Theory of Computing Report

Authors: Alex B. Grilo, Marios Rozos

Despite having an unnatural definition, $\mathsf{StoqMA}$ plays a central role in Hamiltonian complexity, e.g., in the classification theorem of the complexity of Hamiltonians by Cubitt and Montanaro (SICOMP 2016). Moreover, it lies between the two randomized extensions of $\mathsf{NP}$, $\mathsf{MA}$ and $\mathsf{AM}$. Therefore, understanding the exact power of $\mathsf{StoqMA}$ (and hopefully collapsing it with more natural complexity classes) is of great interest for different reasons. In this work, we take a step further in understanding this complexity class by showing that the Stoquastic Sparse Hamiltonians problem ($\mathsf{StoqSH}$) is in $\mathsf{StoqMA}$. Since Stoquastic Local Hamiltonians are $\mathsf{StoqMA}$-hard, this implies that $\mathsf{StoqSH}$ is $\mathsf{StoqMA}$-complete. We complement this result by showing that the separable version of $\mathsf{StoqSH}$ is $\mathsf{StoqMA}(2)$-complete, where $\mathsf{StoqMA}(2)$ is the version of $\mathsf{StoqMA}$ that receives two unentangled proofs.

Authors: Ishaan Kannan, Harald Putterman, Jordan Cotler

Quantum information processing has the potential to substantially enhance how we learn from physical experiments, but coupling a quantum processor to an experimental sample introduces noise that can exponentially degrade learning even when the processor itself is fault-tolerant. In this work, we show that fault tolerance can nevertheless be leveraged to recover exponential speedups by embedding the unknown system into an arbitrarily high-distance quantum code with only constant error overhead and running a fault-tolerant learning algorithm. Using this $\textit{quantum uploading}$ procedure, we prove that both classical shadow tomography and the estimation of cubic observables can be performed exponentially faster than by any adaptive strategy that does not immediately upload the state into encoded memory. These separations hold even when the uploading stage is substantially noisier than the bare experimental interface. To prove them, we introduce the Heisenberg learning tree method, a flexible tool for obtaining learning lower bounds when the limited resource is not quantum replicas but an experimentally motivated constraint such as noise. We numerically illustrate the speedups in an astronomical imaging application, where quantum processing of individual uploaded photons locates an exoplanet obscured by a bright star using orders of magnitude fewer shots than unencoded baselines. Our results establish fault-tolerant quantum computation as a valuable tool for learning from quantum experiments.

Authors: Gorgi Pavlov

We introduce the Banach-Butterfly Invariant (BBT), an influence-adaptive Banach geometry on the Walsh-Hadamard butterfly factorization. For a Boolean function $f:\{-1,+1\}^n\to\{-1,+1\}$ with coordinate influences $\mathrm{Inf}_\ell(f)$, BBT assigns exponent $p_\ell = 1+\mathrm{Inf}_\ell(f)$ to butterfly layer $\ell$, yielding the contraction invariant $μ(f)=\prod_\ell 2^{-\mathrm{Inf}_\ell/(1+\mathrm{Inf}_\ell)}$. We prove a Jensen lower bound $\log_2μ(f) \ge -I(f)/(1+I(f)/n)$ and that $μ$ is strictly Schur-convex in the influence vector (modulo permutation), giving scaling classes $μ\sim 2^{-n/2}$ (parity), $2^{-Θ(\sqrt{n})}$ (majority), $2^{-1/2}$ (dictators). $\log_2μ$ is rational but not polynomial in the Fourier coefficients while $μ$ is algebraic, and $μ$ separates functions with identical total influence (122 pairs at $n=3$). Using the certified $n \le 4$ ternary Walsh-threshold universe from a companion synthesis manuscript as a finite testbed, we compute exact MILP minimum-support certificates for all 65,536 Boolean functions at $n=4$ (mean 6.42, max 9, all-odd by a parity argument) and on 10,000 of the 616,126 NPN-canonical representatives we enumerate at $n=5$ (matching OEIS A000370). Conditional Spearman $ρ(μ,|\mathrm{supp}|)$ at fixed total influence is $+0.571$ in the largest stratum at $n=4$ but reverses to $-0.38$ at $n=5$ under both function-uniform and NPN-canonical sampling: $μ$ is a valid Schur-convex concentration invariant, not a universal monotone predictor of minimum support across $n$. A companion application paper validates a real-valued WHT activation-energy proxy inspired by this theory on five pretrained LLMs at W2A16, cutting wikitext-2 perplexity by 15-58% versus vanilla auto-round; the transfer from Boolean theory to the real-valued proxy is qualitative, not formal.

Authors: Riju Bindu, Hamed Hatami, Hasti Karimi, Robert Robere

We prove new upper and lower bounds on $ε$-approximate sign-rank, a relaxation of sign-rank introduced by Chornomaz, Moran, and Waknine (STOC 2025). We show that every $m \times n$ sign matrix with approximate sign-rank $d$ contains a monochromatic rectangle of size $d^{-O(d)}m \times d^{-O(d^2)}n$, paralleling classical results for exact sign-rank. As an application, we establish a lower bound of $Ω(\sqrt{d/\log d})$ on the $ε$-approximate sign-rank of large-margin $d$-dimensional half-spaces. Prior to our work, the only general lower bound technique known for approximate sign-rank yielded bounds of strength $ε^{-1} - 1$, which are constant for fixed $ε$. A key ingredient is a new geometric theorem on hyperplane avoidance: for any set of $n$ points in general position in $\mathbb{R}^d$, there exist $d$ subsets, each of size $d^{-O(d)} n$, such that no hyperplane simultaneously splits all of them. The proof combines the Forster-Barthe isotropic position theorem with the Bourgain-Tzafriri restricted invertibility principle. We also study the relationship between approximate sign-rank and VC dimension. We prove a lower bound on approximate sign-rank in terms of VC dimension, and exhibit concept classes of VC dimension $2$ with large approximate sign-rank. Finally, we study the approximate sign-rank of the $2^m \times 2^m$ Hadamard matrix $H_m$. The sign-rank of $H_m$ is known to be $Ω(\sqrt{2^m})$ by Forster's classic theorem. Contrasting this, we adapt an argument of Alman and Williams to show that the approximate sign-rank of $H_m$ is at most $m^{O(\sqrt{m} \log(1/ε))}$, and hence the Hadamard matrix does not witness polynomial-strength lower bounds for approximate sign-rank. Using our VC dimension bound, we prove that the approximate sign-rank of $H_m$ is at least $Ω_ε(m)$.

Authors: Mohammad Mahdi Khodabandeh, Igor Shinkar

In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $P = (P_1,\dots, P_n)$ of functions $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ defines a distribution over $\{0,1\}^n$ in the natural way: draw $X$ uniformly at random from $\mathbb{F}_2^m$ and output $(P_1(X),\dots, P_n(X)) \in \{0,1\}^n$. We show that when $P$ is defined by polynomials of degree $d$, the total variation distance of $P$ from the product distribution $\mathrm{Ber}(1/3)^{\otimes n}$ is $1-o_n(1)$, where $o_n(1)$ is a vanishing function of $n$ for any constant degree $d$. For small values of $d$, we show the following concrete bounds. (i) For $d=1$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(n))$. (ii) For $d=2$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(\log(n)/\log\log(n)))$. (iii) For $d=3$ we have $\|P-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-Ω(\sqrt{\log\log(n)}))$. Our results extend the recent lower bound results for sampling distributions, which have mostly focused on local samplers, small depth decision trees, and small depth circuits. As part of our proof, we establish the following result, that may be of independent interest: for any degree-$d$ polynomial $P\colon\mathbb{F}_2^m \to \mathbb{F}_2$ it holds that $\Pr_X[P(X) = 1]$ is bounded away from $1/3$ by some absolute constant $δ= δ_d>0$. Although the statement may seem obvious, we are not aware of an elementary proof of this. The proof techniques rely on the structural results for low degree polynomials, saying that any biased polynomial of degree $d$ can be written as a function of a small number of polynomials of degree $d-1$.

