Theory of Computing Report

Authors: Srinivasan Arunachalam, Arkopal Dutt, Sabee Grewal, Aparna Gupte

Gowers, Green, Manners, and Tao (Annals '25) recently resolved Marton's polynomial Freiman-Ruzsa conjecture. We give an algorithmic counterpart to their result: given uniform sampling and membership-oracle access to a set $A \subseteq \mathbb{F}_2^n$ with doubling constant at most $K$, our algorithm outputs a subspace of size at most $|A|$ whose $K^{O(1)}$ translates cover $A$. The algorithm runs in $\textsf{poly}(n,K)$ time. As applications, we obtain polynomial-time algorithms for a variety of learning problems, including quadratic Goldreich-Levin, improper agnostic tomography of stabilizer states, and tomography of quantum states with bounded stabilizer extent.

Authors: Haoyu Wang, Pei Wu

In this paper, we construct a total Boolean function with $\widetilde{O}(\log n)$ randomized communication protocol, while any monochromatic rectangle has density at most $O(2^{-\mathrm{poly}(n)})$. As a corollary, it gives the first total function separation for $\mathrm{BPP}\not\subseteq\mathrm{P}^{\mathrm{NP}}$ in the communication world. Inspired by Gavinsky's recent work (arXiv:2608.18784), our construction combines the cheat-sheet framework with fully linear PCPs.

Authors: Lorenzo Ciardo, Iris Hebbeker, Gideo Joubert, Jana Kreiß, Antoine Mottet

We prove RE-completeness of the quantum homomorphism problem parameterised by families of graphs derived from the classic metric association schemes. These include Kneser graphs, $q$-Kneser graphs, and the complements of Johnson, Grassmann, and Hamming graphs. Our proof develops a spectral method for establishing non-contextuality of quantum polymorphisms. It combines an equality analysis of Roberson's bound on the projective packing number in terms of Schrijver's theta with a structural argument inspired by Erdős-Ko-Rado theory.

Authors: David Miloschewsky, Supartha Podder

Starting from the entrance of a welded tree, a quantum walk algorithm can find its exit vertex exponentially faster than any classical algorithm. However, it has been an open question whether any quantum algorithm is able to efficiently find a path from the entrance to the exit. We answer this by proving an exponential quantum query lower bound for finding such path. Specifically, for trees of height $n$, any quantum query algorithm requires at least $Ω(2^{n/24})$ queries in order to succeed with constant probability. Our proof uses the compressed permutation oracle technique in order to construct databases which track the graph information an algorithm has learned and forgotten, and show that no efficient quantum algorithm can build an entrance-to-exit path in these records.

Authors: Amir Ali Ahmadi, Abraar Chaudhry, Ijay Narang, Yukai Tang

We show that unless P = NP, there cannot be a polynomial-time (or even pseudo-polynomial-time) algorithm for output feedback stabilization of a linear dynamical system with a linear controller. This settles one of the best-known open problems in control theory. The result holds in both continuous and discrete time. We also present a family of stabilizable linear dynamical systems for which no polynomial-time algorithm can write down a stabilizing controller in its standard representation.

Authors: Yimu Qiao, Lijia Yu, Ruichen Qiu, Xiao-Shan Gao

Transformers have emerged as the dominant architecture in sequence modeling, achieving remarkable success in natural language processing and reasoning tasks. While existing literature has established the Turing completeness of transformers under bounded input length, the reasoning power of a single transformer operating on inputs of unbounded length is not fully explored. In this paper, we theoretically investigate the reasoning limitations of a single transformer and the enhanced capabilities of agent systems. We show that a single fixed finite precision transformer cannot memorize certain Turing machines with inputs of arbitrary length, such as the arithmetic; and a single fixed infinite precision transformer trained with a random algorithm is not Turing complete with probability one under reasonable conditions. To overcome the limitation of a single transformer, we define a formal agent architecture consisting of decision, execution, and memory modules and show that for any Turing machine $\mathbb{T}$, there exists an agent that can memorize $\mathbb{T}$ and is computationally the same as $\mathbb{T}$. Thus, agents are Turing complete.

Authors: Yusuke Kobayashi, Bingkai Lin, Joseph Swernofsky

An instance of {\sc Pinwheel Packing} is a list of positive integers $a_1,\ldots,a_k$. A feasible schedule assigns one task to every integer time so that every interval of $a_i$ consecutive times contains task $i$. The instance is \emph{dense} when $\sum_i1/a_i=1$. We prove that {\sc Dense Pinwheel Packing} is NP-complete even when every period is encoded in unary and equal periods are listed as distinct tasks. Consequently, the usual binary-encoded problem is strongly NP-complete. Kleinberg and Mishra also prove NP-completeness \cite[Corollary~5.1]{KleinbergMishra2026}, but their reduction uses periods of exponential numerical size and therefore yields only weak NP-hardness. Our proof uses a direct reduction from triangle partition in a sparse tripartite graph. If each of the three parts of the source graph has $n$ vertices, the reduction produces $O(n^4\log^3 n)$ explicitly listed tasks, each with period $O(n^4\log^3 n)$; consequently, its full unary encoding has length $O(n^8\log^6 n)$.

Authors: Shyamal Patel

We show that there exists a class of boolean functions C such that $(i)$ there is a distribution-independent statistical query algorithm for learning C that makes a polynomial number of queries of inverse polynomial tolerance and $(ii)$ for any set of functions $Φ_1, \dots, Φ_r$ such that for all $f \in$ C we can write $f(x) = \text{sign} \left( \sum_{i = 1}^r w_i Φ_i(x) \right)$ for some set of weights $w_i \in \mathbb{R}$, we must have that $r \geq n^{ω(1)}$. This gives a superpolynomial separation between dimension complexity and the query complexity of distribution-free learning in the statistical query model, negatively answering a question of Feldman, Kamath, and Srebro [FKS26]. Our construction C is a subclass of DNFs, and the proof is a simple consequence of recent progress on agnostically learning conjunctions [DKR25,CPS26] and the work of Razborov and Sherstov on the sign rank of DNFs [RS10].

