Theory of Computing Report

Authors: Aritra Das, Vincent Froese, Moritz Grillo, Debayan Gupta, Christoph Hertrich, Tharrshann Jayan Logarajah, Georg Loho, Mihir More, Moritz Stargalla

Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult even for shallow ReLU networks. We study this problem for two-layer input-convex neural networks (ICNNs), a restricted architecture where nonnegative output weights enforce convexity. Computing the $L_p$-Lipschitz constant for these networks is equivalent to maximizing the dual norm over a zonotope. While $L_1$- and $L_\infty$-norm maximization on zonotopes admit fixed-parameter and polynomial-time algorithms, respectively, the parameterized complexity of the remaining $L_p$-norms was open. We prove that, for every fixed $p\in (1,\infty)\cap \mathbb{Q}$, maximizing the $L_p$-norm over a zonotope in $\mathbb{R}^d$ is W[1]-hard with respect to the dimension $d$. Moreover, our hardness results imply that brute-force enumeration algorithms are essentially optimal for this problem under the Exponential Time Hypothesis. By duality, the same hardness results hold for computing the $L_p$-Lipschitz constant of two-layer ReLU ICNNs. Our proof first establishes the result for the $L_2$-norm and then transfers the construction to arbitrary fixed $p\in (1,\infty)\cap\mathbb{Q}$ using a suitable Taylor approximation. These results resolve the corresponding questions regarding the parameterized complexity status for zonotope norm maximization and two-layer ICNN Lipschitz constants. Our paper resolves an open problem posted at COLT'25. There are several independent concurrent papers resolving the same problem. Our paper prioritizes a clear exposition of the underlying mathematics and conceptual intuitions behind the proof. Additionally, we explicitly describe our research process including the use of LLMs.

Authors: A. R. Balasubramanian, Dmitry Chistikov, Rupak Majumdar

Many problems in the verification of recursive programs can be reduced to pushdown model checking. In this problem, we are given as input a pushdown automaton (PDA) over a constant-sized stack alphabet, and a description of undesirable behaviors given by an intersection of NFAs, and the problem is to decide if there is a behavior of the PDA that belongs to the set of undesirable behaviors. It is well-known that there is an algorithm for this problem that runs in time $O(n^{2k} |Σ| + n^{3k})$, where $n$ is the maximum number of states of the PDA and the NFAs, $Σ$ is the common input alphabet, and $k-1$ is the number of NFAs. Despite the importance of this problem, no better algorithm is known for it. In this paper, we provide an explanation for this lack of progress using the lens of fine-grained complexity theory. We prove that if the $3k$-Clique hypothesis (resp. combinatorial $3k$-Clique hypothesis) is true, then for any $ε> 0$, there is no algorithm (resp. combinatorial algorithm) that solves this problem in time $O((n^{(ω-1)k} |Σ| + n^{ωk})^{1-ε})$ (resp. $O((n^{2k} |Σ| + n^{3k})^{1-ε})$) where $ω$ is the matrix multiplication exponent. Furthermore, using the combinatorial hypothesis, we also show that pushdown model checking over constant-sized input alphabets cannot be solved in time faster than $O(n^{3(k-1)-ε})$ for any $ε> 0$. Finally, we investigate the possibility of an $O(N^{3k-ε})$ time algorithm for this problem where $N$ is the total bit size of the input. We formulate a new hypothesis, the 2NPDA$(k)$ hypothesis, that helps explain the lack of $O(N^{3k-ε})$ time algorithms for this problem. To corroborate this hypothesis, we show a web of linear-time reductions between the 2NPDA$(k)$ hypothesis, pushdown model checking, and other problems in formal language and automata theory.

Authors: Benedek Nagy

The Post Correspondence Problem is as follows: having a set of dominoes, is there any (maybe repeating) sequence of them such that the words formed by the upper parts and the lower parts by the sequence of dominoes are identical. It is one of the most known problems that is algorithmically undecidable. In this paper, the reverse Post Correspondence Problem is defined, that is, where the two assembled words of the dominos are reversals of each other. Undecidability about this new, modified problem is proven. Further, based on this result, it is also proven that the emptiness problem for 5'->3' String Assembly Systems is undecidable too. 5'->3' String Assembly Systems belong to String Assembly Systems type formal language generating models. The 5'->3' denotes that, in these variants, the derivations of the generated words start from the two extremes. The notation and the new model are bio-motivated as any double stranded DNA has two opposite oriented 5'->3' strands.

Authors: Yota Maeda, Hiroshi Yano

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $ρ$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} \] with success probability $1-δ$. Provided that $ρ$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

Authors: Hari Mohan Pandey

This paper proposes the Coronavirus Optimization Algorithm (COA), a SARS-CoV-2-inspired success-history adaptive evolutionary optimizer for box-constrained continuous global optimization. COA does not model disease transmission; instead, it maps selected coronavirus mechanisms to explicit search operators, including elite-guided attraction, trial-vector generation, adaptive parameter variation, stagnation recovery, and population-size scheduling. The algorithm combines opposition-based initialization, current-to-pbest mutation, binomial crossover, an external archive, success-history adaptation, population reduction, and partial restart. COA is evaluated on 29 CEC 2017 benchmark functions at 10, 30, and 50 dimensions against 15 competitive optimizers. Results show that COA achieves the best overall Friedman rank across all dimensions, with particularly strong performance on composition functions. The findings demonstrate that COA is a compact, transparent, and competitive adaptive evolutionary optimizer, while also highlighting limitations on some hybrid functions and the need for further high-dimensional validation.

Authors: Jiaqi Liu, Yansong Feng, Yanbin Pan

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d=\mathbb{Z}[y]/(y^d+1)$. A deterministic reduction from X3C produces an integral target and squared threshold $Δ$ such that the closest squared distance is exactly $Δ$ in YES instances and at least $Δ+4$ in NO instances. The ideal elements within squared distance $Δ$ are in bijection with exact covers, which also gives $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We further lift these instances to full-rank principal ideals of $\mathbb{Z}[X]/(X^D-1)$, where $D=2d$. The lift preserves principality, doubles the dimension, and scales corresponding squared distances by eight. Hence exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard on principal cyclic ideal lattices. Both results admit uniformly computable fixed-family forms: for each X3C universe size, the principal cyclotomic and cyclic ideals can be fixed before the triple collection is known, with only the targets and thresholds depending on the collection. If exact decision-CVPP were polynomial-time solvable on either family, then $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$; by Karp--Lipton, the polynomial hierarchy would collapse to $Σ_2^{\mathsf P}$. To our knowledge, the cyclic results resolve the exact decision versions of Micciancio's questions for cyclic lattices and fixed cyclic-lattice families.