Authors: Etienne Corman, Yousuf Soliman, Robin Magnet, Mark Gillespie

We introduce an implicit representation of continuous, bijective, orientation-preserving maps between genus zero surfaces with or without boundary. The distortion of these maps can easily be minimized by optimizing the Ginzburg-Landau functional - a ubiquitous model in physics and differential geometry - leading to a simple algorithm for computing bijective correspondences using only standard tools of the tangent vector field toolbox. The method avoids combinatorial mesh modifications and does not require barrier functions to enforce bijectivity making it more robust to noise and simpler to implement. Moreover, the algorithm does not assume a bijective initialization and can untangle non-bijective correspondences generated by computationally cheaper methods such as functional maps. It supports the use of both landmark points and landmark curves to guide the correspondence. The key idea is that a bijection between surfaces defines a two-dimensional mapping surface sitting inside the four-dimensional product space of the two inputs, and this mapping surface can be stored implicitly as the zero set of a complex section - essentially a complex function defined on the product space. Now the distortion of the map can be optimized by minimizing the area of this mapping surface, which amounts to minimizing the Ginzburg-Landau functional of the complex section. We demonstrate the practical benefits of our method by comparing to state-of-the-art correspondence algorithms and show that our implicit representation offers improved stability and naturally supports constraints that are difficult to enforce with explicit map representations.

Authors: Pengfei Wang, Qinghao Guo, Haisen Zhao, Shiqing Xin, Shuangmin Chen, Changhe Tu, Wenping Wang

k-nearest neighbor (k-NN) search is a fundamental primitive in geometry processing and computer graphics. While spatial partitioning structures such as kd-trees are standard, they are often manifold-blind, failing to exploit the intrinsic low-dimensional structure of points sampled from 2-manifolds. Recent advances in dynamic programming-based nearest neighbor search (DP-NNS) leverage incrementally constructed Voronoi diagrams to accelerate queries, where each site p maintains a list of successors that progressively refine its Voronoi cell. However, DP-NNS is restricted to single nearest neighbor (k=1) searches, precluding their adoption in applications that require local neighborhood statistics. In this paper, we generalize the DP-NNS framework to support arbitrary k-NN queries for manifold-aligned data. Our approach is founded on the geometric observation that if p_i is the nearest neighbor of a query q in P, then the second nearest neighbor of q must reside either within the prefix set P_{1:i-1} = {p_1, \dots, p_{i-1}} or within p_i's successor list. By recursively extending this principle, we introduce Manifold k-NN, a recursive algorithmic scheme that significantly outperforms conventional kd-trees for manifold-aligned data. Our method achieves a 1\times--10\times speedup in volume-to-surface query scenarios and inherently supports dynamic prefix queries -- enabling k-NN searches within any subset P_{1:m} (m \leq n) with zero overhead. Furthermore, we extend the framework to support point deletion via local Delaunay updates, providing a complete suite of dynamic operations for point set modification. Comprehensive experiments on diverse geometric datasets demonstrate the efficiency and broad applicability of our approach for modern graphics pipelines. Source code is available at https://github.com/sssomeone/manifold-knn.

Authors: Apurva Mudgal

We give a greedy sweepline algorithm for a Jordan curve and prove that it is maximal in the sense of [1]. Our proof uses Kőnig's lemma.

Authors: Satyabrata Jana, Debabrata Pal, Bodhayan Roy, Sasanka Roy

We study the Witness Set problem, a natural dual to the classical Art Gallery problem. In the Witness Set problem, we are given a polygon $P$ and an integer $k$ as input, and the objective is to determine whether $P$ has a witness set of size at least $k$. A point set $X$ in $P$ is called a witness set if every point in $P$ is visible from at most one point in $X$. For simple polygons, we show that Witness Set lies in both $NP$ and $XP$. This stands in sharp contrast to its dual, the Art Gallery problem, which was recently shown to be $\exists \mathbb{R}$-complete by Abrahamsen et al. and is therefore neither in $NP$ nor admits a polynomial-size discretization unless $NP=\exists \mathbb{R}$. In contrast, we prove that Witness Set for simple polygons admits a finite discretization of size $n^{f(k)}$ for some function $f$. For comparison, even for simple polygons, Efrat and Har-Peled gave an algorithm for Art Gallery running in time $n^{O(k)}$ using tools from real algebraic geometry, and it appears difficult to obtain such algorithms without this machinery. On the other hand, our approach for Witness Set is purely combinatorial and relies on discretization, leading to an $n^{f(k)}$-time algorithm. Although Amit et al. claimed more than fifteen years ago that Witness Set is $NP$-hard, no proof or reference was provided. We show that the discrete version of the Witness Set problem - where the witness set must be chosen from a given finite point set $Q$ (instead of allowing witnesses to be chosen anywhere in the polygon), referred to as Discrete Witness Set - is $NP$-complete, even when the input is restricted to rectilinear polygons with holes. However, for simple polygons, Discrete Witness Set admits a polynomial-time algorithm by Das et al. Thus, it remains an open question whether the Witness Set problem is $NP$-hard.

Authors: Takashi Yoshino

This paper establishes a rigorous geometrical framework for spherical origami, origami using spherical sheets based on spherical geometry. Two settings are treated: origami restricted to the unit sphere ($\mathbb{S}^2$), and three-dimensional folding of spherical sheets in space. For origami on $\mathbb{S}^2$, the definitions of Euclidean origami are systematically extended to the spherical setting, and all seven Huzita--Justin axioms are shown to admit explicit equations in spherical geometry. For three-dimensional folding, equidistant curves are introduced as fold curves, replacing geodesics and enabling a richer family of folds. The framework is validated by successfully constructing computer graphics of spherical origami birds, demonstrating both the theoretical completeness and practical utility of the proposed approach.

Authors: Guangyan Zhou, Bin Wang, Jianxin Wang, Ke Xu

In this paper, we investigate the hitting set problem and demonstrate that solution independence is the crucial property underlying the construction of self-referential instances. As a special case of the hitting set problem, the vertex cover problem lacks the solution independence property. This distinction accounts for its ability to evade exhaustive search, as correlations among candidate solutions can be leveraged to compress the overall search space. In contrast, the dominating set problem on hypergraphs, which is also a special case of the hitting set problem, satisfies the solution independence property, thereby enabling the construction of self-referential instances. Moreover, we prove that these self-referential instances possess an irreducible property, implying that any algorithm for solving such instances must process nearly the entire graph to yield a correct solution.