Authors: Bingkai Lin, Xin Zheng

We study the approximability of \textnormal{\textsc{Set Cover}} parameterized by the target cover size $k$. Let $n$ be the universe size, $m$ the number of available sets, and $|Γ|$ the explicit input length. We prove that, for some absolute constant $c>0$, distinguishing \[ \operatorname{opt}(Γ)\le k \quad\text{from}\quad \operatorname{opt}(Γ)>k\cdot\frac{c\log n}{k^2\log\log n} \] is $\mathsf{W[1]}$-hard. Assuming the Exponential Time Hypothesis, there is also an absolute constant $\varepsilon>0$ for which no deterministic algorithm solves this gap problem in time $f(k)|Γ|^{\varepsilon k}$, for any computable function $f$. For every fixed $α>0$, both hardness results hold even when $n=O((\log m)^{1+α})$, with constants allowed to depend on $α$. For fixed $k$, the gap is within an $O_k(\log\log n)$ factor of the greedy algorithm's guarantee. Under the Strong Exponential Time Hypothesis, we further rule out $o(\log n/\log\log n)$ approximation in time $O(|Γ|^{k-δ})$ for every fixed $k\ge 2$ and $δ>0$. Thus a near-logarithmic hardness factor persists even when the exponent is reduced from exhaustive search by only a fixed constant. The constant in this SETH hardness factor may depend on $k$ and $δ$.

Authors: Zhiyang Dou, Ang Zhao, Chen Peng, Minghao Guo, Haixu Wu, Cheng Lin, Yuan Liu, Junfeng Yao, Xiaohu Guo, Wenping Wang, Wojciech Matusik

Rigid-body interpenetration frequently occurs in procedurally assembled and generated scenes and must be removed before downstream applications such as physical simulation. We present S4R (Scaling for Rigid-Body Interpenetration Resolution), a scale-continuation method for static interpenetration repair. S4R first uniformly shrinks each body about a fixed reference center to a small initial scale, at which the layout is penetration-free, and then restores full scale through a sequence of minimum-norm convex contact quadratic programs (QPs) that target the linearized separation margin during continuation. Resolution thereby replaces one deep correction with a sequence of shallow-contact subproblems. A conservative scale-event bound and frozen-witness gap predictions cut the number of exact mesh queries; the continuation then ends with a full-scale evaluator check and bounded tail refinement. We evaluate S4R on Kubric, HY3D-Bench, and Thingi10K using a shared mesh-level evaluator and a unified per-scene timing protocol. In the main comparisons on all three benchmarks, up to N=5000 bodies, S4R reaches zero reported penetration with displacement that stays small and nearly independent of scene size, and at the lowest wall time within each hardware tier among the compared methods. A GPU implementation extends these results to large-scale scenes. Our code and data can be found on our project page: https://frank-zy-dou.github.io/projects/S4R/index.html.

Authors: Venkatesan Guruswami, Xuandi Ren

We show that $\bigl(\frac{\log n}{\log\log n}\bigr)$-approximate parameterized $k$-SetCover is W[1]-hard, and has no $n^{o(k/\log k)}$-time algorithms under ETH. This improves upon the previous best factors $\bigl(\frac{\log n}{\log\log n}\bigr)^{1/k}$ in (Lin, 2019) and $(\log n)^{1/\operatorname{poly}(k)}$ in (Karthik, Laekhanukit, and Manurangsi, 2019). Here $k$ is the yes-case guarantee and $n$ is the number of candidate sets. While the best approximation ratio is still $O(\log n)$ via the greedy algorithm, closing this $1/k$ gap in the exponent has been a longstanding open problem; we remove this loss via a simple direct reduction. The construction is self-contained and does not rely on the parameterized inapproximability hypothesis (PIH). Starting with sparse parameterized 2-CSP instances (Karthik, Marx, Pilipczuk, and Souza, 2024), we build a monotone CNF formula, which is equivalent to a SetCover instance. To obtain a $k$-versus-$h$ gap, the reduction enumerates all hash functions from $Σ$ to $[2h]$ and all unsatisfiable 2-CSP instances on the same constraint graph with alphabet $[2h]$. For each such instance, it asks for a certificate that the hashed label pairs are not all contained in that instance. Perfect hashing makes this enumeration efficient for $h=\log n/\log\log n$.

Authors: Kristóf Bérczi, Shaddin Dughmi, Vasilis Livanos, José A. Soto, Victor Verdugo

We prove a $1/e$ guarantee for the matroid secretary problem on linear matroids, therefore settling the strong secretary conjecture in this class of matroids. The result holds both when the matroid is known in advance and when a linear representation over a finite field is given online. In the known-matroid model, the result extends more generally to matroids admitting a finitary modular extension. Each element of a fixed optimal basis is selected with probability at least $1/e$. $\mathbf{\text{Concurrent Discovery Disclosure:}}$ The proof of the main result in this manuscript was obtained in a conversation with ChatGPT-6 Astra on Tuesday, September 15, 2026 at 1:02 AM PDT. We then prepared this manuscript for public release, with the intent of uploading it on the morning of Thursday, September 17, 2026. In the early morning hours of September 17, while finalizing the submission, we discovered the manuscript https://arxiv.org/abs/2609.19118 of Abdi, Banihashem, Hajiaghayi, and Mittal, uploaded on September 16, 2026, which contains the same result via an essentially identical approach. We are sharing our manuscript nonetheless in case our exposition is of independent utility to the community, and we hope this experience stimulates broader discussion about concurrent discovery in the AI era.

Authors: Debarati Das, Evangelos Kipouridis, Tomasz Kociumaka

For every $0 < \varepsilon \le 1$, we give a randomized $(3+\varepsilon)$-approximation to weighted edit distance when the costs form a metric on the alphabet augmented with a gap symbol. For strings of total length $N$, the running time is $\widetilde{O}(N^{8/5}/\varepsilon^{16/5})$, where $\widetilde{O}$ suppresses factors polynomial in $\log(N/\varepsilon)$. The dependence on $N$ matches that of the fastest known $(3+\varepsilon)$-approximation for unit-cost edit distance. The algorithm never underestimates the edit distance and achieves the approximation guarantee with inverse-polynomial failure probability in $N$. The running time bound assumes constant-time exact arithmetic operations and metric queries, and it is independent of the numerical range of the edit costs. We build on three tools: the sampling framework of Chakraborty, Das, Goldenberg, Koucký, and Saks (J. ACM, 2020), with subsequent refinements by Andoni (2020); Kuszmaul's removal of inexpensive characters (ICALP 2019); and Klein's data structure for distances in planar graphs (SODA 2005). Our new ingredients include, among others, a decomposition of one string into pieces of bounded length with highly structured total deletion costs. This decomposition lets us compare all pieces against a small family of substrings of the other string.