Authors: Vincent Delecroix, Vincent Despré, Camille Lanuel, Hugo Parlier, Monique Teillaud

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.

Authors: Emerson G. Escolar, Yuta Shimada

In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree $1$ and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data.

Authors: Jiaheng Wang

It is shown in this manuscript that a random graph $G$ drawn from the Erdős--Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

Authors: Rolando D. Somma, Ronald de Wolf

The guided Hamiltonian problem is the following: given access to the unitary $U=e^{i H}$ for some Hamiltonian $H$, and given access to a unitary that prepares a guiding state promised to have overlap at least $γ>0$ with the ground space of $H$, estimate the ground-state energy of $H$ within additive error $δ> 0$ and success probability at least $1-\varepsilon $, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? An upper bound $O(\log(1/\varepsilon)\log(1/γ)/γδ)$ was known, and was improved to $O(\log(1/\varepsilon)/γδ)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $δ,γ,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $Ω(\log(1/\varepsilon)/γδ)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/γ^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $δ$ between its first and second eigenvalue; and for ground-state preparation, where $δ$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $Ω(\log(1/\varepsilon)/γ\sqrtδ)$ for this case.

Authors: Stacey Jeffery, Freek Witteveen

In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap $γ$ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary $U$. We give a quantum algorithm that estimates the largest eigenphase of a unitary $U$ in this guided setting using a factor of $\log\frac{1}γ$ fewer queries to $U$ than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.

Authors: Ainesh Bakshi, Alex Conway, Hanna Komlós, William Kuszmaul, Alek Westover

Affine modular linear hashing is one of the simplest classical hash families. For a prime $p > u$, the hash function is obtained by choosing $s,t$ uniformly from $\mathbb{Z}_p$ and mapping each key $x \in \{0,\ldots,u-1\}$ to one of $n$ bins by $h(x) = [(sx+t) \bmod p] \bmod n$. Despite its simplicity, the maximum load of linear hashing remains poorly understood. For $n$ keys hashed into $n$ bins, the best known upper bound is $O((n \log n)^{1/3})$, whereas the best known lower bound is only $Ω(\log n / \log\log n)$. We prove a lower bound of $\exp(Ω(\log n / \log\log n))$ for universes of size $n^{1+o(1)}$. Surprisingly, there is a key set for which this load holds not just in expectation, but for every random seed. The proof is driven by two simple reductions: one transfers lower bounds from a real version of linear hashing to modular linear hashing, and the other transfers arithmetic Kakeya constructions to real hashing. We further show that, for sufficiently large $p$, the expected maximum loads in the modular and real settings are essentially the same, giving an alternative route to an $n^{1/3+o(1)}$ upper bound. Finally, we show that any uniform subpolynomial upper bound for either setting would imply a polynomial-length arithmetic Kakeya conjecture and hence the Kakeya conjecture for upper Minkowski dimension.

Authors: Gabriel Marques Domingues, Minh Hang Nguyen, Shay Solomon

We study the \emph{fully dynamic edge orientation problem}, focusing on \emph{worst-case} time bounds. An undirected graph undergoes edge insertions and deletions, and the goal is to maintain an orientation with small {\em maximum outdegree} (hereafter, outdegree) and small worst-case update time. The outdegree of any orientation is at least $α-1$, where $α$ is the graph's \emph{arboricity}, i.e., the minimum number of forests into which its edge set can be partitioned. When $α= O(1)$, it is long known that both the outdegree and the worst-case update time can be bounded by $O(\log n)$. Despite numerous follow-ups, no $o(\log^3 n)$ worst-case update time is known for maintaining constant outdegree, even for very basic graph families---with a notable exception, \emph{forests}. For forests, a \emph{simple folklore} algorithm maintains outdegree 2 via \emph{random walks}: When an insertion creates a vertex of outdegree 3, the algorithm repeatedly chooses a uniformly random outgoing edge until reaching a vertex of outdegree at most 1, and then flips the resulting directed path. As the underlying graph is cycle-free, the path length is easily shown to be $O(\log n)$ in expectation, and also with high probability for polynomially long update sequences. We prove that this simple random walk paradigm extends to \emph{outerplanar graphs}. Our algorithm maintains constant outdegree with $O(\log n)$ worst-case update time, where the time bound holds in expectation, and also with high probability for polynomially long update sequences. We give a \emph{tight analysis}: outdegree 4 is achievable with $O(\log n)$-length paths, while outdegree 3 incurs $\mathtt{poly}(n)$-length paths. We also extend the argument to $K_{2,t}$-minor-free graphs, for any $t \ge 2$, with the outdegree bound depending only on $t$ and with the same update time guarantees. The locality of [...]

Authors: Karthekeyan Chandrasekaran, Krishna Kalathur

We show that there exists a polynomial-time algorithm to find a stable matching in network hypergraphic preference systems. The key connection that drives the algorithm was discovered by chatting with ChatGPT-5.6 Sol Max. We verified it independently and present the details in our own words.

Authors: Júlia Baligács, Bartłomiej Bosek, Yann Disser, Andreas Emil Feldmann, Grzegorz Gutowski, Katarzyna Kępińska, Paweł Putra, Anna Zych-Pawlewicz

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

Authors: Enrico Olivucci, Mariia Sobchuk, Sehmimul Hoque, Jeffrey Hnybida, Kyungho W. Kim, Ala Shayeghi, Pooya Ronagh

We prove the first quantum--classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-βE}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $α=e^{βΔ}$, where $Δ= \max E-\min E$, every classical algorithm---querying the value, gradient, or any higher-order derivatives of the log-density---requires $Ω(α)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrtα\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{Ω(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.

