Papers

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.

The round-robin procedure is a simple and well-studied fair division mechanism where agents pick goods in turns. Motivated by draft mechanisms in sports leagues, we investigate strategic behaviour in online round-robin for subadditive agents. This gives rise to an extensive-form game, and we study the computational problem of computing a subgame perfect Nash equilibrium (SPNE). We show that for just two submodular agents, computing an SPNE is $\mathsf{PSPACE}$-hard. Even for the class of $\mathit{OXS}$ utilities, which are a special case of submodular utilities, computing an SPNE remains $\mathsf{NP}$-hard for a small number of agents. We complement our computational results with normative results. We show that for just three additive agents, there exist instances where every equilibrium violates EF1. This separates the online and the direct revelation games. On the positive side, we show that for additive agents every equilibrium allocation is proportional up to one good (PROP1) and for two additive agents it is also EF1. Finally, by showing that round-robin is bossy at equilibrium, we prove that the number of equilibrium allocations can be exponential even if agents have lexicographic preferences.

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.

We study the challenging problem of learning depth-three circuits in the mistake-bound model of (realizable) online learning, which is a more difficult model than distribution-free PAC learning. Prior algorithms for this problem, due to Servedio and Tan [ST17], could only learn polynomial-size depth-three circuits of poly$(n)$ size over $\{0,1\}^n$ with a running time of $2^{n - \Omega(n/\log n)}$, and hence they ran in time $N^{1-o(1)}$ where $N=2^n$ is the running time of a naive memorization-based approach. In this work we substantially improve on the [ST17] result: for any constant $\gamma\geq1$, we give an algorithm that learns depth-three circuits of size $n^\gamma$ with running time \[ 2^{n-c_\gamma n}, \] where $c_\gamma>0$ depends only on $\gamma$ and not on $n$. Hence we achieve a polynomial savings over the naive approach for learning any polynomial-size depth-three circuit. The main driving force behind our improvement is an improved bound on the approximate degree of width-$k$ CNFs. Inspired by Szegedy [Sze04] and Magniez et al. [MNRS11], the rough idea of our construction is to use a Chebyshev polynomial to efficiently amplify the spectral gap of a carefully designed random walk. This is combined with a random-restriction-like approach to separately learn different subfunctions corresponding to different assignments to a randomly chosen set of variables, using the Perceptron algorithm over a specially designed feature space. A simplified warmup instantiation of our approach achieves $c_\gamma = \exp(-O(\gamma))$; by augmenting this warmup with further ingredients we obtain the sharp form of our result, which achieves $c_\gamma=\Omega(1)/\gamma$.

We prove that counting perfect matchings is $\#P$-complete under polynomial-time Turing reductions on each of three classes of simple, unweighted graphs: monotone graphs, unit interval graphs, and chordal permutation graphs. The monotone result settles the exact-counting complexity left open by Dyer, Jerrum, and Müller (JACM 2017), complementing their rapid-mixing theorem. Inspired by quantum circuits, our reductions implement a circuit simulation using globally coupled matching-transfer operators. The key construction is an exact projection, implemented by a polynomial-length sequence of normalized transfers, that restores tensor-product locality and makes encoded gates composable. Interpolation-based cancellation then reduces circuit evaluation to unweighted perfect-matching counts in all three classes. We also place Dyer and Müller's class QChains within the distance-hereditary graphs and give an $O(n^2)$-arithmetic-operation counting algorithm for the latter, improving the $O(n^4)$ bound obtainable from Curticapean and Marx (SODA 2016). Together with prior results, these advances complete the exact-counting classification of the graph classes in Dyer and Müller's diagram (SIDMA 2019).

We study data structure lower bounds for the Approximate Matrix Powering (AMP) problem. Given a substochastic, symmetric matrix $\mathbf{M}\in\mathbb{R}^{n\times n}$ and parameters $k$ and $\alpha$, the goal is to preprocess $\mathbf{M}$ so as to answer entry queries $(u,v)\mapsto \mathbf{M}^{k}[u,v]$ up to additive error $1/n^{\alpha}$. We focus on AMP in the succinct and systematic regime, in which the data structure stores $\mathbf{M}$ verbatim, uses an additional $r$ bits of redundancy, and must answer queries by probing only a small number of entries of $\mathbf{M}$. Our main conceptual contribution is a general framework for proving probe–redundancy trade-offs for systematic data structures. We introduce the query-with-sketch model and develop a min-entropy-based approach that lifts conditional min-entropy bounds in the absence of redundancy to probe lower bounds in the presence of redundancy. We then establish these min-entropy bounds using problem-specific analytic and algebraic tools, for the downstream applications to AMP and its variants. As a consequence, our results provide new unconditional evidence toward a conjecture of Patrascu and Roditty (2010) on the space required for constant-time set-disjointness queries.