Authors: Xiaoyu Chen, Heng Guo, Eric Vigoda, Xiongxin Yang

We give an FPRAS for the permanent of an $n\times n$ $0/1$ matrix with running time $\widetilde{O}(n^{3.5}\varepsilon^{-2})$. Our algorithm extends to a strongly polynomial FPRAS for arbitrary nonnegative matrices, as in previous works. Jerrum, Sinclair, and Vigoda (2004) gave the first FPRAS for the permanent of a nonnegative matrix. The running time was subsequently improved to $\widetilde{O}(n^7)$ by Bezáková, Štefankovič, Vazirani, and Vigoda (2008), and recently to $\widetilde{O}(n^6)$ by Chen, Vigoda, and Yang (2026). We introduce a multicommodity-flow bound inspired by electrical flows, replacing the usual path-length factor by routing energy. For a boosted version of the classical JSV chain, we prove a relaxation-time bound of $O(n^3\log n)$ and show that stationary trajectories of this length estimate all stationary hole-pattern probabilities, yielding an $\widetilde O(n^5)$-time FPRAS algorithm. Our new hole-weighted slide (HWS) chain improves both bounds to $O(n^2\log n)$, yielding an $\widetilde O(n^4)$-time algorithm. Finally, we obtain the claimed $\widetilde O(n^{3.5})$ running time by using a subset of $\widetilde{O}(\sqrt{n})$ checkpoint temperatures in an iterated sequence of warm-starts to obtain initializations at every temperature.

Authors: Yu Cong, Chao Xu, Yi Zhou

We consider a large-scale incentive allocation problem where the entire trade-off curve between budget and profit has to be maintained approximately at all times. The application originally comes from assigning coupons to users of ride-sharing apps, where each user can have a limit on the number of coupons assigned to them. We consider a more general form, where the coupons for each user form a matroid, and the set of coupons assigned to each user must be an independent set. We show the entire trade-off curve can be maintained approximately in near real time.

Authors: Roie Levin

We show an information-theoretic lower bound of $Ω\left(\frac{\log n \log m}{\log \log n + \log \log m}\right)$ for online set cover against randomized algorithms, for all sufficiently large $m$ and $n$ satisfying $\log^2 n \leq m \leq 2^n$.

Authors: Sitan Chen, Liye Wang

A popular selling point of diffusion large language models (dLLMs) is their capacity for parallelism: the ability to generate sequences of text far more efficiently than autoregressive models, which require one forward pass per token. Yet among the many competing paradigms for dLLMs, from masked to uniform to Gaussian diffusion, principled understanding of how these different proposals compare in parallelism remains limited. In this work, we initiate a fine-grained comparison of the capacity for parallelism among these three leading approaches and prove the following: - Uniform and Gaussian diffusion can sample in a number of forward passes which scales with the dual total correlation of the underlying distribution, a measure of intrinsic complexity which can be much smaller than the context length. Previously, it was only known how to achieve this using masked diffusion. - For a certain family of random empirical measures, we show that $\widetildeΘ(\sqrt{d})$ forward passes are necessary and sufficient to sample using uniform or Gaussian diffusion, yet there exist approximate score oracles for which $\widetildeΩ(d)$ forward passes are needed for masked diffusion. This establishes the first provable separation in parallelism between the three prevailing dLLM paradigms. Contrary to popular intuition that masked diffusions are harder to parallelize because they must commit to token values, the latter separation instead comes from the fact that the critical windows in masked diffusion sampling are asymptotically narrower than those in uniform and Gaussian diffusion sampling.

Authors: Heng Guo, Hongyang Liu, Xiongxin Yang, Yitong Yin, Yiyao Zhang

In this note, we give a simple analysis of a non-adaptive simulated annealing algorithm for estimating the partition function of Gibbs distributions. This yields the most efficient reduction of this kind so far. We also establish lower bounds for both general and non-adaptive algorithms, showing that our algorithm is optimal over a broad range of parameters.

Authors: Eric Angel, Evangelos Bampas, Evripidis Bampis, Vincent Chau, Johanne Cohen, Alexander Kononov, Yizheng Zhang

The Minimum Vertex Cover problem is a fundamental combinatorial optimization problem, aiming to identify a minimum subset of vertices in a graph such that every edge is incident to at least one vertex in this subset. Among its variants, the Min-Power-Cover problem stands out due to its practical applications, such as camera placement at intersections: in an edge-weighted graph, an edge is covered if one of its endpoints is assigned a power value at least as large as the edge's weight. In this paper, we introduce the Emergency Vertex Cover (Em-VC) problem where an edge may be covered not only by its endpoints, but also by a distant vertex, provided the vertex is given sufficient power to "cover" the cumulative weight of the edges along a shortest path to one of the edge's endpoints plus the weight of the edge. Em-VC is motivated by different practical scenarios, e.g. the need for urban disaster response, where ensuring accessibility to all road segments (edges of the graph) is crucial for effective aid delivery. We prove that Em-VC is NP-hard, derive lower bounds, and design a polynomial-time algorithm for its continuous version. Moreover, we present a 4/3-approximation algorithm for the discrete case and identify several special graph classes for which the problem can be solved in polynomial time.

Authors: Koppány István Encz, Monaldo Mastrolilli, Eleonora Vercesi

We propose a framework for the systematic study of integrality gaps of combinatorial optimization problems with respect to a fixed linear programming formulation. The method, called \emph{integrality gap preserving reduction}, consists of iteratively shrinking the input universe of the problem while guaranteeing that gap-maximizing instances remain selected. When the subset of remaining instances becomes specific enough, we calculate the integrality gap explicitly. Besides applying integrality gap preserving reductions to three well-known optimization problems via their standard linear programming formulations (weighted vertex cover problem, multiple knapsack problem, and unrelated machine scheduling problem), we analyse the restricted assignment problem via its configuration LP relaxation. We prove that the integrality gap is equal to $1$ for three ``easy'' subclasses of the problem that are either solvable in polynomial time or admit a PTAS (e.g., the all-one processing time case). For some remaining cases, we improve the current lower bound using our technique.

Authors: Matic Požar

Computing influence spread under the Independent Cascade (IC) model is #P-hard, and influence maximization is commonly approached using Monte Carlo or reverse-reachable-set sampling. We study IC diffusion on bounded-treewidth graphs. Using probability distributions over separator reachability relations, we obtain exact influence evaluation in $O(n2^{O(w^2)}\operatorname{poly}(w))$ time for a graph with $n$ nodes and treewidth $w$. Our main contribution is an exact all-marginal-gains algorithm. We introduce variable artificial source edges and show that, at a deterministic seed set, the derivative with respect to each source-edge probability equals the corresponding greedy marginal gain. Reverse-mode differentiation therefore computes all marginal gains simultaneously with the same asymptotic complexity as one exact influence evaluation. This yields an exact implementation of classical greedy influence maximization in $O(Kn2^{O(w^2)}\operatorname{poly}(w))$ time, linear in graph size for fixed $w$ and seed budget $K$. We also show that the separator-relation representation has tight $2^{Θ(w^2)}$ state complexity within exact context-independent compositional separator summaries. This contrasts with the NP-hardness of globally optimal IC influence maximization already on graphs of treewidth one and pathwidth two. Experiments on synthetic bounded-treewidth networks are consistent with linear scaling for fixed width and show that runtime is largely insensitive to propagation and seed-activation probabilities. In demanding diffusion regimes, the method substantially outperforms reverse-reachable-set and optimized Monte Carlo greedy baselines while computing greedy marginal gains exactly.