Authors: Guillaume Ducoffe

A vertex in a graph is called central if it minimizes its maximum distance to the other vertices. The radius of a graph $G$ is the largest distance between a central vertex and the other vertices, and it is denoted by $rad(G)$. In the center problem, we are asked to find a central vertex. We study the fine-grained complexity of the center problem on graphs with small Gromov hyperbolicity. Roughly, the Gromov hyperbolicity of a graph represents how close, locally, it is to a tree, from a metric point of view. It has applications in the design of approximation algorithms. In particular, there is a linear-time algorithm that for every $δ$-hyperbolic graph $G$ outputs some vertex at distance at most $rad(G) + 5δ$ to the other vertices [Chepoi et al, SoCG'08]. However, a linear-time algorithm for computing a central vertex is known only for $0$-hyperbolic graphs, whereas its existence was ruled out for $2$-hyperbolic graphs under the Hitting Set Conjecture of [Abboud et al, SODA'16]. Our main contribution in the paper is a linear-time algorithm for computing a central vertex in the class of $\frac 1 2$-hyperbolic graphs. Furthermore, we rule out the existence of such an algorithm for $1$-hyperbolic graphs, under the Hitting Set Conjecture, thus completely settling all the cases left open.

Authors: Vedat Levi Alev, Daniel Frishberg, Mihalis Sarantis, Prasad Tetali

We prove an $\widetilde O(n^2)$ bound for the \emph{relaxation time} and the \emph{log-Sobolev time} (inverse log-Sobolev constant) of the classical triangulation flip chain on a convex $(n+2)$-gon, implying a mixing time of $\widetilde O(n^2)$. The previous state of the art for the mixing time of this chain, due to Eppstein and Frishberg, was $\widetilde O(n^3)$, while the best known lower bound on the mixing time, due to Molloy, Reed, and Steiger, is $Ω(n^{3/2})$. Our relaxation time bound makes significant progress towards Aldous' conjectured bound of $Θ(n^{3/2})$ for the relaxation time. We improve upon the analysis of Eppstein and Frishberg by further developing the framework of \emph{transport flows} introduced in the work of Chen et al. In this light, our results can be seen as a more efficient way of using combinatorial decompositions to obtain functional inequalities for Markov chains. We hope our ideas will find other applications in the future.

Authors: Samuel Boardman

Given an undirected graph representing similarities between a set of items and an additive measure evaluating the items, we treat the position of a special subset of items in an ordinal ranking through a collection of combinatorial optimization problems in which items may be combined if they are similar. The objective for these problems is to either maximize or minimize the absolute or relative rank of the special subset, with a meta-goal of assessing the robustness of the rank, even in the presence of a well-defined criterion. We classify the computational complexity of all four problems, mostly finding worst-case hardness, then find exact and approximate solutions to special cases and variants of the problems. These structured cases are inspired by several real-world examples and may be used to assess commonly cited facts across disparate domains, as we demonstrate for sources of greenhouse gas emissions that contribute to climate change.

Authors: Ashwin Jacob, Arpit Kumar, Diptapriyo Majumdar

Vertex deletion to hereditary graph class is well-studied in parameterized complexity. Vertex deletion to the scattered graph classes has gained attention in recent years. In this paper, we consider (Proper-Interval, Tree)-Vertex Deletion, the input to which is an undirected graph $G = (V, E)$ and an integer $k$. The goal is to pick a set $X \subseteq V(G)$ of at most $k$ vertices such that $G - X$ is a simple graph and every connected component of $G - X$ is a proper interval graph or a tree. When parameterized by the solution size $k$, (Proper-Interval, Tree)-Vertex Deletion has been proved to be fixed-parameter tractable by Jacob et al. [JCSS-2023, FCT-2021]. In this paper, we consider this problem from the perspective of polynomial kernelization. We provide a first nontrivial polynomial kernel for (Proper-Interval, Tree)-Vertex Deletion, with $O(k^{33})$ vertices.

Authors: Binh-Minh Bui-Xuan, Michel Habib, Fabien de Montgolfier, Renaud Torfs

It is well-known since the seventies of last century that Depth First Search (DFS) can be used to compute strongly connected components [RE. Tarjan. SIAM Journal on Computing, 1972] and Breadth First Search (BFS) can be used to compute distance in graphs [GY. Handler. Transportation Science, 1973]. We furthermore demonstrate that these standard graph searches are powerful enough to recognize and certify several well-structured graph classes. Specifically, we provide a single DFS approach for recognizing and certifying trivially perfect graphs that is significantly simpler than previous methods using [FPM. Chu. Information Processing Letters, 2008]. We further show that a single BFS can recognize split graphs and bipartite chain graphs, and we improve upon the triple LexBFS algorithm for proper interval graphs [DG. Corneil. Discrete Applied Mathematics, 2004] by proposing a two consecutive BFS recognition scheme. These results are underpinned by characterizations using vertex orderings that avoid specific patterns [L. Feuilloley, M. Habib. SIAM Journal on Discrete Mathematics, 2021]. Finally, we provide a structural study of connected proper interval graphs, proving that their characterizations via special orderings are unique up to reversal and the permutation of true twins.

Authors: Arpon Basu, Joshua Brakensiek, Aaron Putterman

We study the problem of Hamiltonian sparsification: given a parameter $\varepsilon \in (0,1)$ and an $n$-qubit Hamiltonian $H$ which is the sum of $r$-local positive semi-definite (PSD) terms $H_1, \dots H_m$, our goal is to compute a sparse set $L \subseteq [m]$, along with weights $w: L \rightarrow \mathbb{R}_{\geq 0}$ such that for every state $|ψ\rangle\in \mathbb{C}^{2^n}$, $$ \sum_{i \in L} w(i) \langle ψ| H_i | ψ\rangle \in (1 \pm ε) \sum_{i = 1}^m \langle ψ| H_i | ψ\rangle $$. When the set $L$ is significantly smaller than $m$, this reduces the number of terms in the underlying system, while still ensuring that the behavior of the system is essentially unchanged. We show that many Hamiltonians indeed are sparsifiable to a number of terms much smaller than $n^r$, including: (a) Hamiltonians where each term is an $r$-local Pauli string, (b) Hamiltonians where each term is an $r$-local random operator of rank $R$, for $R \geq 2^{r-1}+1$, and (c) Hamiltonians where each term is an arbitrary $r$-local operator of rank $\geq 2^r -1$ (a.k.a. Quantum SAT). Taken together, our results show that the sparsifiability of Hamiltonians is a robust phenomenon, contrary to prevailing belief (see for instance, Aharonov-Zhou ITCS 2019, QIP 2019). Our results find applications, for instance, to better (semi-)streaming algorithms for quantum Max-Cut, answering a question left open by Kallaugher and Parekh (FOCS 2022). In fact, our results even codify that quantum systems are often easier to sparsify than their classical counterparts.