A decomposable randomized encoding (DRE) for a function $f$ allows $n$ parties, using shared randomness, to encode their individual inputs locally so that the collection of encodings reveals $f(x_1,\ldots,x_n)$ and nothing else. DREs are widely used in efficient multiparty computation. Their main complexity measure is size, the total bit length of the local encodings. Yet the optimal DRE size remains poorly understood even for the $n$-bit OR function. We prove the first superlinear lower bound for OR and, more generally, for every non-periodic symmetric function. Under an additional symmetry assumption, we prove a sharp $\Omega(n\log n)$ lower bound for OR, matching the classic construction of Feige, Kilian, and Naor (STOC 1994). We also prove the first $\Omega(n^2)$ lower bound on DRE size for non-explicit Boolean functions.

For a constraint satisfaction problem defined by a relation $R$, its non-redundancy $\text{NRD}(R,n)$ is the size of largest instance (as a function of the number $n$ of variables) for which no constraint is implied by the rest. Its chain length $\text{CL}(R,n)$ is the largest such instance where the constraints can be ordered so that no constraint is implied by the preceding ones. Clearly $\text{CL}(R,n) \ge \text{NRD}(R,n)$ but so far no asymptotic separation was known between these quantities. We exhibit an explicit arity $4$ relation for which $\text{CL}(R,n) \ge \omega(\text{NRD}(R,n))$.

Eternal Vertex Cover (EVC) is a turn-based attacker-defender game on an undirected graph $G$. To begin with, the defender places $k$ guards on vertices of $G$. The attacker, on their turn, can choose an edge $e$ not already occupied at both endpoints to "attack". The edge $e$ is defended if a guard moves along the edge $e$. The defender, on their turn, can move any subset of guards. A guard can only move to a neighboring vertex. The minimum number of guards needed to indefinitely defend against any sequence of attacks is called the eternal vertex cover number, generalizing the classic vertex cover number. Determining this number is NP-hard in general, motivating the study of parameterized and approximation algorithms. The problem is known to be FPT when parameterized by the cover number, but structural parameters remain relatively unexplored in the literature. In this work, we explore structural parameterizations for EVC. We show that EVC is FPT parameterized by the cluster vertex deletion number, which generalizes the previously studied parameterization by vertex cover number. We next study the problem parameterized by vertex integrity, which is the smallest number of vertices we need to delete from $G$ so that the resulting graph is a disjoint union of constant-sized components. We first show that Eternal Vertex Cover is XP parameterized by vertex integrity. Then, we develop a polynomial-time approximation algorithm, which computes an additive $6k+1$ ($g(k)$) approximation, where $k$ is equal to the cluster vertex deletion number (vertex integrity). Finally, we show a FPT algorithm for when the deletion set produces "nice" connected components, which are components that are bounded in size and satisfy a technical condition.

This paper is the first work at the intersection of game theory and property testing, giving algorithms and lower bounds for efficiently testing whether an allocation mechanism is incentive compatible (IC). We propose distinguishing whether a mechanism is $\epsilon$-far from being IC, i.e., when it observes many monotonicity "violations." Conceptually, inspired by the literature on Boolean function monotonicity testing, we construct a tester for discrete single-parameter allocation rules. Technically, our work is the first to consider monotonicity testing of vector-valued functions on the hypergrid. We give a $\tilde{O}(n/\epsilon)$-query algorithm to test whether a function (representing n-player allocation mechanisms) is coordinate-wise monotone versus $\epsilon$-far from it. We also show a matching lower bound: the class of coordinate-wise monotone vector-valued functions on a Boolean hypercube or hypergrid requires $\tilde{\Omega}(n/\epsilon)$ queries to test whether it is $\epsilon$-far from monotonicity, and this holds even if the tester is two-sided and allowed to make adaptive queries. Finally, we extend our upper bound to and give a tester of the same query complexity for pricing functions of allocation mechanisms. This requires overcoming the technical challenge that the path in function space to the closest IC mechanism may involve interdependent changes to both the price and the allocation rule.