Authors: Sourav Chakraborty, Debarshi Chanda, Arijit Ghosh, A. Pavan, Chhaya Trehan, N. V. Vinodchandran

Most existing graph streaming algorithms assume the ideal scenario where each edge arrives only once. Real-world graph streams, such as communication or transaction logs, often contain many repeated occurrences of the same edge. In general, the algorithms developed for the single-edge arrival case can fail when edges can arrive multiple times. Motivated by this, we study the {\em repeated-edge arrival graph streaming model} where an edge is allowed to arrive multiple times. In this work, we study the triangle counting problem in the repeated-edge arrival model: approximate the number of triangles in the underlying {\em simple graph} despite arbitrary edge repetitions. We design the first algorithms for triangle counting with optimal space complexity. In particular, we present a single-pass algorithm that computes an $(\varepsilon,δ)$-approximation of the number of triangles with optimal space complexity. We introduce {\em right-to-be-forgotten graph streaming} (RFGS) model, where a forget operation can cause all previous occurrences of an edge to disappear. We show that our single-pass algorithm can be extended to the RFGS model with optimal space complexity. Finally, we present optimal constant-pass algorithms that compute an $(\varepsilon,δ)$-approximation of the number of triangles and cliques for the repeated-edge arrival graph streams.

Authors: Vanessa Kosoy

In previous papers, we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of "zipline programs" (a variant of straight-line programs), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.

Authors: Ling Li, Daniel Gibney, Sharma V. Thankachan, Rahul Shah, Grigorios Loukides, Solon P. Pissis

There is increasing interest in queries about the context of a string $P$ in a longer text $T$, i.e., the set of all string pairs $(L,R)$, with $|L|=|R|=q$, for a given $q$, such that the string $LPR$ occurs in $T$. Such contextual queries are important in several domains but are challenging to answer efficiently. This is because the length of $T$ in applications is massive and existing indexes do not directly encode the context of a given $P$, which is key for answering retrieval queries efficiently. Our work introduces the ZigZag Trie (ZZT), a new full-text index to specifically address these challenges. This index reorganizes the text so that, for any $P$, all possible strings $L$ and $R$ growing symmetrically around $P$ are grouped into a common subtree of the index, allowing their efficient retrieval. We show how to construct the ZZT of $T$, which has size $\mathcal{O}(n)$ where $n=|T|$, in $\mathcal{O}(n\log n)$ time and $\mathcal{O}(n)$ space. On top of ZZT, we design specialized indexes that, for a query pattern $P$, answer four new types of contextual queries: (I) finding the longest string $LPR$ that occurs at least $τ$ times in $T$, for a fixed $τ$; (II) finding the longest string $LPR$ that occurs in at least $τ$ texts of a text collection, for a fixed $τ$; (III) reporting the total number of distinct contexts of $P$ in $T$; and (IV) retrieving, for a given $q$, the $k$ pairs $(L,R)$ of $P$ with the highest scores according to a given scoring function. Our indexes answer queries of type I, II, and III in optimal time, and of type IV in near-optimal time. Moreover, their size, construction space, and construction time are linear or near-linear in $n$, given ZZT. Using real billion-letter datasets, we show that our indexes answer queries orders of magnitude faster than baselines and perform similarly or better in index size and construction space and time.

Authors: Zhao Song, Song Yue

Marcus, Spielman, and Srivastava [MSS15] established the existence of Kadison--Singer partitions. We provide polynomial-time algorithms for the Kadison--Singer problem. For Hermitian matrices $A_1,\ldots,A_m\in\mathbb C^{n\times n}$ of rank at most one, we give two algorithms that find signs $σ\in\{\pm1\}^m$ satisfying $\|\sum_i σ_iA_i\|\le C\|\sum_i A_i^2\|^{1/2}$. The deterministic algorithm achieves $C=3.3443$ using $\widetilde O(mn^2+n^{56})$ arithmetic operations. The randomized algorithm achieves $C=4.8628$ using $\widetilde O(mn^2+n^{5.88})$ arithmetic operations in expectation. For vectors satisfying $\sum_i a_ia_i^*=I$ and $\|a_i\|^2\leα$, the algorithms yield partitions $[m]=I_1\cup I_2$ satisfying $\|\sum_{i\in I_j}a_ia_i^*-I/2\|\le (C/2)\sqrtα$ for $j=1,2$.

Authors: Naonori Kakimura, Yoshihiko Terai

A splay tree is a self-adjusting binary search tree that allows access, insertion, and deletion to be performed in amortized $O(\log n)$ time, where $n$ is the number of stored elements.The sequential access theorem states that, when the elements of a splay tree are accessed in increasing order, the amortized cost per operation becomes a constant. In this paper, we show that the upper bound for this constant is at most $5.5$ by refining the existing analysis and introducing a new potential function. Furthermore, we complement our result by showing that there exists a splay tree for which the constant is lower-bounded by almost $4$.

Authors: Dylan J. Altschuler

We give an efficient algorithm with improved algorithmic guarantees for the (offline) Beck--Fiala problem when the sets have bounded size. Let $A$ be an arbitrary matrix $A\in\{0,1\}^{m\times n}$ with at most $d$ ones per column and at most $s$ ones per row. Let $\log^*$ denote the iterated logarithm and $\ell_j$ denote the $j$-fold composition of log. Assume $s\le\exp(O(\sqrt d))$. We provide an efficient algorithm that, for arbitrary sparsity $d$, gives $O(\sqrt d(1+\log^*n))$ discrepancy. Moreover, if $d\ge\ell_j(n)$ for a fixed integer $j\ge1$, the algorithm gives $O_j(\sqrt d)$ discrepancy. The proof is a bootstrapping scheme using the Bansal-Jiang algorithm.