Authors: Leon Kullmann, Phuoc Lucky Trinh, Leon Kellerhals, Mitja Krebs, André Nichterlein, Stefan Schmid

The classical Minimum Linear Arrangement (MLA) problem has been studied extensively. It is known to be NP-hard and it admits an $O(\sqrt{\log n}\log\log n)$-approximation [Feige and Lee, IPL, 2007]. MLA can be defined as follows as design problem: Given a graph $G$ with vertex set $V(G)$, design a path $H$ on the same vertex set that minimizes the linear arrangement cost $\sum_{uv\in E(G)}\textrm{dist}_H(u,v)$, where $\textrm{dist}_H(u,v)$ indicates the distance of $u$ and $v$ in $H$. We initiate the study of the generalization in which $H$ is allowed to be a caterpillar graph of maximum degree at most $Δ$. Caterpillars are the simplest generalization of paths, having pathwidth one and interpolating between paths and stars via the degree parameter $Δ$. We give an algorithm that lifts any $α$-approximation for MLA to an $(α+3-2/(Δ-1))$-approximation for our problem, thus obtaining an $O(\sqrt{\log n}\log\log n)$-approximation for our more general problem as well. Moreover, we derive a $4$-approximation whenever MLA is polynomial-time solvable, in particular, for trees. Complementing these results, we prove NP-hardness for every constant $Δ\geq 2$, and, in stark contrast to MLA, show it remains NP-hard on trees when $Δ$ is part of the input.

Authors: Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, Shuyi Yan

We study the shortest-path problem on graphs with positive real-valued edge weights. Given a source vertex $s$ and a target vertex $t$, the goal is to calculate the length of the shortest path from $s$ to $t$. We are particularly interested in instances that can be solved in sublinear time. Recently, Haeupler, Hladík, Rozhoň, Tarjan, and Tětek proved that (a version of) the bidirectional Dijkstra's algorithm is instance-optimal on positively weighted multigraphs, both directed and undirected, considering the number of vertices and edges queried by the algorithm. However, multigraphs are not the canonical setting for the shortest-path problem. The problem is typically formulated on simple graphs without loops and parallel edges. They therefore left as an open problem whether bidirectional Dijkstra remains instance-optimal on simple weighted graphs. We answer this question, but for simple graphs, the answer is more complex, depending on the setting. We show that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order. In contrast, under the order-dependent model, where incident edges have a given order, we show that bidirectional Dijkstra is not instance-optimal. For simple directed weighted graphs, we show that bidirectional Dijkstra is not instance-optimal under either the order-oblivious or the order-dependent model, being off by a factor of $Θ(m/n)$ in both cases. We further show that no algorithm can have instance-optimality ratio $o(m/n)$ under the order-dependent model, or under the order-oblivious model when $m=O(n\sqrt{n})$. On the positive side, the above results imply that bidirectional Dijkstra is instance-optimal up to logarithmic factors on all sparse directed and undirected graphs satisfying $m/n=\log^{O(1)} n$.

Authors: Taha El Ghazi, Jonas Ellert, Chien-Chung Huang, Tatiana Starikovskaya

Clustering is one of the most fundamental tools in data analysis, allowing large datasets to be summarized by a small number of representative points. Given a metric space $(\mathcal{X}, \mathbb{d})$ and a set $S$ of $n$ points in this space, the continuous $k$-median clustering problem asks to find a set $C$ of $k$ points that minimizes the objective function $\sum_{s\in S} \mathbb{d}(s,C)$. When $\mathcal{X} = Σ^\ell$ is the set of strings of length $\ell$ and $\mathbb{d}$ is the Hamming distance, the continuous $k$-median clustering problem is known to be W[1]-hard when parameterized by $k$. In this work, we present the first $(1+\varepsilon)$-approximation algorithm for this problem with FPT runtime $2^{\mathrm{poly}(\varepsilon^{-1},k)} \cdot n\ell \mathrm{polylog} \; n$. An additional feature of the algorithm is that it can be implemented in streaming, requiring only $\tilde{O}_\varepsilon(\ell k + k^2)$ space. As an auxiliary tool of independent interest, we show the first streaming algorithm for computing an $\varepsilon$-coreset for continuous $k$-median clustering under the Hamming

Authors: Soheil Kolouri

Partial optimal transport compares two measures while leaving part of the mass unmatched, which is what makes it robust to outliers, occlusion, and clutter. The quantity of interest is usually the whole profile - the optimal cost at every transported cardinality - because the right amount to transport is rarely known in advance, and on the real line the PAWL algorithm returns that profile in $O(N\log N)$. Much data is periodic rather than linear: angles, phases, orientations, time of day, hue, and every direction obtained by projecting onto a great circle. On the circle the same problem acquires a global circulation, or equivalently an optimized cut, which the naive exact method handles by running the line algorithm once per support gap, at $O(N^{2}\log N)$. We show that this factor $N$ is unnecessary. The line structure survives in cut-free form, and a free-gap invariant supplies, at every step, a cut at which all previous local updates remain valid line updates. This yields PAWC: an exact $O(N\log N)$ time, $O(N)$ memory algorithm returning all $K+1$ costs, nested active sets and plans in one run, together with a single gap that is simultaneously optimal for every cardinality. Slicing over great circles extends it to $\mathbb{S}^{d-1}$. Empirically the whole profile costs $0.56$ms at $N=4096$ against $1.5$s for a single transported fraction from a general solver; on occluded, cluttered mpeg-7 shapes, holding the descriptor fixed and varying only the cost, it retains $66\%$ of the clean-data retrieval score against $16\%$ for balanced circular OT, and on $\mathbb{S}^{2}$ it halves the fitting error of spherical sliced Wasserstein against contaminated targets, synthetic and real. Code is available at https://github.com/mint-vu/Partial_Wasserstein_on_Circles.

Update: I haven’t joined Anthropic or taken leave from the university.

That means I’ll be teaching as usual this fall, and you can look forward to your regular installment of live lecture blogs. Unlike past years, I’m assigned to my grad seminar in the fall, not the spring. Next semester I’ll be teaching our undergraduate probability class. That is assuming that the University is still here in the spring and hasn’t been put out of business by some new AI super tutor released by my friends across the bay. I mean, it would be the year 2027, and popular forecasts suggest AIs will be able to do everything taught in a CS degree by late 2026. By the end of the spring semester, those same prognosticators predict those AIs will go rogue, and we’ll find ourselves in the reality forecast by James Cameron in his 1984 prophecy, Terminator. Why should I bother dusting off my copy of Bertsekas and Tsitsiklis when the forecasts tell me I should work harder at the gym to prepare myself for robot enslavement in the salt mines?

Actually, you know what would be a good way to prep for the robot apocalypse? Why don’t we spend a semester talking about forecasting and why people are obsessed with being certain about the future?

Answering that question could potentially be a great way to shape a graduate course. We could spend a semester digging into not only how people forecast, but why they forecast. We could split our time in half, looking at the particularities of different domains where people make forecasts, and then looking into the tools they have settled on as mathematical culture.

If you look at the places where forecasts are common, they all have different purposes. A local weather report is very different from a prediction of the end of the world. The former tells you if you should pack an umbrella on the way to work. The latter tells you whether you need to lobby your government to radically change its planned energy buildout. We all believe the evidence supporting forecasts of rain is far more certain and reliable than forecasts of nature’s end. But the costs of being wrong couldn’t be more different.