Prefix-free parsing provides an efficient way to construct compressed representations of large and repetitive pangenomes and naturally induces a graph representation known as a prefix-free graph. In this work, we initiate a theoretical study of the problem of constructing prefix-free graphs of minimum size, where the size accounts for both the total length of distinct segment labels and the paths representing the input sequences. We show that selecting an optimal set of trigger words is NP-hard, already when triggers consist of single characters. Using a synchronized-code reduction, we extend this hardness result to every fixed trigger length and further show that the problem remains NP-hard over an alphabet of size three. We then establish a structural connection between prefix-free graphs and de Bruijn graphs. In particular, we show that every compacted de Bruijn graph can be realized as a prefix-free graph and derive a hierarchy relating the sizes of minimum pangenomic graphs, minimum prefix-free graphs, compacted de Bruijn graphs, and de Bruijn graphs. Finally, we give an exact fixed-parameter algorithm running in $O(2^q n)$ time, where $q$ is the number of distinct candidate trigger words and $n$ is the total pangenome length. Our results characterize both the computational limitations and the structural properties of optimizing prefix-free graph representations and provide a theoretical foundation for the design of compact graph representations of repetitive pangenomic data.

Conner, Gesmundo, Landsberg and Ventura (2019) determined the largest stabilizer dimension of concise $n\times n \times n$ tensors that are binding, and they determined the corresponding maximizing tensors to be the null algebra tensors. They left as an open problem to extend this to all concise $n \times n \times n$ tensors (i.e. dropping binding). We solve this problem: We prove that the largest stabilizer dimension of concise $n\times n\times n$ tensors is $n^2 + 1$ and the maximizers are the null algebra tensors (as in the binding case) and the skew symmetric tensor $e_1 \wedge e_2 \wedge e_3$. As part of our approach we obtain upper bounds on the stabilizer dimension of matrix tuples under left-right action (generalized Kronecker quiver representations), which we think are of independent interest.

Communication complexity studies how much information must be exchanged to solve a problem whose input is split among several parties. The classical setting deals with Boolean inputs split between two parties. We study an algebraic variant, where the inputs are vectors over a field $\mathbb{F} \in \{\mathbb{R}, \mathbb{C}\}$. Alice and Bob have inputs $X\in \mathbb{F}^n$ and $Y\in \mathbb{F}^n$, respectively. We consider two kinds of tasks: the polynomial evaluation problem (compute the value of a polynomial $g\in \mathbb{F}[X,Y]$), and the set-recognition problem (decide whether (X,Y) is in $S$, for $S\subseteq \mathbb{F}^{n} \times \mathbb{F}^n$). In both settings, Alice and Bob send evaluations of polynomials depending only on their own inputs. In the set-recognition problem, a referee receives the messages and may apply polynomial tests to the messages received so far; the outcomes of these tests determine acceptance or rejection. The protocols may be deterministic or probabilistic. We study: - Upper bounds and reductions: We give non-trivial upper bounds for a range of natural polynomial evaluation and set-recognition problems and prove reductions between different problems, which help organize the landscape of the model. - A lower bound framework and tight lower bounds: Our main technical contribution is a general framework for proving lower bounds for algebraic set-recognition problems. We prove several probabilistic lower bounds for natural problems, giving tight or near-tight characterizations of their algebraic communication. - Applications of the framework: Finally, we give two applications of our framework: proving lower bounds for a class of left-to-right algebraic algorithms (algebraic scanners) and a more general algebraic computational setting inspired by the BSS model.