Authors: Jakob Nogler, Virginia Vassilevska Williams

Given a graph and two fixed vertices $s$ and $t$, the Replacement Path Problem (RP) is to compute for every edge $e$, the distance between $s$ and $t$ when $e$ is removed. There are two natural extensions to RP: (1) Single Source Replacement Paths (SSRP): Given a graph $G$ and a source node $s$, compute for every vertex $v$ and every edge $e$ the $s$-$v$ distance in $G \setminus \{e\}$. That is, we do not fix the target anymore. (2) $2$-Fault Replacement Paths (2-FRP): Given a graph $G$ and two nodes $s$ and $t$, compute for every pair of edges $e, e'$ the $s$-$t$ distance in $G \setminus \{e, e'\}$. That is, we consider two failures instead of one. Previously, there was no known reduction between SSRP and 2-FRP. It seemed plausible that 2-FRP would be computationally harder because there are no settings where 2-FRP admits a faster algorithm than SSRP. In directed unweighted graphs there is a provable gap in complexity, and in undirected graphs many of the known 2-FRP algorithms in a variety of settings are much slower than those for SSRP in the same setting. The main contribution of this paper is a tight reduction from undirected $2$-FRP to undirected SSRP, showing that contrary to prior intuition, 2-FRP is not harder than SSRP. As our reduction is weight-preserving, we obtain the first algorithms for $2$-FRP that match the best-known runtimes for SSRP: (1) $\tilde{O}(M n^ω)$ for weights in $[1, M]$ [GVW19], improving upon $O(Mn^{2.87})$ [CZ24]; (2) $n^3/2^{Ω(\sqrt{\log n})}$ for weights in $[1, \text{poly}(n)]$ [GVW19], improving over the previous $n^3\text{polylog}(n)$ running time [VWWX22]; (3) $\tilde{O}(mn^{1/2}+n^{2})$ combinatorial time for unweighted graphs [CC19], and more generally for rational weights in $[1, 2]$ [CM20], improving upon $\tilde{O}(n^{3-1/18})$ [CZ24]. We complement these upper bounds with tight lower bounds under fine-grained hypotheses.

Authors: Irene Aldridge

This paper proposes an eigenvalue-based small-sample approximation of the celebrated Markov Chain Monte Carlo that delivers an invariant steady-state distribution that is consistent with traditional Monte Carlo methods. The proposed eigenvalue-based methodology reduces the number of paths required for Monte Carlo from as many as 1,000,000 to as few as 10 (depending on the simulation time horizon $T$), and delivers comparable, distributionally robust results, as measured by the Wasserstein distance. The proposed methodology also produces a significant variance reduction in the steady-state distribution.

Authors: Huayi Wang, Jingfan Meng, Jun Xu

Approximate nearest neighbor search under universal L_p metrics (ANNS-U-L_p) is an important and challenging research problem, as it requires answering queries under all possible p (0

Authors: Christian Coester, Romain Cosson

We study the design of computationally efficient randomized algorithms for the $k$-server problem. Existing randomized algorithms with the best known competitive ratios are, on the one hand, inherently implicit and, on the other hand, employ a rounding scheme that maintains a distribution over exponentially many configurations. In this work, we introduce a derandomization framework that transforms any randomized $k$-server algorithm on a hierarchically separated tree into one that uses only $O(\log k)$ random bits for request sequences of arbitrary length; hence maintaining a distribution over only polynomially many server configurations. Leveraging this black-box derandomization, we obtain the first polynomial-time randomized $k$-server algorithm on arbitrary $n$-point metrics with a polylogarithmic competitive ratio. Our results also have implications for the advice complexity of the $k$-server problem.

Authors: Satoru Iwata, Haruto Konno

Symmetric submodular function minimization admits purely combinatorial algorithms using special orderings of the ground set. Extending the minimum-cut algorithm of Nagamochi and Ibaraki (1992), Queyranne (1998) showed that the maximum adjacency ordering yields a pendent pair, which can be used to find a nontrivial minimizer. Nagamochi (2010) later introduced the minimum degree ordering, which yields a flat pair and leads to the identification of extreme sets. Despite the apparent similarity between these two algorithms, their connection remained unclear. In this paper, we introduce yet another ordering called minimum capacity ordering, and extend it to a one-parameter family of orderings, called $α$-orderings, that unifies these two previously known orderings. We prove a general inequality for $α$-orderings, and our framework recovers the known pendent-pair and flat-pair results as special cases, corresponding to $α= -1$ and $α= 1$, respectively. For each $α\in [-1, 1]$, the last two elements of an $α$-ordering form a contractible pair, i.e., a pair whose contraction preserves the existence of a nontrivial minimizer, which leads to a contraction algorithm that finds a nontrivial minimizer of a symmetric submodular function in $O(n^3)$ oracle calls, where $n$ is the cardinality of the ground set. In addition, we discuss the ranges of $α$ that ensure $α$-ordering to obtain these special pairs.

Authors: Tal Wagner

We study query time bounds for the fundamental problem of estimating the kernel mean $\frac1{|X|}\sum_{x\in X}\mathbf{k}(x,y)$ of a query $y$ in a finite dataset $X\subset\mathbb{R}^d$ up to a prescribed additive error $\varepsilon$. The best known bounds for the Gaussian kernel are $O(d/\varepsilon^2)$, $\widetilde O(d+1/\varepsilon^4)$, and $\widetilde O(d+Δ^2/\varepsilon^2)$, where $Δ$ is the diameter of a region containing the points. We prove the new bound $\tilde O(d+\varepsilonΔ^2+1/\varepsilon^3)$, which improves over the previous ones in regimes with small error $\varepsilon$ and intermediate diameter $Δ$. At the center of our proof is a new fast spherical embedding theorem in the sense introduced by Bartal, Recht and Schulman (2011), which limits the embedded data diameter while preserving local Euclidean distances and avoiding ``distance collapse'' at larger scales. This fast embedding theorem may be of independent interest.

Four‑year fellowship at Imperial College London for early‑career researchers. Open across CS. Sponsorship available for outstanding Theory of Computing candidates. Full freedom to define and drive an independent research agenda, with no obligatory teaching; salary + research budget (~£45k), strong potential towards permanence. (Departmental deadline 22 June.)

Website: https://www.imperial.ac.uk/research-and-innovation/research-office/funder-information/research-fellowships/irf/
Email: iddo.tzameret@gmail.com

By shacharlovett

Lutz (1987) introduced resource-bounded category and showed the circuit size class SIZE($\frac{2^n}{n}$) is meager within ESPACE. Li (2024) established that the symmetric alternation class $S^E_2$ contains problems requiring circuits of size $\frac{2^n}{n}$. In this note, we extend resource-bounded category to $S^E_2$ by defining meagerness relative to single-valued $FS^P_2$ strategies in the Banach-Mazur game. We show that Li’s $FS^P_2$ algorithm for the Range Avoidance problem yields a winning strategy, proving that $SIZE(\frac{2^n}{n})$ is meager in $S^E_2$. Consequently, languages requiring exponential-size circuits are comeager in $S^E_2$: they are typical with respect to resource-bounded category.

Authors: Jing-Yuan Wei

We study the problem of locating violating principal minors in structured matrix families that lie near the boundary of P-matrices and admit sparse violations under perturbation. Viewing violation search as an information acquisition problem, we show that, despite strong underlying structure, the location of a violation is globally encoded and not accessible through local queries. This leads to an information-theoretic bottleneck: each query reveals only vanishing information about the violating subset, so that polynomially many queries accumulate insufficient information to identify it. Using mutual information and Fano's inequality, we show that any algorithm restricted to polynomially many queries cannot recover the violating subset with constant success probability. Our analysis highlights a fundamental distinction between structure and accessibility: even highly structured problems can be computationally intractable when the information required to locate a solution is not accessible through the available queries.