What impact do these forecasts have? Why do we forecast the weather, and what hidden technology is needed to make these forecasts accurate? Why does Congress demand that we forecast future budgetary consequences of proposed laws, even though we know we can’t predict the actual structural shocks that will render those forecasts moot? Why are people so obsessed with predicting the rise of superintelligent robots? Is it more than a way to justify their greed and obsessive 996 work conditions?

We’d learn a lot from a comparative study. I’d like to look at astronomy, meteorology, climate science, seismology, macroeconomics, government, epidemiology, public opinion research, and millenarianism to see what they have in common and how they differ. Forecasts let people externalize their beliefs about the likelihoods and consequences of various scenarios. Some forecasts are made for mundane planning. Some forecasts are made to literally gamble. Some forecasts communicate possible futures that others might not be considering. Some forecasts are made to be self-fulfilling, to manifest a change in the world the forecaster desires. Some are made to be self-negating, to convince people to act to avoid worst-case scenarios. I’m interested in understanding the threads that link all of these different purposes together.

Though I’m much more interested in the why, the how has some fun tidbits too. The how is about creating certainty about the future by quantifying it. Uncertainty becomes certain once we turn it into an interval, right? In talking about the how-of-forecasting, I could tie together a lot of loose ends in methods that I’ve blogged about over the years.

We might determine the conditions under which pattern recognition (aka machine learning aka AI) becomes a forecast. We could look at how forecasts are evaluated post-hoc with scoring rules and calibration tests and why people think those are good evaluations. We could look at methods for cost-benefit analysis and uncertainty quantification, and how people justify their modeling assumptions to make decisions. We could learn about tools from dynamical systems that move from simple moving averages to complex simulations. We could examine how statistical tools can be applied to extrapolate from the present to the future. And we could see how these sorts of metrics and models tie your hands algorithmically into unsurprising answers.

This sounds like a fun class to me. I predict I’ll teach this class this fall and live blog it here, starting this Thursday. If you’re a Berkeley graduate student whose research depends on forecasts, email me if you’d like to join the course.[footnote: If you do email, please send me a description of your background and why you’re interested.] If you’re not local, I’ll post a syllabus and webpage this week, and I’ll do my best to keep all of the material public. I predict it will be fun.

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By Ben Recht

Authors: Michal Hospodár, Galina Jirásková, Viktor Olejár, Juraj Šebej, György Vaszil

The 17th International Conference on Automata and Formal Languages (AFL 2026) was held in Košice, September 7-10, 2026. It was organized by the Faculty of Science, P. J. Šafárik University in Košice and the Mathematical Institute of the Slovak Academy of Sciences. This volume contains the texts of the three invited lectures and the 18 papers selected by the Program Committee from a total of 27 submissions. The accepted papers, submitted by authors from 14 countries, cover all important areas of automata and formal language theory and their applications such as grammars and acceptors for strings, trees, and graphs, algebraic theories and algorithms for automata and languages, decision problems, relations to complexity theory, picture description and analysis, and other fields.

Authors: Venkatesan Guruswami, Shilun Li, Mihir Singhal

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - ε$ is $(p, C_{p,q}/ε)$-average-radius list-decodable with probability at least $1 - q^{-Ω(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/ε$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/ε)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

Authors: Kirill Osipov

We develop a resource representation for step recursion in which mutable-state width and recursion descent are explicit and independent parameters. A width bound $u$ controls the size of the encoded machine state, while an effective descent $ρ$ determines the available recursion depth $δ_ρ(u)$. For generalized-inverse descents, we derive the depth directly from generator growth and characterize the increasing sequences that can occur as generator orbits. We then connect this depth--width geometry to standard finite-branching computation. Every deterministic bounded-state dynamics is realizable by a single ordinary bounded step recursion over a fixed finite numerical basis. Using deterministic, existential, universal, or alternating aggregation on the same local dynamics yields the corresponding machine semantics. After closure under the width reparameterizations needed to absorb fixed local cost, the resulting language classes are exactly the machine time--space classes on profiles $(δ_ρ(u),u)$. Finally, profile domination quotients effective descents by admissible width reparameterization. Some depth curves collapse, yet polynomial widths support an explicit infinite strict hierarchy between the canonical polynomial- and exponential-depth profiles. Thus descent remains a nonredundant resource coordinate after polynomial width reparameterization; standard complexity classes are calibration points.

Authors: Sijia Dai, Minming Li, Xiaowei Wu, Yong Zhang

The house allocation problem is a classical one-sided matching problem that concerns the assignment of a set of $m$ houses to $n$ agents according to their preferences, where each agent is assigned exactly one house. Among the various objectives studied in this setting, envy-freeness is one of the most widely adopted fairness criteria. As envy-free house allocations do not always exist, we address this challenge by introducing subsidies and aim to compute allocations that achieve envy-freeness with minimum total subsidy. For binary instances, we show that a total subsidy of at most $(n-1)$ suffices to guarantee envy-freeness in house allocation, and this bound is tight. Building on the known NP-hardness for general utilities, we further show that computing an allocation that minimizes the total subsidy is NP-hard, even under binary utilities. However, when there are only a bounded number of types of agents with binary utilities, the problem can be solved in polynomial time. Finally, we present a polynomial time algorithm that computes the minimum subsidy required to achieve envy-freeness for two types of agents with general utilities.

Authors: Zhefan Duan, Huawei Wu

We study the deterministic norm-query complexity of computational problems in finite-dimensional vector spaces over $\mathbb{Q}_p$ equipped with an arbitrary ultrametric norm. For orthogonalization, we prove that no uniform finite query bound depending only on the dimension exists: for every deterministic algorithm that produces an $N$-orthogonal basis for every ultrametric norm $N$, the number of norm queries is unbounded as $N$ varies. We then study the Longest Vector Problem (LVP) for a rank-$m$ $p$-adic lattice. By adapting a brute-force search to the general norm-query setting and eliminating the scalar redundancy among nonzero coefficient vectors modulo $p$, we obtain an algorithm using exactly $(p^m-1)/(p-1)$ norm queries for $m\ge 2$, and prove that no deterministic norm-query algorithm can use fewer queries in the worst case. Finally, we consider the Closest Vector Problem (CVP). Apart from the trivial cases in which no norm query is needed, we prove that the deterministic worst-case norm-query complexity of the CVP is unbounded, even when the lattice and the target vector are fixed.