We give new bounds for list sizes of random linear codes at capacity, max loads of linear hash functions, and Furstenberg sets, over every finite field $\mathbb{F}_q$. 1. Random linear codes over $\mathbb{F}_q$ with rate $1 - H_q(p) - \epsilon$ are $(p, O(q H_q(p)/\epsilon))$-list decodable with high probability for all values of $p, q, \epsilon$, including the high error regime. This nearly matches the list size lower bound of $H_q(p)/\epsilon$ due to Guruswami, Li, Mosheiff, Resch, Silas, and Wootters [IEEE Trans. Inf. Theory 2022]. Our bound is the first uniform improvement for $q > 2$ since Guruswami, Håstad, and Kopparty [STOC 2010]. 2. Linear hash functions over $\mathbb{F}_q$ hashing $n$ balls to $n$ bins achieve maximum load $O(q \ln \ln q / {\ln q}) \cdot \ln n / {\ln \ln n}$, both in expectation and with probability $1-o(1)$. This nearly matches the lower bound of $\ln n / {\ln \ln n}$. Previously, only a polylogarithmic upper bound was known for $q > 2$, due to Alon, Dietzfelbinger, Miltersen, Petrank, and Tardos [J. ACM 1999]. We reduce list decodability and linear hashing to strong Furstenberg set lower bounds, which we prove using a new polynomial method of multiplicity gaps. While previous polynomial methods analyze a set $S$ by studying polynomials that vanish on it, we consider polynomials that vanish everywhere, but with higher multiplicity inside $S$ than outside.

We refine Gavinsky's framework (arXiv:2608.18784) for exponential separations between quantum and randomized communication complexity of total functions and obtain larger separations: polylogarithmic quantum communication versus $\tilde\Omega(\sqrt n)$ randomized communication with two quantum messages, and versus $\Omega(n^{1-\varepsilon})$ for every fixed $0<\varepsilon<1$ with more quantum messages.

We prove that exact Euclidean SVP is NP-hard under deterministic polynomial-time many-one reductions for full-rank cyclic integer lattices, equivalently full-rank ideals of $R_N:=\mathbb{Z}[X]/(X^N-1)$ in the coefficient norm. Hardness holds with $N=q-1$ for a varying odd prime $q$. As an application, we prove the same hardness for the algebraic class of NTRU-form lattices $\{(x,z)\in R_N^2:Hx\equiv z\pmod{QR_N}\}$, where $H,Q$ are unrestricted inputs. The decision problems are NP-complete, and the exact search problems are NP-hard under polynomial-time Turing reductions. No hardness claim is made for cryptographic NTRU parameter subclasses or key-generation distributions.

We prove that, for every constant $\varepsilon>0$ and every function $f:\Sigma^n\to\{0, 1\}$, there is a coalition of $O(n/\sqrt{\log n})$ coordinates and a target output $b\in\{0, 1\}$ such that, after the remaining coordinates are sampled uniformly and independently, the coalition can choose its values to make the output equal to $b$ with probability at least $1-\varepsilon$. The bound is independent of the alphabet size and also holds for monotone Boolean functions on $[0,1]^n$, resolving a conjecture of Friedgut (Combinatorics, Probability and Computing, 2004). Unlike the Boolean cube setting, where Kahn, Kalai, and Linial (FOCS, 1988) give a coalition bound of $O(n/\log n)$, no sublinear bound independent of the alphabet size was previously known. In collective coin flipping, our result gives the first sublinear bound on the number of bad players needed to force a fixed output with probability at least $1-\varepsilon$ in any one-round protocol with independent uniform messages, regardless of the message length. A key ingredient in our proof is an encoding that lets us relate the influence of a function on a product space to the $p$-biased influence of the encoded function. We then rely on a structure theorem of Hatami (Annals of Mathematics, 2012) for functions with small $p$-biased influence to bias the encoded function.

While quantum interactive proof systems using at least three messages can achieve perfect completeness, as shown by Kitaev and Watrous (STOC 2000), whether perfect completeness is achievable for one- and two-message quantum proof systems has remained open. For the one-message case, whether $\sf QMA$ can achieve perfect completeness was posed as an open problem in Watrous (FOCS 2000) and Aharonov and Naveh (2002); for the two-message case, the corresponding problems were (implicitly) posed in Jain, Upadhyay, and Watrous~(FOCS 2009) and Kobayashi, Le Gall, and Nishimura (SICOMP, 2019). In this work, we establish that ${\sf QIP}(2)$, ${\rm qq}\text{-}{\sf QAM}$, $\sf QAM$, and $\sf QMA$ can achieve perfect completeness. Here ${\rm qq}\text{-}{\sf QAM}$ denotes the class of promise problems admitting two-message quantum-public-coin quantum interactive proof systems in which the verifier's only message consists of half-EPR pairs. Our main technical contributions are the follows: 1. For $\sf QMA$ (and directly for $\sf QAM$), an exactly constructible block-encoded matrix whose kernel certifies yes instances, constructed from the acceptance operator induced by the verification circuit. This also yields an alternative proof of ${\sf QCMA}={\sf QCMA}_1$. 2. For ${\sf QIP}(2)$ (and implicitly ${\rm qq}\text{-}{\sf QAM}$), a new turn-halving transformation that preserves completeness and ensures that the resulting proof system retains at least two messages, provided that the terminal state before the final measurement is efficiently preparable.