Authors: Eike Neumann, Margret Tembo

We study the problem of deciding universal termination of linear and affine loops over the reals in the bit-model of real computation. We show that both problems are as close to decidable as one can expect them to be: there exist sound partial algorithms that halt on all problem instances whose answer is robust under all sufficiently small perturbations. We further show that in each case the set of non-robust problem instances has Lebesgue measure zero.

Authors: Adam Kurpisz, Lucas Slot, Mikhail Zaytsev

We analyze the sum-of-squares rank of unweighted instances of the Minimum Knapsack (MK) problem, i.e., minimization of $\sum_{i=1}^n x_i$ for 0/1 variables under the constraint $\sum_{i=1}^n x_i \geq q$, with $q \in \mathbb{R}$. Such instances have long served as a testbed for understanding the limitations of lift-and-project methods in Boolean optimization. For example, both the Lovász-Schrijver and Sherali-Adams hierarchies require (maximal) rank $n$ to solve them, already when $q=1/2$ is constant. The SOS hierarchy requires only \emph{sublinear} rank $O(\sqrt{n})$ to solve unweighted MK when $q=1/2$. On the other hand, when $q$ is allowed to vary with~$n$, the SOS rank of the problem may become linear. Interestingly, this is known to happen both when $q$ is large, and when $q$ is very small ($0

Authors: Kevin Buchin, Mark Joachim Krallmann, Frank Staals

Let $S$ be a set of $n$ points in $\mathbb{R}^2$. Our goal is to preprocess $S$ to efficiently compute the smallest enclosing disk of the points in $S$ that lie inside an axis-aligned query rectangle. Previous data structures for this problem achieve a query time of $O(\log^6 n)$ with $O(n \log^2 n)$ preprocessing time and space by lifting the points to 3D, dualizing them into polyhedra, and searching through their intersections. We present a significantly simpler approach, solely based on 2D geometric structures, specifically 2D farthest-point Voronoi diagrams. Our approach achieves a deterministic query time of $O(\log^4 n)$ and, via randomization, an expected query time of $O(\log^{5/2} n \log\log n)$ with the same preprocessing bounds.

Authors: Patrizio Angelini, Sabine Cornelsen, Giordano Da Lozzo, Fabrizio Frati, Philipp Kindermann, Ignaz Rutter, Johannes Zink

We consider upward-planar layered drawings of directed graphs, i.e., crossing-free drawings in which each edge is drawn as a y-monotone curve going upward from its tail to its head, and the y-coordinates of the vertices are integers. The span of an edge in such a drawing is the absolute difference between the y-coordinates of its endpoints, and the span of the drawing is the maximum span of any edge. The span of an upward-planar graph is the minimum span over all its upward-planar drawings. We study the problem of determining the span of upward-planar graphs and provide both combinatorial and algorithmic results. On the combinatorial side, we present upper and lower bounds for the span of directed trees. On the algorithmic side, we show that the problem of determining the span of an upward-planar graph is NP-complete already for directed trees and for biconnected single-source graphs. Moreover, we give efficient algorithms for several graph families with a bounded number of sources, including st-planar graphs and graphs where the planar or upward-planar embedding is prescribed. Furthermore, we show that the problem is fixed-parameter tractable with respect to the vertex cover number and the treedepth plus the span.

Authors: Julia Chuzhoy, Sanjeev Khanna, Junkai Song

In the fully dynamic maximal matching problem, the goal is to maintain a maximal matching in a graph undergoing an online sequence of edge insertions and deletions. The problem has been studied extensively in the oblivious-adversary setting, where randomized algorithms with polylogarithmic worst-case and constant amortized update time have been known for some time. A major challenge in this area has been designing an algorithm with non-trivial update time against an adaptive adversary. In a recent breakthrough, Bernstein, Bhattacharya, Kiss, and Saranurak (STOC 2025; hereafter, BBKS25) obtained the first algorithms with sublinear update time for this setting: namely, a randomized algorithm with $\tilde{O}(n^{3/4})$ amortized update time, and a deterministic algorithm with $\tilde{O}(n^{8/9})$ amortized update time. Our main result is a deterministic algorithm for fully dynamic maximal matching with amortized update time $n^{1/2+o(1)}$. A powerful tool in dynamic matching is the use of matching sparsifiers: sparse subgraphs that preserve enough information to recover matchings with desired properties. Sparsifiers, such as the EDCS data structure, have been successfully used for approximate maximum matching. For maximal matching, however, this paradigm is not as natural, since maximality must hold with respect to the entire graph. Nevertheless, BBKS25 showed that EDCS can be repurposed as a verification-and-repair mechanism for fully dynamic maximal matching against adaptive adversaries. We introduce a new deterministic framework, referred to as the subgraph system, which, in contrast to EDCS, is purpose-built for verification and maintenance of maximality. It is also designed to allow efficient recursive refinements leading to stronger and stronger parameters, that yield our deterministic algorithm with $n^{1/2+o(1)}$ amortized update time.

Authors: Babak Ghanbari, Robert Šámal

We present a near-linear-time algorithm that, given a bridgeless cubic graph, finds a perfect matching intersecting every 3-edge-cut in exactly one edge. This improves over a cubic algorithm of Boyd et al. for the same problem, and over our previous algorithm, which worked only for 3-edge-connected graphs. The main ingredient is a cactus representation of the 2-edge-cuts, together with an efficient update procedure under 2-cut reductions.

Authors: Zichun Ye, Runqi Wang, Xuchuang Wang, Xutong Liu, Shuai Li, Mohammad Hajiesmaili

Machine unlearning aims to unlearn data points from a learned model, offering a principled way to process data-deletion requests and mitigate privacy risks without full retraining. Prior work has mainly studied unsupervised / supervised machine unlearning, leaving unlearning for sequential decision-making systems far less understood. We initiate the first study of a foundational sequential decision-making problem: offline stochastic multi-armed bandits (MAB). We formalize the privacy constraint for offline MAB and measure utility by the post-unlearning decision quality. We conduct a systematic study of both single- and multi-source unlearning scenarios under two data-generation models, the fixed-sample model and the distribution model. For these settings, our algorithmic design is built on two canonical base algorithms: Gaussian mechanism and rollback, and we propose adaptive algorithms that switch between them according to the data regime and privacy constraint. We further introduce a mixing procedure that elucidates the rationale behind these baselines. We provide performance guarantees across the above settings and establish lower bounds under both dataset models. Experiments validate the predicted tradeoffs and demonstrate the effectiveness of the proposed methods.

Authors: Moshe Lewenstein, Ely Porat

We study the "set parameterized matching" problem, a generalization of the classical parameterized matching problem introduced by Baker. In set parameterized matching, both the pattern and text are sequences where each position contains a set of characters rather than a single character. Two set-strings parameterized match if there exists a bijection between their alphabets that maps one to the other set-wise. Boussidan introduced this problem for the case of equal-length set-strings. We present a randomized algorithm running in $O(N + M)$ time with high probability, where $N$ is the text size and $M$ is the pattern size. Our approach employs a novel three-layer hashing scheme based on Karp-Rabin fingerprinting that addresses the challenges of (1) the size blowup in representations of the problem, (2) set-to-set matching, and (3) the dynamic nature of encodings of text substrings during pattern scanning.