Authors: Hongmin Li

We study fixed-layout local verification with target-dependent local tests. Let $M$ be a random variable on $\{0,1\}^K$, and let $S$ record the test selected at each coordinate. For each $s\in\operatorname{supp}(S)$, let $F_s$ be the corresponding target fiber and set $D_{\mathrm{fib}}=\max_s\operatorname{VCdim}(F_s)$. We prove $H(M\mid S)\le \log_2\!\left(\sum_{j=0}^{D_{\mathrm{fib}}}\binom Kj\right)$. A fiber that shatters $d$ coordinates yields a weak relaxed locally decodable code with message length $d$ and block length $d+P$ over the original proof alphabet. For a uniform $K$-bit target and fixed proof alphabet, $Q$, and $σ$, the Goldberg--Gur--Saraogi lower bound implies that $I(M;S)\leγK$, for fixed $γ<1$, forces $P=Ω\!\left(K^{1+1/a}/(\log K)^{2+2/a}\right)$, where $a=\lceil Q/σ\rceil$. If $P\le K(\log K)^c$, then $I(M;S)\ge K-O\!\left(K^{a/(a+1)}(\log K)^{3+ac/(a+1)}\right)=K-o(K)$. Any discrete verifier state $T$ determining $S$ satisfies the same information lower bound. Bounded-randomness adaptive branches can be simulated nonadaptively by exposing their decision trees. A branch using at most $r$ random bits and $q$ adaptive proof queries yields a decoder with perfect completeness and at most $1+2^{r+1}\sum_{j

Authors: Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, Madhu Sudan, Sophus Valentin Willumsgaard

We study low-degree testing for group-valued functions over a Boolean slice. Specifically given a degree parameter $d$ and oracle access to a function $f:\{0,1\}^n_{n/2}\to G$ where $\{0,1\}^n_k$ denotes the set of vectors in $\{0,1\}^n$ of Hamming weight $k$ and $G$ is an Abelian group, the low-degree testing problem asks us to distinguish the case where $f$ is a polynomial of degree at most $d$ (with coefficients from $G$) or is $\varepsilon$-far from the set of all such polynomials. Classical works in this area considered functions with domain $\mathbb{F}_q^n$ and range $\mathbb{F}_q$. More recent works have considered the setting where the domain is the Boolean cube [Bafna, Srinivasan, Sudan (Random Struct. Algorithms 2020), Amireddy, Srinivasan, Sudan (RANDOM 2023)], or when the domain is the slice (i.e., $\{0,1\}^n_{k}$) and the range is $\mathbb{F}_2$ [David, Dinur, Goldenberg, Kindler and Shinkar (SIAM J. Comput. 2017), Kalai, Lifshitz, Minzer and Ziegler (FOCS 2024)]. Each of the changes introduces new challenges in designing and analyzing low-degree tests and this happens again in our setting with domain being a slice and range is general. Our main theorem gives a test that makes $O_d(1)$ queries to $f$ and accepts degree-$d$ functions while rejecting functions that are $\varepsilon$-far with probability $Ω(\varepsilon)$. The central proof idea is to reduce this low-degree testing problem to the problem of low-degree testing on the cube. Specifically we show how to randomly embed the $n/2$-dimensional cube $\{0,1\}^{n/2}$ in the $n$-dimensional slice while nearly preserving the proximity of $f$ to the space of degree-$d$ polynomials on this cube. While the embedding is simple and natural, the analysis involves a careful induction with a novel use of a basis of degree-$d$ polynomials on slices (from a work of Anstee, Rónyai and Sali (Graphs and Combinatorics 2002)).

Authors: Bowen Liu, Yizhou Wang, Lingqian Meng

Triangle-based measures, commonly referred to as badness functions, are widely employed to quantify the extent to which a distance matrix deviates from an ideal geometric configuration. Different formulations of these functions may capture distinct facets of local non-uniformity, and their behavior is often influenced by the underlying distance metric chosen for evaluation. In practical settings, although a canonical badness function may be conceptually preferred, factors such as computational cost, algorithmic constraints, or data-specific characteristics frequently necessitate the adoption of modified versions-for instance, approximate forms or alternatives defined under different distance metrics. This gives rise to a central question: to what degree do these variants retain the structural consistency properties of their original counterparts? To address this issue, we develop a systematic correlation-based framework for evaluating structural consistency. As an illustrative instantiation of this framework, we compute badness sequences from a set of representative distance matrices alongside randomly generated triangle configurations, which are designed to cover variants that may arise under diverse practical scenarios. We then assess pairwise similarities among these sequences using four correlation coefficients. The experimental outcomes indicate that certain badness variants exhibit a notably high degree of structural consistency, whereas others reveal complementary behavioral patterns; moreover, the choice of distance metric exerts a considerable influence on the observed trends. These findings offer practical insights for the informed selection of distance metrics and triangle badness function variants in tasks including geometric reconstruction, triangulation, and structural analysis of pairwise distance data.

Authors: Zeping Yi, Yongjun Wanga, Baoshan Wang, Jian Zhang, Songyi Liu

Deploying wireless nodes to maximize coverage area within a given region is an important challenge in wireless sensor networks, UAV path planning, base station placement and other industrial fields. This practical problem can be mathematically equivalent to an optimal circle covering problem. Although theoretical optimal configurations exist for simple cases in mathematics, the NP-hard nature of this problem makes it computationally prohibitive for complex polygons with numerous nodes. Existing approaches are usually designed for regular domains, while those applicable to irregular polygons often suffer from poor initialization, excessive coverage overlap and failure to constrain nodes within the boundary, leading to low coverage efficiency and long runtime. To address these issues, we propose an improved quasi-physical dynamic algorithm (IQPD) for wireless node deployment in arbitrary convex polygons. Our contributions are threefold: (1) proposing a structure-preserving initialization that maps a hexagonal close packing pattern into the target polygon via scaling and affine transformation, ensuring near-optimal initial node distribution; (2) constructing a refined virtual force model by incorporating friction and a radius-expansion optimization mechanism to reduce coverage area overlap; (3) developing a boundary encircling strategy leveraging normal and tangential gradients to reposition nodes deployed outside boundaries after initial optimization. Extensive experimental results demonstrate that our method consistently outperforms other new metaheuristic algorithms across diverse convex polygon shapes, including randomly generated data and real-world scenarios. Our method achieves the highest coverage rate and node utilization rate among all compared algorithms, greatly improving wireless coverage efficiency.

