the input graph has a Hamiltonian path from $s$ to $t$, then with probability at least $\left(1-\frac{1}{1+(2^d-1)^2}\right)^{n-1}$ one can select one group at each vertex other than $s$ so that the selected arcs contain an odd number of such paths.