Authors: Shuchi Chawla, Kristin Sheridan

We consider the problem of finding the value of a maximum flow over time in a network with uniform edge lengths where the edge capacities change at specific time instants. To solve this problem, we show how to construct a condensed version of a Time Expanded Network (cTEN) whose standard max flow value is the same as the max flow over time on the original network. In particular, for a graph with $n$ nodes, $m$ edges, and $μ$ {\em critical times} where some edge capacity changes, we obtain a cTEN with $O(n^2μ)$ nodes and $O(μmn)$ edges. This implies that the problem can be solved in $O(μ^2n^3m)$ time using the combinatorial max flow algorithm of Orlin [Orl13], or in $O(μ^{(1+o(1))}(nm)^{1+o(1)}\log (UT))$ time using the algorithm of Chen et al. [CKL+22], where $U$ is the maximum capacity of any edge and $T$ is the time horizon. We focus on graphs that experience many time changes across the period of interest, as in such graphs the $μ$ term dominates the runtime.

Authors: Jeffery Li, Jayson Lynch, Liva Olina, Cecilia Chen, Andrew Lucas, Neil Thompson

In nearly every discipline, scientific computations are limited by the cost and speed of computation. For example, the best-known exact algorithms for the canonical Traveling Salesman Problem would take centuries to run on an instance of size 1 million. A natural response to such limits is to try to find new algorithms or to parallelize existing ones, but many algorithms are already at their theoretically-optimal level and parallelization is often impossible or prohibitively expensive. Starting in the 1960's, computer scientists pursued another solution: allowing solutions to have a small amount of error (i.e. approximating them). In this paper, we survey 118 of the most important algorithm problems in computer science, quantifying the gains and tradeoffs from approximation that have been discovered over the history of the field. Overall, only $\approx$20\% of problems have benefited from approximation. However, those with good approximate algorithms can be dramatically faster to compute with little cost to accuracy. For example, a quarter of computationally intractable problems (e.g. those that take exponential time to compute) have polynomial time approximate algorithms. Approximation also increases the number of algorithms that can run in linear time by 23\%, opening up new computational opportunities for those working in the big data regime. This work also sheds light on what should be expected from progress in AI, where approximation is at the heart of how deep learning works.

Authors: Kiarash Banihashem, MohammadTaghi Hajiaghayi, Mahdi JafariRaviz, Danny Mittal

The standard oracle model for matroid algorithms assumes that each independence query can be answered in constant time, regardless of the size of the queried set. While this abstraction has underpinned much of the theoretical progress in matroid optimization, it masks the true computational effort required by these algorithms. In particular, for natural and widely studied classes such as graphic matroids, even a single independence query can require work linear in the size of the set, making the constant-time assumption implausible. We address this gap by introducing a size-sensitive cost model where the cost of a query $Q$ scales with $|Q|$. Nearly linear-time oracle implementations exist for broad families of matroids, and this refined abstraction therefore captures the true cost of query evaluation while allowing for a more faithful comparison between general matroids and their natural special cases. Within this framework we study three fundamental algorithmic tasks: finding a basis of a matroid, approximating its rank, and approximating its partition size. We establish tight results, proving nearly matching upper and lower bounds that show the optimal query cost is (up to logarithmic factors) quadratic in the size of the matroid. On the algorithmic side, our upper bounds are realized by explicit procedures that construct the desired solution. On the complexity side, our lower bounds are unconditional and already hold even for weaker distinguishing formulations of the problems. Finally, for matroids with maximum circuit size at most $c$, we show that the quadratic barrier can be broken, providing an algorithm that calculates the maximum-weight basis with expected query cost $\mathcal{O}(n^{2-1/c} \log n)$.

Authors: Arend-Jan Quist, Marc Farreras Bartra, Alexis de Colnet, John van de Wetering, Alfons Laarman

Tensor networks and circuits are widely used data structures to represent pseudo-Boolean functions. These two formalisms have been studied primarily in separate communities, and this paper aims to establish equivalences between them. We show that some classes of tensor networks that are appealing in practice correspond to classes of circuits with specific properties that have been studied in knowledge compilation as \emph{tractable circuits}. In particular, we prove that matrix product states (tensor trains) coincide with nondeterministic edge-valued decision diagrams and that tree tensor networks exactly correspond to structured-decomposable circuits. These correspondences enable direct transfer of structural and algorithmic results; for example, canonicity and tractability guarantees known for circuits yield analogous guarantees for the associated tensor networks, and vice versa.

In secret-key private information retrieval (SK-PIR), the client in an offline phase processes the database using a short secret key. In the online phase the client could then use the secret key to make queries to the server, without revealing the entries accessed, and using only sublinear communication $o(N)$ in the database size $N$. While (non-SK) PIR requires public-key cryptography, recent work provides evidence that SK-PIR may not. In particular, Chen, Ishai, Mour, and Rosen (STOC 26) construct SK-PIR with communication $N^{\varepsilon}$, for any $\varepsilon$, from high-noise LPN, which is not known to imply public-key cryptography. We construct SK-PIR with online communication $\tilde{O}(\sqrt{N)}$, under the minimal assumption of one-way functions. More generally we can achieve client-to-server communication $\tilde{O}(N_c)$ and server-to-client communication $\tilde{O}(N_s)$ as long as $N_c \cdot N_s \geq N$. Our construction is simple and is based on garbled circuits satisfying an uncorrelated input encoding property. We show that this property is satisfied by point and permute schemes from the literature.

Michael Rabin passed away on April 14,2026. I blogged about him here. 

My post listed results of his that proved upper and lower bounds on problems. My point was that he proved upper and lower bounds for MANY different levels- from decidable to regular. And I am sure I left out some of his results. 

Here are some things I did not mention.

1) Rabin and Scott shared the Turing Award in 1976.  My not mentioning it raises the following question:

If I want to say someone has an impressive set of results which is a better way:

listing the awards they've won, or

listing  their results. 

I leave this to the reader. 

2) I had Rabin for two graduate courses at Harvard: Algorithms and Complexity Theory. He was a great teacher and gave insights into the results, some of which he had either proven or worked on.

3) I recalled thanking him in my PhD Thesis. So I dusted it off to see what I had said: 

The many courses I have taken at Harvard and MIT have helped me create this thesis. I am especially indebted to Michael Rabin, Mike Sipser, and Michael Stob for their excellent courses in algorithms, complexity theory, and recursion theory. Their pedagogy has been an inspiring example of what good teaching can and should be.

What is the probability that all three great teachers were named Michael? I do not know, however, I suspect Michael Rabin could have told me. 

By gasarch

The Question

I recently restarted working on this CS theory blog. One reason that I took a bit to get it going again is really stupid. Really silly. And perhaps I should not talk about it. It does not make me look smart. But here it goes anyway.

My email system stopped working recently. I got connected to the help desk. I told the help desk person H that I was not getting lots of messages.This was something that H could not figure out. Why were some messages being lost and others were not. H sent some test message to me and they never seemed to get there. H had set what I could see as what they could also see. What was making messages get lost? That led H to get me a ticket for a better expert at the help desk and said: “please wait a while and then use the ticket to get better help”. I thanked H and started to wait.

The Solution

While waiting my dear wife—Kathryn Farley asked me how the help desk was going. She knew I had finally had talked to the help desk. She was just interested in hearing if I had finally fixed things—were messages now getting there?