Authors: Nicolás Honorato-Droguett

Given $n$ points on a line or closed cycle and a threshold $r>0$, the connectivity-maintenance problem is to move the points so that every gap between consecutive points is at most $r$, whereas the points-spreading problem requires every gap to be at least $r$. Li and Wang [CCCG 2015; CGT 2025] and Chen, Gu, Li, and Wang [SWAT 2012; DCG 2013] gave $O(n)$-time algorithms for the cyclic versions of min-max points-spreading and min-max connectivity-maintenance, respectively. Ghadiri and Yazdanbod [CCCG 2016] gave an $O(n\log n)$-time algorithm for the linear version of min-sum points-spreading. In this paper, we show that the two problems can be reduced in linear time to each other for both objectives and on both linear and cyclic domains. As an implication, min-sum connectivity-maintenance is solvable in $O(n\log n)$ time on both domains. Finally, we extend the reduction to points on a line with individual thresholds when their initial order is preserved.

Authors: Lin F. Yang

Let $k_ε(n)$ be the smallest number of real linear measurements needed by a randomized, oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pmε$, with probability at least $2/3$. For every fixed $0<ε<1$, the proved result is $$ \frac{n^2}{(\log n)^{A_ε}} \;\le\; k_ε(n) \;\le\; C_ε\frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)} $$ for all sufficiently large $n$, where $A_ε,C_ε$ depend only on $ε$. Previously, the best bounds for general linear sketches were $Ω(n)$ and the trivial $O(n^2)$ upper bound (Li, Nguyen, Woodruff, 2019). The theorem therefore nearly resolves the open measurement-complexity question left by that work: the displayed lower and upper bounds are tight up to polylogarithmic factors. In particular, the complexity is $n^{2-o(1)}$, and for every fixed $c>0$, $O(n^{2-c})$ measurements are impossible. The upper bound is obtained by a fixed Gaussian sketch whose decoder combines implicit low-rank recovery with moment estimation on a high-stable-rank residual. The lower bound constructs moment-matched spectra, randomizes their singular vectors, and compares every low-dimensional observation through an odd-order tensor estimate and a Fisher-information path argument.

Authors: Cassandra Marcussen, Edward Pyne, Ronitt Rubinfeld

We give a catalytic logspace algorithm for the problem of graph spectral sparsification. Given an undirected graph $G$ on $n$ vertices and $\varepsilon>0$, our algorithm outputs an $\varepsilon$-spectral sparsifier of $G$ with $O(n\varepsilon^{-2}\log n)$ edges, matching the effective resistance sampling of Spielman and Srivastava (STOC 2008). This gives a new, natural problem in catalytic logspace that is not known to be in deterministic $\mathbf{NC}$ or $\mathbf{SC}$. Our main contribution is an entirely new technique in the compress--or--random paradigm for catalytic logspace that we believe will have further applications. We first analyze effective-resistance sparsification using a pessimistic estimator that can itself be computed in catalytic logspace. The estimator is motivated by the viewpoint of graph quasirandomness and immediately gives a simple, deterministic greedy algorithm for graph sparsification. Subsequently, we show that such a pessimistic estimator can be transformed into an algorithm that performs an in-place compression of a string with bad potential. Our algorithm is based on using the potential function to define a measure over strings, and implementing arithmetic coding using this measure in-place. This compression technique is substantially distinct from all prior tools in the field of catalytic computation.

Authors: Noga Alon, Shay Moran, Shlomo Moran

Consider the following problem of learning an unknown linear order on $n$ items. In each round, the learner guesses a complete ordering of the items and receives either confirmation that the guess is correct or a counterexample: a pair of items in the wrong order. The goal is to identify the unknown order using as few queries as possible. We study this problem when up to $k$ of the returned counterexamples may be untruthful, where $k$ is not known in advance. We determine the optimal query complexity up to constant factors: \[ Θ(n\log n + nk). \] Thus, while the noiseless complexity matches the classical complexity of sorting, each untruthful counterexample incurs an additional cost of order $n$. The upper bound is based on a geometric representation of permutations and Grünbaum's theorem, while the lower bound combines sorting arguments with a Condorcet-type construction. We also study the case where the target ranking has a low-dimensional geometric representation: each item is represented by a point in $\mathbb{R}^d$, and the ranking is obtained by projecting the points onto an unknown direction. For these classes we give an upper bound of $O(d^2\log n+dk)$ and a lower bound of $Ω(d\log n+dk)$, leaving a factor of $d$ gap in the noiseless term.

Authors: Yuta Inoue, Ken-ichi Kawarabayashi, Rintaro Matsuo, Atsuyuki Miyashita, Bojan Mohar, Tomohiro Sonobe

A graph $G$ is \emph{apex} if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.

Authors: Robert Krauthgamer, Asaf Petruschka, Nir Petruschka

Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of $n$-point sets in $\ell_\infty$ and $\ell_2$ spaces of high dimension $d$. For $\ell_\infty$, we design a padded-decomposition algorithm that runs in time $\tilde{O}(nd^2)$, which is near-linear in $n$, and achieves padding parameter $\tilde{O}(\log n)$. Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of $\ell_\infty$ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For $\ell_2$, we design a separating-decomposition algorithm that achieves near optimal separation $\tilde{O}(\sqrt{\log n})$ in almost-linear time $n^{1+o(1)}$. Our bounds improve over known algorithms with similar running time by a factor $Ω(\sqrt{\log n})$, and the techniques have additional applications to spanners and nearest-neighbor search.

Authors: Or Zamir

Consider the canonical universal hash family $h(x)= ((ax+b)\text{ mod } p)\text{ mod } m$, where $a,b$ are chosen uniformly from $\mathbb Z_p$, which we call linear hashing, being used to hash $n$ elements into $m=Θ(n)$ buckets. For any universal family, the expected size of the largest bucket is at least $Ω(\log n / \log\log n)$ and at most $O(\sqrt{n})$. The only improvement upon these trivial bounds for linear hashing is a 2019 upper bound of $\tilde{O}(n^{1/3})$ by Knudsen. We show that for any $p$ sufficiently larger than $n$, there is a set of $n$ keys whose expected maximum load is $n^{Ω(1/\log\log n)}$, proving linear hashing does not have a polylogarithmic maximum load. We extend the same bounds to the classical multiply-shift hash family of Dietzfelbinger, Hagerup, Katajainen, and Penttonen. We prove an equivalence between the maximum load problem to a density variant of arithmetic Kakeya sets. We then complete the lower bound using a construction of Green and Ruzsa of a small set containing long arithmetic progressions with every difference in a prescribed range. Surprisingly, our equivalence also implies that any substantial improvement over Knudsen's upper bound would imply new results about standard arithmetic Kakeya sets.