Authors: Jingyang Zhao, Yuxi Liu, Mingyu Xiao

The metric traveling salesman problem (TSP) is a fundamental problem in combinatorial optimization that asks for a minimum-cost tour covering all clients in a metric graph. The metric multiple-depot TSP (MD-TSP) is a natural extension, where the graph contains depots and clients, and the objective is to compute a minimum-cost set of tours covering all clients, with each tour starting and ending at the same depot. When the number of depots is part of the input, an adaptation of the Christofides--Serdyukov heuristic yields an approximation ratio of $2$. In this paper, we introduce a $(1+1/\sqrt{2})$-approximation algorithm. Like the Christofides--Serdyukov heuristic, our algorithm first computes a rooted spanning forest (RSF), then a matching to correct its odd degrees, and finally obtains a solution by shortcutting. However, instead of using a minimum-cost RSF, we construct an RSF by a primal-dual algorithm for a natural cut relaxation. The algorithm grows rootless components and the component containing all depots at different rates, adding an edge when its dual constraint becomes tight. Vertex labels record the times at which clients first become connected to a depot. The two-speed growth provides a joint bound on the forest cost and two label-dependent terms that also arise in bounding the parity-correction cost. Balancing the coefficients of these two terms by setting both to $\sqrt{2}-1$ yields the claimed approximation ratio.

Authors: Chandra Chekuri, Richard Ueltzen, Jan Vondrak

We consider the question of designing a universal family of sets $F \subset 2^{[n]}$ such that for any function $f:2^{[n]} \to R_{\geq 0}$ in a certain class, we have $$\max_{S \in F} f(S) \geq c(n) \cdot \max_{S \subset [n]} f(S).$$ We prove that there is a family of subpolynomial size such that for any nonnegative submodular function, $c(n) = Ω(\frac{\log \log n}{\log n})$, and there is a family of logarithmic size such that $c(n) = Ω(\frac{1}{\log n})$. We also prove that pairwise independence (which achieves a constant factor for graph cut functions), or even $k$-wise independence, does not imply a bound better than $O(\frac{1}{\sqrt{\log n}})$ for submodular functions. On the other hand, we prove that for any polynomially representable subclass of nonnegative submodular functions (such as the matroid connectivity functions for matroid representable over $F_q$), a constant-factor universal family of polynomial size always exists. For absolute XOS functions (a class that we introduce, in the form $f(S) = \max_i |\sum_{j \in S} w_{ij} + c_i|$ where $w_{ij}, c_i \in R$), we design a family of polynomial size such that $c(n) \geq \sqrt{\frac{\log n}{n}}$, and prove that there is no polynomial-size family achieving a factor better than $O(\sqrt{\frac{\log n}{n}})$.

Authors: Xiaoyu Chen, Kuikui Liu

Local-to-global techniques for establishing spectral gaps have played a central role in the modern theory of Markov chain mixing times and the theory of high-dimensional expanders. One of the most striking results in this burgeoning literature is that a spectral gap for the global down-up walk on the facets of a pure simplicial complex can be reduced to sufficiently strong spectral expansion of just the codimension-2 links of the complex, a phenomenon colloquially referred to as "trickle-down". These types of theorems have had many important applications, including rapid mixing of the exchange walk on the bases of any matroid. In this primarily expository article, we give streamlined proofs of two such theorems in the literature, one by Oppenheim (2018) and one by Leake and Oveis Gharan (2025), via an integrated Bochner method. Moreover, in the latter setting, we quantitatively strengthen the dependence of the global spectral gap on the dimension of the complex and the spectral influence, resolving an open question of Leake and Oveis Gharan. Disclaimer: The proofs were developed through a couple of rounds of interaction with GPT-5.6 Sol Ultra. We later discovered that Guo and Zhang (2026) had independently proven the same strengthening of the trickle-down theorem of Leake and Oveis Gharan using an extremely similar argument, also found by GPT-5.6 Sol Ultra. The focus of their paper is the complexity of approximating the partition function of spin systems on planar graphs, not on the trickle-down phenomenon itself. In contrast, our motivation is primarily expository, and we hope to bring Bochner-type methods and their connections with the trickle-down phenomenon to the attention of a wider community of researchers.

Authors: Evan Unit Lim

Figured-bass realization can be described as a sequence of choices constrained both within each sonority and between successive sonorities. This paper gives an explicit mathematical model of a restricted, examination-style four-part realization problem. Pitch spelling, range, chord membership, doubling, omission, spacing, crossing, overlap, melodic motion, consecutive perfect intervals, and selected resolution requirements are expressed as predicates. We distinguish hard constraints from optional preference costs. Four labeled notes are represented visually as the vertices of a quadrilateral and computationally as one ordered voicing state. Legal progressions become paths through a layered graph. We prove that feasibility and minimum-cost realization are polynomial-time problems for a fixed number of voices with explicit finite note domains and fixed local rules. For fixed ranges, a fixed note alphabet, and adjacent-event rules, the number of graph operations is linear in the number of events. Worked two-, four-, and eight-beat examples illustrate legality, optimization, and the failure of a greedy choice. The result concerns the stated formal model; it is not a claim that every musical judgment is captured by local predicates.

A fully-funded PhD position is available with Sajin Koroth at UVic starting Jan 2027. Research focuses on theoretical CS (circuit & communication complexity, quantum info). A solid TCS background is required. As the official deadline has passed, please email your CV, transcripts, and background summary to skoroth@uvic.ca by Sept 30, 2026.

Website: https://web.uvic.ca/~skoroth/
Email: skoroth@uvic.ca

By shacharlovett

The UC San Diego Department of Computer Science and Engineering (CSE) invites applications for tenure-track faculty positions at the Assistant Professor rank. The department is looking for exceptional candidates in all areas of Computer Science and Engineering.

Website: https://apol-recruit.ucsd.edu/JPF04649
Email: nbarr@ucsd.edu

By shacharlovett

October 17-18, 2026 Rochester, NY https://www.cs.rochester.edu/u/shossei2/eagl2026website/index.html Submission deadline: October 1, 2026 Registration deadline: October 1, 2026 The purpose of this annual workshop is to bring together researchers in theoretical computer science, who work in the vicinity of the eastern great lakes region. For 2026, this event is held at the University of Rochester.