Kathryn has a PhD in performance studies from Northwestern in performance studies. She is now achieving her goal of being a professional actress in TV, film, and live theater. Here is an image of Kathryn from “The Gilded Age” which she began filming in January.

I explained the email problem to her. She solved it immediately.

Kathryn noticed that the email program was ordering the messages not in date order as I usually did. Rather it was ordering them in alphabet order on the sender’s name. So a message from asmith@some-where would be way before xfred@some-else. This made it look like certain messages were never received at all. We reset the order to date based and all the missing messages now appeared. Pretty cool.

Kathryn made me proud. Her insight was brilliant. It solved a problem that the help expert H could not do. I certainly could not do it. But she could see right away what would fix it.

Perhaps I Should Explain The P=NP Question To Her?

Does this mean that sometimes we all have a bad view of a problem? Can we need some cool insight to see how a problem can be solved? Perhaps I wondered. So many of us think that P is not equal to NP and could we all be missing some easy solution. Not clear to me.

By rjlipton

In this work, we continue the line of research on the complexity of distributions (Viola, Journal of Computing 2012), and study samplers defined by low degree polynomials. An $n$-tuple $\mathcal{P} = (P_1,\dots, P_n)$ of functions $P_i \colon \mathbb{F}_2^m \to \mathbb{F}_2$ defines a distribution over $\{0,1\}^n$ in the natural way: draw $X$ uniformly at random from $\mathbb{F}_2^m$ and output $(P_1(X),\dots, P_n(X)) \in \{0,1\}^n$. We show that when $\mathcal{P}$ is defined by polynomials of degree $d$, the total variation distance of $\mathcal{P}$ from the product distribution $\mathrm{Ber}(1/3)^{\otimes n}$ is $1-o_n(1)$, where $o_n(1)$ is a vanishing function of $n$ for any constant degree $d$. For small values of $d$, we show the following concrete bounds. (i) For $d=1$ we have $\|\mathcal{P}-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(n))$. (ii) For $d=2$ we have $\|\mathcal{P}-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(\log(n)/\log\log(n)))$. (iii) For $d=3$ we have $\|\mathcal{P}-\mathrm{Ber}(1/3)^{\otimes n}\|_{TV} \geq 1-\exp(-\Omega(\sqrt{\log\log(n)}))$. Our results extend the recent lower bound results for sampling distributions, which have mostly focused on local samplers, small depth decision trees, and small depth circuits. As part of our proof, we establish the following result, that may be of independent interest: for any degree-$d$ polynomial $P\colon\mathbb{F}_2^m \to \mathbb{F}_2$ it holds that $\Pr_X[P(X) = 1]$ is bounded away from $1/3$ by some absolute constant $\delta = \delta_d>0$. Although the statement may seem obvious, we are not aware of an elementary proof of this. The proof techniques rely on the structural results for low degree polynomials, saying that any biased polynomial of degree $d$ can be written as a function of a small number of polynomials of degree $d-1$.

The partial derivative method is a central tool in algebraic complexity, underlying lower bounds for multilinear formulas, bounded depth circuits, and algebraic branching programs. A key feature of this measure is its subadditivity and submultiplicativity, which are usually used to upper bound the measure. However, proving lower bounds requires bounding the measure of explicit polynomials from below, and in some cases, a sharp estimate is required. For example, a frequently used fact is that the dimension of the space spanned by order $k$ partial derivatives of a product of $n$ linearly independent linear functions is ${n\choose k}$. Beyond the linear case, however, not much is known about the behavior of the (general) partial derivative measure under multiplication. In particular, it has been conjectured that for algebraically independent polynomials $g_1,\dots,g_r \in \mathbb{C}[\mathbf{x}]$, the partial derivative complexity of the product $\prod_{i=1}^{r}{g_i(\mathbf{x})}$ grows exponentially with $r$ (see Question 42 in (Chaugule et al., 2023)), but prior to this work such bounds were only known when the $g_i$'s are linear polynomials, or satisfy additional restrictions. In this paper, we show a lower bound of $\exp(\Omega(r^{1/6}))$ for the measure of a product of $r$ linearly independent quadratic polynomials. This is the first result to show such a lower bound on the partial derivative measure of a product of nonlinear polynomials, without any further restrictions. Interestingly, we only assume linear independence, which is weaker than algebraic independence. Our proof relies on algebraic-geometric and combinatorial techniques, combining the Jacobian approach of (Chaugule et al., 2023) together with the theory of wide algebras introduced in (Ananyan and Hochster, 2018; Oliveira and Sengupta, 2022; Garg et al., 2023). To our knowledge, this is the first use of wide-algebra techniques for proving lower bounds on partial derivative complexity, and one of the first applications of these techniques outside the context of Sylvester-Gallai type problems.

Authors: Mingyang Liu, Gabriele Farina, Asuman Ozdaglar

Most familiar equilibrium concepts, such as Nash and correlated equilibrium, guarantee only that no single player can improve their utility by deviating unilaterally. They offer no guarantees against profitable coordinated deviations by coalitions. Although the literature proposes solution concepts that provide stability against multilateral deviations (\emph{e.g.}, strong Nash and coalition-proof equilibrium), these generally fail to exist. In this paper, we study an alternative solution concept that minimizes coalitional deviation incentives, rather than requiring them to vanish, and is therefore guaranteed to exist. Specifically, we focus on minimizing the average gain of a deviating coalition, and extend the framework to weighted-average and maximum-within-coalition gains. In contrast, the minimum-gain analogue is shown to be computationally intractable. For the average-gain and maximum-gain objectives, we prove a lower bound on the complexity of computing such an equilibrium and present an algorithm that matches this bound. Finally, we use our framework to solve the \emph{Exploitability Welfare Frontier} (EWF), the maximum attainable social welfare subject to a given exploitability (the maximum gain over all unilateral deviations).

Authors: Susanna F. de Rezende, David Engström, Yassine Ghannane, Kilian Risse

We prove superpolynomial length lower bounds for the semantic tree-like Frege refutation system with bounded line size. Concretely, for any function $n^{2-\varepsilon} \leq s(n) \leq 2^{n^{1-\varepsilon}}$ we exhibit an explicit family $\mathcal{A}$ of $n$-variate CNF formulas $A$, each of size $|A| \le s(n)^{1+\varepsilon}$, such that if $A$ is chosen uniformly from $\mathcal{A}$, then asymptotically almost surely any tree-like Frege refutation of $A$ in line-size $s(n)$ is of length super-polynomial in $|A|$. Our lower bounds apply also to tree-like degree-$d$ threshold systems, for $d \approx \log\bigl(s(n)\bigr)$, that is, for $d$ up to $n^{1-\varepsilon}$. More generally, our lower bounds apply to the semantic version of these systems and to any semantic tree-like proof system where the number of distinct lines is bounded by $\exp\bigl(s(n)\bigr)$.