Authors: Andrew Au

The Bitner--Ehrlich--Reingold algorithm generates the binary reflected Gray code with constant work per codeword, using a focus-pointer array. Its compact update is easy to state but gives little indication of why such pointers should exist. This note reconstructs a geometric route to the algorithm. The sequence of flipped bit positions is the ruler sequence, OEIS A007814. We realize its finite prefixes as in-order traversals of recursively expanding trees, decorate each activation with its nearest ancestor to the right, contract all stack operations between successive outputs into successor jumps, and prepare the bounded level-indexed stack by predicting its future slot values. This gives and proves a branch-based loopless generator. The construction grew from an earlier informal public exposition by the author. Viewing horizontal position in the tree as time then explains each stack-slot write as a prediction for the next activation at the same level. Finally, the two possible future roles---an inherited continuation for a right child and a default continuation for a left child---are scheduled together, leading to the two assignments of the published focus-pointer algorithm. The account is a reconstructed derivation, not a claim about the historical reasoning of Bitner, Ehrlich, or Reingold.

Authors: David Wajc

In this brief note, we close the asymptotic gap between known upper and lower bounds for fractional online matching under edge arrivals. We prove that the optimal competitive ratio for this problem is $1/2+Θ(1/n)$. The algorithm was suggested and analyzed by OpenAI's ChatGPT Sol based on a single prompt. The presentation was streamlined over a few hours, based on a back and forth discussion with the author, who assumes responsibility for any errors.

Authors: Paweł Górecki, Agnieszka Mykowiecka, Jarosław Paszek

The classical duplication episode clustering (EC) model introduced by Guigó et al. in the 1990s provides a foundational approach for inferring genomic duplication events crucial to understanding genome evolution. This model clusters single gene duplications from a collection of gene trees at locations in the species tree to minimize the total number of such locations, called duplication episodes. Here, we introduce NetEC, a novel extension of this problem to phylogenetic networks. To solve NetEC, we first develop a polynomial-time dynamic programming (DP) algorithm for testing whether a given set of network nodes can serve as episode locations. We then propose a main inference algorithm that utilizes this DP component to optimize the episode count; while the feasibility test runs in polynomial time, the full optimization has exponential worst-case complexity, and an optional heuristic mode is provided for larger instances. We also propose an extended episode analysis procedure that identifies additional genomic duplication candidates below reticulation nodes, complementing the main algorithm by resolving potential upward clustering of duplications induced by reticulation. We evaluate our method on simulated data and on an empirical Pandanales dataset comprising over 29,000 gene trees, demonstrating exact and accurate inference of genomic duplication events even in the presence of multiple reticulations.

Authors: John Burke, Ciaran McGoldrick

Current approaches to quantum search fail to deeply exploit extant structure in the underlying Hilbert space. Decomposing the search by this structure empowers new strategies and formulations for quantum search and algorithm design. We present a new decomposition technique acting directly on this structure by recursively decomposing the Hilbert space and constructing the search operator from reflections over the resulting partition. When initial and target states factorise over this partition, dynamics reduce to a single rotation in a two-dimensional plane at each level, with angle given by a scalar recurrence. This recurrence avoids error accumulation from separately bounding success probabilities at each level, yielding an exact state description enabling treatment of the recursion as a whole. We obtain the target state deterministically and derive oracle and non-oracle costs independently of the search setting. For unstructured search, our approach attains the simultaneously optimal $Θ(\sqrt{N})$ oracle and non-oracle gate counts. For spatial search on $d$-dimension grids, it recovers the $O(\sqrt{N})$ time for $d\geq3$ and the $O\bigl(\sqrt{N}(\log N)^{3/2}\bigr)$ bound of Aaronson and Ambainis for $d=2$. The exact description of the recursion extends over our decomposition to new subdivision structures and provides a new approach for applying and analysing recursion in quantum algorithm design.

Authors: Wenxuan Tao, Jianan Wang, Fen Zuo

Quantum states with conserved total spins, or spin eigenfunctions, are important for studying quantum chemistry and quantum manybody physics problems. A typical class of spin eigenfunctions are Dicke states, which attain maximal spins. While we already have many efficient quantum algorithms to prepare Dicke states, it is not yet clear if we could do so for arbitrary spin eigenfunctions deterministically. Generalizing Bärtschi and Eidenbenz's elegant algorithms for Dicke state preparation, we successfully prepare arbitrary spin eigenfunctions characterized by branching paths and binary spin trees. As a byproduct, we also develop the corresponding classical algorithms to reconstruct all these spin states.

Authors: Ganesh P Kumar

The uniform car-parking process is the one-dimensional random sequential adsorption of unit cars on a segment of finite length $s$: cars arrive at uniformly random positions and park wherever they fit, until no gap admits another. This paper develops the exact finite-$s$ theory. The joint density of the parked positions is resolved into jamming cells, on each of which it is a rational function, and evaluated by a subset recursion in $O(2^n n)$ operations; the marginal and gap order statistics are obtained as hyperlogarithms whose weight is fixed by the number of coordinates integrated out; and the absorption count and the aggregate quantities are treated through the integral equation descending from Rényi.

Authors: Julia Baligacs, Bartłomiej Bosek, Paweł Putra, Marek Sokołowski, Anna Zych-Pawlewicz

We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total number of reallocations, also known as the recourse budget. Despite extensive work, the exact recourse complexity of the problem remains unsettled: the best lower bound is $Ω(n \log n)$, whereas the best upper bound is $\mathcal{O}(n \log^2 n)$, where $n$ denotes the number of online vertices. Tight upper bounds of $\mathcal{O}(n \log n)$ are known only for restricted graph classes, such as forests. The best known upper bounds are attained by a very simple and natural algorithm SAP, which after each arrival applies a shortest augmenting path, and it is conjectured to be optimal. All known upper bound analyses of this algorithm do not depend on the particular maximum matching maintained by the algorithm. Consequently, they also apply to a more difficult problem, which we call the malicious matching setting: after each arrival, the maintained matching is replaced by a worst-case maximum matching for the next step. This led to the conjecture that the malicious setting still admits an $\mathcal{O}(n \log n)$ recourse bound, in line with the conjectured optimal complexity of the original model. Our main result is an $Ω(n \log^2 n)$ lower bound for the malicious matching setting, thus disproving the conjecture. Together with the previous upper bound, this settles the asymptotic recourse complexity of the malicious variant of the problem. We complement our lower bound with an upper bound of $\mathcal{O}(n \log n)$ for expander graphs.