By shacharlovett

Cryptographic combiners take $n$ candidate schemes for some primitive $P$, and realize it securely provided that at least $k$ candidates are secure. Intuitively, we expect that if we plug $n$ independent instances of a $\delta$-weak candidate for $P$ into the combiner, assuming $\delta \ll (n-k)/n$, security should be amplified and the resulting candidate should exhibit a significantly smaller weakness. This intuition relies on the implicit “all-or-nothing” assumption that each candidate fails with probability $\delta$ and is otherwise perfectly secure, allowing us to bound the failure probability of the combiner using a simple binomial tail bound. However, this intuition often fails for standard security notions where, for example, a weak candidate may consistently leak partial information rather than exhibit a clean all-or-nothing failure. Recently, Applebaum, Bitansky and Geier (CRYPTO 2026) showed that indistinguishability combiners inherently act as security amplifiers. However, this general result incurs a multiplicative loss of roughly $2^{n-k}$ in the error parameter relative to the natural all-or-nothing bound. Moreover, the combiner-is-amplifier approach is fundamentally restricted to the regime $\delta < 0.5$. Consequently, both the resulting error rate and the amplification threshold $\delta_0$ are suboptimal. In this work, we overcome these limitations by establishing a new specialized framework of bias-resilient indistinguishability combiners. This formulation allows us to achieve the optimal all-or-nothing bound without the exponential penalty. We observe that bias-resilience provides a unifying abstraction for security amplification across different primitives, neatly capturing prior ad hoc results such as those for weak PRGs and weak NIZK. As our main application, we use this framework to establish a generalized XOR lemma over prime fields $\mathbb{F}_p$, showing that the sum modulo $p$ of independent weakly pseudorandom elements becomes computationally indistinguishable from uniform. This improves upon a recent work by Shimizu and Yasunaga (STOC 2026) by achieving a sample complexity that is independent of the field size. Finally, we explore the idealized notion of an all-or-nothing amplifier. We establish a tight characterization of the multiplicative penalty incurred when applying such an amplifier to candidate schemes that only guarantee standard weak indistinguishability error.

The permanent and mixed discriminant of positive matrices are classic problems for which we do not expect an efficient algorithm for exact computation. Thus much work has been done to understand how well we can bound and approximately compute these quantities. One line of research in this area begins with the results of the first author, where van der Waerden lower bounds of $\frac{n!}{n^n}$ are proven for doubly stochastic inputs for both problems, using a simple proof via stable polynomials. Along with the bound itself, the same techniques are used to show that the permanent and mixed discriminant are uniquely minimized at a certain natural symmetric input. In this paper, we generalize those results in two ways. First, we extend the unique minimization results beyond doubly stochastic inputs to other marginals which are near doubly stochastic. This yields the first such unique minimization results for the mixed discriminant beyond the doubly stochastic case. We also discuss why one cannot hope similar results to hold in general for all marginals. Second, we extend the unique minimization result for real stable polynomials to strongly log-concave (aka Lorentzian) polynomials in the doubly stochastic case. This captures an analogous previous result on unique minimization for the mixed volume. Finally, we discuss various open problems related to these results.

In this paper, we construct a total Boolean function with $\widetilde{O}(\log n)$ randomized communication protocol, while any monochromatic rectangle has density at most $O(2^{-\mathrm{poly}(n)})$. As a corollary, it gives the first total function separation for $\mathrm{BPP}\not\subseteq\mathrm{P}^{\mathrm{NP}}$ in the communication world. Inspired by Gavinsky’s recent work (arXiv:2608.18784), our construction combines the cheat-sheet framework with fully linear PCPs.

Gowers, Green, Manners, and Tao (Annals '25) recently resolved Marton's polynomial Freiman–Ruzsa conjecture. We give an algorithmic counterpart to their result: given uniform sampling and membership-oracle access to a set $A \subseteq \mathbb{F}_2^n$ with doubling constant at most $K$, our algorithm outputs a subspace of size at most $|A|$ whose $K^{O(1)}$ translates cover $A$. The algorithm runs in $\mathrm{poly}(n,K)$ time. As applications, we obtain polynomial-time algorithms for a variety of learning problems, including quadratic Goldreich–Levin, improper agnostic tomography of stabilizer states, and tomography of quantum states with bounded stabilizer extent.

We give an elementary proof of the Komlos conjecture by simplifying the recent proof of Guo, Fang, and Lu. We show that any vectors $v_1,\ldots,v_n\in\mathbb{R}^d$ with $\|v_i\|_2\le1$ admit signs $\varepsilon_i\in\{-1,1\}$ such that $\|\sum_{i=1}^n\varepsilon_i v_i\|_\infty\le36$. The proof uses only elementary combinatorial and probabilistic arguments and basic calculus.

Sandia National Labs invites applications for the Gil Herrera Fellowship in Quantum Information Science. We encourage candidates working in quantum algorithms, complexity, information, or related areas of computer science or mathematics to apply.

Website: https://www.sandia.gov/careers/careers/students-and-postdocs/fellowships/gil-herrera-fellowship-in-quantum-information-science/
Email: odparek@sandia.gov

By shacharlovett

ITMS 2026

Two years ago I attended the International Manufacturing Technology Show in Chicago's McCormick Place and found a rather limited focus on artificial intelligence among the exhibitors. ITMS is back in town so I went again this week. A quiet respite from all the AI/math angst, though that will come in full view when mathematicians take over the same venue in January.

I picked up my badge, the last to say "Illinois Tech" as I registered for a free academic pass well before the layoffs. Of course, you get to see all sorts of neat machines that make stuff but I tried to focus on where artificial intelligence plays a role. This time you could see AI everywhere, though as one exhibitor said, more of a marketing scheme than deep use of modern artificial intelligence. Real artificial intelligence did make appearances: vision recognition for robotic arms, backend software such as bid, invoice and document generation, predictive maintenance, and a variety of robotics, though mostly arms for assembling, cutting and even welding. 

I didn't see much of humanoid robots, digital twinning, use of large language models or manufacturing on demand. When do we get to the point that I can describe a product and have it designed, made and shipped to me quickly?

Not soon. I talked with someone from a small company that takes CAD designs and gets them ready for the manufacturing process. I asked him about automating the design phase and he said they leave that to ChatGPT. But OpenAI and Anthropic were nowhere to be found and Microsoft, Google and Amazon had scaled-down exhibits from two years ago.

Two years ago I remarked on the big booths for European and Asian manufacturers. This year had noticeably fewer giant foreign machinery stands. I'm guessing tariffs and trade uncertainty have dampened the influx of foreign suppliers.

China has gone all in on AI and manufacturing. I can imagine a Chinese slogan:

The US uses AI to make theorems, China uses AI to make products.

By Lance Fortnow

Hi there, argmin readers! As the fall semester picks up, posting volume will, too. So I’m going to commit to writing short descriptive headers to help you sort through the different threads. Today’s post is a live blog of Class 7 of my graduate seminar “Forecasting: A Critical Retrospective.” The syllabus and list of past posts is here.

In Kathryn Schulz’s New Yorker article, “The Really Big One,” she often cites figures about the chances of earthquakes.

“[T]he odds of the big Cascadia earthquake happening in the next fifty years are roughly one in three. The odds of the very big one are roughly one in ten.”

She described how these numbers arose by counting historical events and turning them into probabilities of the future. In the first week of class, we discussed this alchemy for clear discrete events. Coin flips, free throws, or elections have known times at which they occur. Only their outcome is uncertain. When we try to predict time, we need a new protocol: survival analysis.