Authors: Sanyam Agarwal, Markus Bläser, Mridul Gupta

Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over all possible multiplication orders in the Leibniz formula for the determinant. He used the symmetrized determinant to design algorithms estimating the permanent of a matrix. To this end, he showed that there is a $O(n^{r+3})$ algorithm computing $\sdet$, where $r$ is the dimension of the algebra, and is therefore polynomial-time computable for fixed $r$. In this work, we study the algebraic properties and complexity of $\sdet$. While most of the properties of the ordinary determinant don't generalize to $\sdet$ defined on non-commutative algebras, we show that the principal minor expansion of the $\sdet$ is analogous to the ordinary determinant. Second, we prove that there exists a polynomial-sized algebra such that computing the symmetrized determinant is $\sharpP$-hard. Third, we show that the associated polynomial family is $\VNP$-complete over a suitable polynomial-dimensional algebra in the non-commutative setting. Further, when seen as a family of polynomials over the matrix algebra, it is also $\VNP$-complete in the commutative setting. This places the symmetrized determinant among the natural complete families arising from algebraic computation.

Authors: Yupan Liu, Pei Wu

Stoquasticity, originating in sign-problem-free physical systems, gives rise to $\sf StoqMA$, introduced by Bravyi, Bessen, and Terhal (2006), a quantum-inspired intermediate class between $\sf MA$ and $\sf AM$. Unentanglement similarly gives rise to ${\sf QMA}(2)$, introduced by Kobayashi, Matsumoto, and Yamakami (CJTCS 2009), which generalizes $\sf QMA$ to two unentangled proofs and still has only the trivial $\sf NEXP$ upper bound. In this work, we initiate a systematic study of the power of unentanglement without destructive interference via ${\sf StoqMA}(2)$, the class of unentangled stoquastic Merlin-Arthur proof systems. Although $\sf StoqMA$ is semi-quantum and may collapse to $\sf MA$, ${\sf StoqMA}(2)$ turns out to be surprisingly powerful. We establish the following results: - ${\sf NP} \subseteq {\sf StoqMA}(2)$ with $\widetilde{O}(\sqrt{n})$-qubit proofs and completeness error $2^{-{\rm polylog}(n)}$. Conversely, ${\sf StoqMA}(2) \subseteq {\sf EXP}$ via the Sum-of-Squares algorithm of Barak, Kelner, and Steurer (STOC 2014); with our lower bound, our refined analysis yields the optimality of this algorithm under ETH. - ${\sf StoqMA}(2)_1 \subseteq {\sf PSPACE}$, and the containment holds with completeness error $2^{-2^{{\rm poly}(n)}}$. - ${\sf PreciseStoqMA}(2)$, a variant of ${\sf StoqMA}(2)$ with exponentially small promise gap, cannot achieve perfect completeness unless ${\sf EXP}={\sf NEXP}$. In contrast, ${\sf PreciseStoqMA}$ achieves perfect completeness, since ${\sf PSPACE} \subseteq {\sf PreciseStoqMA}_1$. - When the completeness error is negligible, ${\sf StoqMA}(k) = {\sf StoqMA}(2)$ for $k\geq 2$. Our lower bounds are obtained by stoquastizing the short-proof ${\sf QMA}(2)$ protocols via distribution testing techniques. Our upper bounds for the nearly perfect completeness case are proved via our new rectangular closure testing framework.

Authors: Hunter Monroe

We study when a sound arithmetic theory $\mathcal S{\supseteq}S^1_2$ with polynomial-time decidable axioms efficiently proves the bounded consistency statements $Con_{\mathcal S{+}φ}(n)$ for a true sentence $φ$. Equivalently, we ask when $\mathcal S$, viewed as a proof system, simulates $\mathcal S{+}φ$. The paper's two unconditional contributions constrain possible characterizations. First, for finitely axiomatized sequential $\mathcal S$, if $EA{\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}φ}$, then $\mathcal S$ interprets $\mathcal S{+}φ$, implying ${\mathcal S}{\vdash^{n^{O(1)}}}Con_{\mathcal S}(p(n)){\rightarrow}Con_{\mathcal S{+}φ}(n)$ for some polynomial $p$, and hence ${\mathcal S}{\vdash^{n^{O(1)}}}Con_{\mathcal S{+}φ}(n)$. Second, if $\mathcal S$ fails to simulate $\mathcal S{+}φ$ for some true $φ$, then for all sufficiently large $k$ it also fails for $φ_{BB}(k)$ asserting the exact value of the $k$-state Busy Beaver function. Informally, any argument showing that $\mathcal S$ fails to simulate some $\mathcal S{+}φ$ also yields unprovable $φ_{BB}(k)$ witnessing the same obstruction. These results suggest that relative consistency strength is a serious candidate for governing when simulation is possible, while leaving open whether it is the correct criterion. The paper's central conjectural proposal is that the above sufficient condition is also necessary: if $EA{\not\vdash}Con_{\mathcal S}{\rightarrow}Con_{\mathcal S{+}φ}$, then for every constant $c{>}0$, ${\mathcal S}{\not\vdash^{n^c}}Con_{\mathcal S{+}φ}(n)$. Under this proposal, hardness follows in canonical cases where $φ$ is $Con_{\mathcal S}$ or a Kolmogorov-randomness axiom. The latter yields further conjectural consequences and extensions.

Authors: Siu On Chan, Tommaso d'Orsi, Jeff Xu

Under what condition is a random constraint satisfaction problem hard to refute by the sum-of-squares (SoS) algorithm? A sufficient condition is t-wise uniformity, that is, each constraint has a t-wise uniform distribution of satisfying assignments, as shown by the lower bounds of Kothari, Mori, O'Donnell, and Witmer (STOC 2017). This condition is also necessary for random CSPs given by a predicate and uniformly random literals, due to the constant-degree SoS refutation of Allen, O'Donnell, and Witmer (FOCS 2015). For higher degree, Raghavendra, Rao, and Schramm (STOC 2017) gave a refutation for Boolean random CSPs with uniformly random literals, matching the lower bounds optimally in terms of the three-way tradeoff between constraint density, SoS degree, and strength of refutation. Two long-standing open problems are to find a more general sufficient condition for SoS lower bounds, and to refute similar random CSPs not involving literals. We show that for a general random k-CSP, the necessary and sufficient hardness condition is not t-wise uniformity, but t-wise independence. We generalize the optimal three-way tradeoff to any random k-CSP, without assuming a Boolean domain or uniformly random literals. Our analysis involves new Kikuchi matrices for odd order and for asymmetric tensors. It also uses the global correlation rounding technique of Barak, Raghavendra, and Steurer (FOCS 2011). To avoid the running-time penalty of this technique, we also give a spectral refutation algorithm.

Authors: Argyrios Deligkas, John Fearnley, Alexandros Hollender, Themistoklis Melissourgos

We study the problem of computing a competitive equilibrium with approximately optimal bundles in Fisher markets with separable piecewise-linear concave (SPLC) utility functions, meaning that every buyer receives a $(1-δ)$-optimal bundle, instead of a perfectly optimal one. We establish the first intractability result for the problem by showing that it is PPAD-hard for some constant $δ> 0$, assuming the PCP-for-PPAD conjecture. This hardness result holds even if all buyers have identical budgets (competitive equilibrium with equal incomes), linear capped utilities, and even if we also allow $\varepsilon$-approximate clearing instead of perfect clearing, for any constant $\varepsilon < 1/9$. Importantly, we show that the PCP-for-PPAD conjecture is in fact required to show hardness for constant $δ$: showing PPAD-hardness for finding such approximate market equilibria in a broad class of markets encompassing those generated by our hardness result would prove the conjecture. This is the first natural problem where the conjecture is provably required to establish hardness for it.