Authors: Amir Carmel, Yakov Kosoburd, Robert Krauthgamer

The 3SUM problem asks, given sets $A,B,C$ of integers, whether there exist $a\in A$ and $b\in B$ whose sum belongs to $C$. In the preprocessed variant with unknown $C$, one preprocesses sets $A$ and $B$, each of size $n$, and subsequently answers a query specified by subsets $A'\subseteq A$, $B'\subseteq B$ and a target set $C'$ of size $O(n)$, by solving the 3SUM instance $(A',B',C')$. Kirkpatrick, Kuszmaul, Mathialagan, and Vassilevska Williams [ICALP 2026] gave the first algorithm with subquadratic space for this problem, achieving $\tilde{O}(n^{3/2+ε})$ query time using $\tilde{O}(n^{2-2ε/3})$ space, for every $ε\in[0,1/2]$. Their algorithm employs separate mechanisms for heavy and light targets, and for each heavy target it stores explicitly the list of pairs $(a,b)$ summing to it; these lists dominate the space bound. We present a unified construction that uses a single mechanism for all queries. Instead of storing these lists of pairs, we recover them on demand by leveraging the Fiat--Naor data structure [SICOMP 1999] to invert the function $(a,b)\mapsto (a+b\bmod p)$. This simplification improves the space bound to $\tilde{O}(n^{\max(2-ε, 11/6-ε/3)})$, while maintaining the same query time. Moreover, our construction is the first to achieve subquadratic space while supporting adaptively chosen queries.

Authors: Mina Khoshmehr, Fernando Beltran

In many practical deployments of differential privacy, queries do not arrive all at once. We study online differentially private query answering under a finite zero-concentrated differential privacy (zCDP) contract. In this setting, queries arrive sequentially, carry different accuracy thresholds, and may overlap with information already released. We formulate this setting as residual privacy budgeting: for each arriving query, the mechanism first credits reusable support from previous DP outputs and then spends new budget only on the remaining support required to satisfy the current threshold. The controller separates feasible cases, where the minimal residual support is allocated exactly, from scarcity cases, where a weighted shortfall-conservation optimiser assigns limited support according to query difficulty. We define the weight using the Query Influence Factor (QIF), a diagnostic signal for query difficulty and instability rather than query importance. For scalar Gaussian exact reuse, inverse-variance fusion justifies additive support. We prove zCDP composition, residual minimality, 1-competitiveness against the offline optimum in the feasible regime, and avoidable expenditure for allocators that ignore released history. A scarcity impossibility result shows that no online allocator can guarantee a competitive ratio better than 1/n in threshold satisfaction, contextualising the QIF scarcity layer as a design choice for an inherently hard online problem.

Authors: Sihan Wang

We prove that for every fixed inverse temperature $β< 1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_β\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $β$. The main ingredient is a deterministic criterion for optimal-order Poincaré inequalities in general Ising models, established via the integrated Bakry-Émery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincaré inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincaré inequality were generated by GPT-5.6 Sol Ultra.

Authors: Mohsen Ghaffari, Cheng Jiang

A longstanding limitation of known techniques for parallel derandomization was that they incurred at least polylogarithmic overhead in work. For instance, for fundamental and frequently used problems such as maximal independent set, maximal matching, and $(Δ+1)$-coloring, where $Δ$ denotes the maximum degree of the graph, the best-known deterministic parallel algorithms with polylogarithmic depth used $Ω((m+n)\mathrm{poly}(\log n))$ work on $n$-vertex, $m$-edge graphs; see, e.g., Luby [FOCS '88]. Consequently, at least $\mathrm{poly}(\log n)$ processors were needed for these algorithms to outperform straightforward single-processor algorithms. Recently, Ghaffari and Grunau [FOCS '25] introduced a new parallel derandomization method that substantially reduced the overhead from $\mathrm{poly}(\log n)$ to $\mathrm{poly}(\log\log n)$, achieving work bounds of $O((m+n)\mathrm{poly}(\log\log n))$. In this paper, we settle this line of research by obtaining linear work bounds of $O(m+n)$, thereby achieving truly work-efficient parallel derandomization.

Authors: Janina Reuter

We present an algorithm with a natural geometric interpretation for computing the determinant of a matrix $B\in\mathbb{Z}^{d\times d}$. It improves upon the current fastest deterministic algorithms by a factor of $d^{ω(1)+1-ω(2)}\approx d^{0.1213}$, where $ω(k)$ denotes the exponent required for multiplying a $d\times d$ matrix with a $d\times d^k$ matrix. Our approach builds on a recent result of Klein and Reuter (STOC 2025), who introduced a novel algorithmic idea for lattice basis computation that can be viewed as extending the Euclidean algorithm from $\mathbb{Z}$ to $\mathbb{Z}^d$. By adapting their techniques, we compute the determinant with the same bit complexity as applying the generalized Euclidean algorithm to an input matrix $A\in\mathbb{Z}^{d\times 2d}$ with $\|A\| = \|B\|$, namely $\tilde{O}(d^{ω(2)}\log\|B\|)$. Prior to this work, the fastest deterministic algorithm for computing the determinant required $\tilde{O}(d^{ω(1)+1}\log\|B\|)$ bit operations.

Authors: Rikuya Hamai, Yuto Nakashima, Shunsuke Inenaga

Fine and Wilf [Proc. Amer. Math. Soc. 1965] showed that any string of length at least $p+q-d$ with periods $p$ and $q$ also has period $d=\gcd(p,q)$. For parameterized strings, Apostolico and Giancarlo [Discrete Appl. Math. 2008] proved an analogue with length bound $p+q$, assuming that the two induced bijections commute. Ideguchi et al. [SPIRE 2023] removed this assumption and gave the bound $p+q+\min(p,q)(σ-1)$, where $σ$ is the number of distinct letters. This was later improved by Hamai et al. [SPIRE 2024] to $p+q+\min(p,q)(σ-2)$, which was used to bound the number of non-equivalent parameterized squares. In this paper, we establish the optimal Fine--Wilf type bound for parameterized strings. Namely, if a string $s$ containing $σ$ distinct letters has parameterized periods $p$ and $q$ and satisfies $|s| \ge p+q+(σ-3)d+1$, where $d=\gcd(p,q)$, then $d$ is also a parameterized period of $s$. We also give matching lower-bound instances, proving that our bound is optimal for any $σ\geq 2$.