Survival analysis is something I learned (and I think most people learn) in medical statistics. The name kind of gives it away: who do you think is surviving here other than patients in medical studies? In medicine, survival analysis captures the proportion of individuals still alive after a potentially life-extending treatment. Or, just as commonly, we flip this around and ask what proportion of subjects have not experienced a bad event yet.

In a randomized clinical trial, all patients start their timers at the same time—when they are randomized. The trialists gather the times from randomization until the bad events, and then estimate a probability distribution on the time until an event occurs. This distribution is over times, and is specified by a curve that models the chance the time to a bad event is greater than T. For example, this could be a distribution of how long it takes for cancer to progress under some new treatment. Or it could be how long until someone contracts an infection in a vaccine study. A survival curve lets you make probabilistic forecasts. For every time, you can look at the estimated proportion of individuals who have not yet experienced a bad event and call that the prognosis. “90% of patients experience no bad outcomes in the year following treatment.”

Here’s the most famous survival curve of all time. Who remembers this one?

These curves are estimated using a nonparametric method called the Kaplan-Meier estimator. The Kaplan-Meier curves give a rough shape of the survival distributions and let clinicians compare the relative effectiveness of treatments. When the treatment and control curves are far apart, it suggests something meaningful differs between the treatment and control conditions.

We can apply the same survival analysis ideas to other time-to-event forecasts. Let’s caricature how we might do it for earthquakes. For a single fault, you can imagine a major earthquake as a “reset” of the tension in the earth. Each earthquake gives you a time to start counting until the next one. If we assume every earthquake follows identical geodynamics, we can treat each earthquake like a patient in a trial, now estimating a survival curve for the time to the next earthquake. This is a crude model, as it assumes a total reset of conditions, but it’s a starting point.

Once we have this model, our historical record gives us a path to estimate the survival curve and forecast the probability of an earthquake in the next T years. Although we could build a Kaplan-Meier curve here, we could also pose an explicit model of how the probability changes over time and fit the parameters.

Any probability distribution over nonnegative numbers can serve as a model for the time to event and thus be turned into a survival curve. A common distribution in earthquakes is the exponential distribution. That is, the model is that the probability that a new major earthquake happens within T years after the first one is:

The nice thing about the exponential distribution is that it only has one parameter to estimate from data, and the maximum likelihood estimate is super simple. It’s

This formula gives us a straightforward program. Look at the historical record and compute the average waiting time between events, W. If you want probability forecasts over time windows, treat this average time as the inverse of the parameter of the exponential distribution. In this model, the probability that there will be a new event in T years is

This model is too simplistic, but it’s the first back-of-the-envelope calculation people do, and it’s where all the figures in Schulz’s New Yorker article come from.

This exponential survival model is the same as modeling earthquakes as a Poisson process. You can get fancy and make your model more sophisticated to capture more physical reality, specializing parameters to the particulars of each fault. You can model the survival function with some other distribution, be it log-normal, Weibull, or whatever. However, every modeling assumption you make adds more parameters to fit from data, and earthquakes don’t occur frequently enough to fit that many parameters to reasonable precision. If you have only forty events, you should probably estimate only one parameter.

Whatever modeling you do, survival analysis gives us another apparatus for turning counts into chances. How precise you think those chances are now rests on a whole lot of untestable modeling assumptions. What is the chance those assumptions are wrong?

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By Ben Recht

For every fixed $k\ge3$, we construct an explicit total Boolean function in the $k$-player number-on-forehead model with public-coin randomized communication complexity $O_k(1)$ and nondeterministic communication complexity $\Omega_k(n)$, where $n$ is the number of bits on each forehead. This extends the explicit three-player separations of Kelley, Lovett, and Meka (STOC 2024) and Kelley and Lyu (FOCS 2025) to every fixed number of players, and as a side product improves the three-player nondeterministic lower bound from $\Omega(n^{1/2})$ to the optimal $\Omega(n)$. Our construction is based on algebraic geometry codes.

We show that $\bigl(\frac{\log n}{\log\log n}\bigr)$-approximate parameterized $k$-SetCover is W[1]-hard, and has no $n^{o(k/\log k)}$-time algorithms under ETH. This improves upon the previous best factors $\bigl(\frac{\log n}{\log\log n}\bigr)^{1/k}$ in (Lin, 2019) and $(\log n)^{1/\text{poly}(k)}$ in (Karthik, Laekhanukit, and Manurangsi, 2019). Here $k$ is the yes-case guarantee and $n$ is the number of candidate sets. While the best approximation ratio is still $O(\log n)$ via the greedy algorithm, closing this $1/k$ gap in the exponent has been a longstanding open problem; we remove this loss via a simple direct reduction. The construction is self-contained and does not rely on the parameterized inapproximability hypothesis (PIH). Starting with sparse parameterized 2-CSP instances (Karthik, Marx, Pilipczuk, and Souza, 2024), we build a monotone CNF formula, which is equivalent to a SetCover instance. To obtain a $k$-versus-$h$ gap, the reduction enumerates all hash functions from $\Sigma$ to $[2h]$ and all unsatisfiable 2-CSP instances on the same constraint graph with alphabet $[2h]$. For each such instance, it asks for a certificate that the hashed label pairs are not all contained in that instance. Perfect hashing makes this enumeration efficient for $h=\log n/\log\log n$.

Given any fixed constant $0<\varepsilon<1$ and a matrix space $\mathcal{B}=\langle B_1,\ldots,B_m\rangle\le\mathbb{Q}^{n\times n}$, we give a deterministic $NC^3$ algorithm that outputs a matrix \(A\in\mathcal{B}\) such that $rank(A)\geq (1-\varepsilon) crk(\mathcal{B})$, where $crk(\mathcal{B})$ denotes the maximum rank of a matrix in $\mathcal{B}$. This complements the recent breakthrough of Chatterjee, Ghosh, Gurjar, Raj, and Thierauf (ECCC, TR26-100), who gave an $NC$ algorithm for computing the noncommutative rank of symbolic matrices. For commutative rank, Bl\"{a}ser, Jindal, and Pandey previously gave a deterministic polynomial-time approximation scheme (ToC, 2018). Our algorithm follows a different route from the subspace-design approach of Chatterjee, Ghosh, Gurjar, Raj, and Thierauf. It has two main ingredients. First, using a polynomial-size $4$-wise independent family together with operator scaling (Gurvits'04, Garg-Gurvits-Oliveira-Wigderson'20), we obtain a scalar matrix whose rank is an absolute constant fraction of $crk(\mathcal{B})$. Second, we boost this constant-factor approximation to a $(1-\varepsilon)$-approximation by analyzing the associated Schur complements through Smith normal form over a discrete valuation ring. This mainly helps in iteratively reducing the rank deficit.

Braverman, Rao, Raz, and Yehudayoff (SICOMP 2014) showed that there is an explicit pseudorandom generator (PRG) that fools standard-order regular read-once branching programs (ROBPs) with seed length $$ O(\log n \cdot \log \log n + \log n \cdot \log(wd/\epsilon)), $$ where $w$ is the width of the program, $n$ is the length, $d$ is the alphabet size, and $\epsilon$ is the error of the generator. To prove it, they prove a bound on the error of the INW generator (Impagliazzo, Nisan, and Wigderson, STOC 1994) in terms of the spectral expansion parameters of the expander graphs used to construct the generator. Then they plug in standard explicit constructions of sparse spectral expanders. In this paper, we prove that Braverman, Rao, Raz, and Yehudayoff's analysis is optimal. That is, if some instantiation of the INW generator fools standard-order regular ROBPs and the proof of correctness doesn't use any properties of the underlying graphs except bounds on their spectral expansion parameters, then the seed length of the generator is at least $$ \Omega(\log n \cdot \log \log n + \log n \cdot \log(wd/\epsilon)), $$ provided $w \in [6, 2^{n^{0.99}}]$, $\epsilon \in [2^{-n^{0.99}}, 0.01]$, and $d \leq \mathrm{poly}(n)$. A lower bound of $\Omega(\log n \cdot \log(w/\epsilon))$ was already known even for the special case of fooling permutation ROBPs (Hoza, Pyne, and Vadhan, Algorithmica 2024). Our contribution is to prove that the $\log n \cdot \log \log n$ and $\log n \cdot \log d$ terms are unavoidable if one wishes to fool regular programs.

Authors: Michal Dvořák, Ioannis Kakatelis, Dušan Knop

Minimum-cost spanning tree game (MSTG) is a cooperative game played on an undirected edge-weighted graph $(G,w)$ representing the network, where each vertex corresponds to a player and each edge has an associated cost~$w$. A distinguished vertex $s \in V(G)$ represents the supply or source. For any coalition of players $S$, the characteristic cost function $c(S)$ is defined as the minimum cost of a spanning tree with respect to $w$, connecting exactly the vertices in $S \cup \{s\}$. In this paper we study the computational complexity of deciding core membership for MSTG. In general, deciding whether a given allocation is in the core is \textsf{coNP}-hard~(Faigle et al.,International Journal of Game Theory,1997). We study the core recognition problem under the name {\sc MSTG Core Non-Membership}. We extend the hardness to graphs which are very close to being planar. On the positive side, we present several algorithmic results within the framework of parameterized complexity. We show that {\sc MSTG Core Non-Membership} is fixed-parameter tractable when parameterized by the support size of the allocation. Turning into structural parameters of graphs, we show that the problem admits an FPT algorithm parameterized by treewidth and signed neighborhood diversity. Last but not least, we investigate kernelization. While in general graphs, under standard complexity-theoretical assumptions, {\sc MSTG Core Non-Membership} does not admit a polynomial kernel parameterized by the vertex cover number, we design a cubic kernel in planar graphs. Furthermore, in general graphs, we obtain quadratic kernel for signed neighborhood diversity and linear kernel for the parameter feedback edge number.

Authors: Antoine Vinciguerra

In a seminal work, Buhrman et al.\ (STOC 2014) introduced catalytic computation and proved that uniform $TC^1$ circuits are computable in catalytic logspace, the class of problems solvable in space $s$ with an additional catalytic tape of size $c$, a tape whose initial content must be restored at the end of the computation. A central ingredient of their proof is the register program model. Namely, they constructed a uniform family of register programs that computes $x^n$ using $n$ registers and four accesses to $x$. Since then, determining the number of registers and input accesses required to compute a polynomial of a given degree has become a central question in the study of catalytic computation. On one hand, we prove that the four-access bound of Buhrman et al.\ is optimal: every passive-output register program computing a polynomial of degree greater than three requires at least four input accesses, independently of the number of registers. On the other hand, we show that their register bound is not optimal. For every $t\geq2$ and every field $K$ of characteristic $0$ or greater than $2t-1$, we construct a register program for $x^{2t-1}$ with four input accesses and $t$ registers. Our proofs rely on derivations and their exponential operators. This approach represents a register program as a series of exponential derivation operators, reducing register restoration to an operator identity. Finally, we use the uniform family of register programs to improve known trade-offs for catalytic streaming algorithms and register programs for matrix powering. The generalization of the lower-bound methods and the construction of the uniform family of register programs were developed with assistance from ChatGPT 5.6.

Authors: Stefan Göller, Amaldev Manuel

We study the circuit complexity of regular languages in terms of unbounded fan-in Boolean circuit families. We characterize the regular languages of constant circuit complexity in terms of the one-variable fragment of first-order logic with regular predicates, in terms of the pseudovariety of stamps $\mathbf{QEJ}_\mathbf{1}$, suitable word congruences and regular expressions. We analogously characterize the neutral letter regular languages of constant circuit complexity. Our lower bound result implies that the class of regular languages of sublogarithmic circuit complexity coincides with the one of constant circuit complexity. In addition we show that deciding whether a regular language, given as a nondeterministic finite automaton, has constant circuit complexity is $\mathbf{PSPACE}$-complete. We introduce a strong notion of reduction, called rational truth-table reduction, that is tailored towards algebraically defined classes of languages. We show that, for a class of functions we call mild, rational truth-table reductions preserve both upper and lower bounds on circuit complexity. We show that the class of regular languages, whose circuit complexity is bounded by a mild function, is in fact a length-multiplying variety of languages. Slightly extending the class of regular languages of constant circuit complexity, we analogously characterize the class of regular languages that are in the pseudovariety $\mathbf{QEACom}$. For these we derive logarithmic circuit complexity upper bounds.

Authors: Pierre Senellart, Anton Gnatenko

We show that descriptive complexity can serve as a foundation for formalizing computational complexity results in a proof assistant, by constructing a Lean library centered around the following concepts: decision problems are isomorphism-invariant predicates on finite structures; complexity classes are defined by their logical characterization; membership is shown by definability witnesses; hardness is shown by first-order reductions from a known hard problem. We also establish bridges to traditional machine models such as (non)deterministic Turing machines. The library proves 73 completeness results, on 68 problems or problem families, over 14 different classes; relations between the classes established inside the logic and not by machine simulation, among them NL = coNL and the Abiteboul-Vianu theorem; and unconditional lower bounds, among them $\mathrm{FO}(\leq) \subsetneq \mathrm{FO}(\leq, \mathrm{TC})$ and the failure of order-free FO(IFP) to capture PTIME.