We establish the worst-case hardness of several ideal lattice problems (including SVP and CVP) in the $\ell_2$ norm by providing a dimension-preserving, deterministic polynomial time reduction from their generic lattice versions. The reduction constructs an ideal lattice in the canonical embedding of a number field that approximates some input lattice up to scaling and orthogonal transformation. The integers defining the ideal and the ambient number ring, in particular its discriminant, are all polynomial in bit length relative to the generic input lattice. Furthermore, the ideal is invertible, the ring is monogenic, and the number field is totally real. If the number ring is also required to be a full ring of integers, the reduction conjecturally succeeds in bounded-error quantum polynomial time.

A long-standing open question in query complexity asks whether there is a total Boolean function f with R(f) << C(f), where R(f) and C(f) denote its bounded-error randomized query complexity and certificate complexity, respectively. We construct a function with R(f) = O~(sqrt{C(f)}), which is optimal up to log factors. The same function also has $Q(f) = O~(C(f)^{1/4}), where Q(f) is the bounded-error quantum query complexity of f, which is also nearly optimal.

We determine the exact time exponent of space-optimal reversible pebbling on chains as $1.331742379256310\ldots$. The growth rate of space-optimal reach exists as a limit and admits a variational formula. The same exponent governs complete computations at minimal space, uniformly in the chain length.

We study the minimum coefficient degree $N_{k,\F}(G)$ of a Nullstellensatz certificate for Bayer's $k$-coloring equations, where the characteristic of $\F$ does not divide $k$. If $J$ is a \HJ\ join of non-$k$-colorable graphs $G,H$ and $m=\max\{N_{k,\F}(G),N_{k,\F}(H)\}$, then $N_{k,\F}(J)\leq m+k$. When deletion of the selected edge makes each input $k$-colorable, we also have $N_{k,\F}(J)\geq m$; the degree congruence then gives $N_{k,\F}(J)\in\{m,m+k\}$. This partially answers a question of Li, Lowenstein, and Omar. For three-coloring over $\F_2$, we construct an infinite $4$-critical family of exact degree seven, attaining the bound at input degree four. In contrast, every graph constructed from $K_4$ solely by \HJ\ joins has degree $O(\log n)$ and a certificate with polynomially many terms: joins preserve treewidth at most three, and balanced separators yield low-degree certificates. Additional vertex identifications are excluded from this obstruction. We classify all single identifications of the $25$-vertex base graph; exactly $36$ preserve degree seven, producing $24$-vertex $4$-critical graphs of treewidth four. A compressed self-join at adjacent true twins prevents degree loss and gives a repeatable rule adding four vertices per round. The rule does not establish degree amplification or preservation of criticality. Exact witnesses and standalone verification programs accompany the finite results.

We prove that the quantum monad in dimension $2n$ admits no natural transformation to the polymorphism clone of linear equations modulo $n$. Consequently, for every $n\geq 2$, there exists an unsatisfiable system of linear equations over $\mathbb{Z}_n$ whose constraint system game admits a perfect finite-dimensional quantum strategy. As a corollary, we completely characterise pseudo-telepathic constraint languages in finite dimension. The proof combines a result of Harding, Jager, and Smith on group-valued measures on subspaces of Hilbert spaces with the polymorphism-minion characterisation of quantum pseudo-telepathy.

An explicit graph is given with left size $N$, left degree $\widetilde O(\log^2 N)$, right size $(1+o(1))K$ and $1$-expansion up to~$K$, meaning that every left subset of size $K' \le K$ has at least $K'$ neighbors. Let $\C(x)$ be the minimal length of a program that prints~$x$ (i.e., the central concept in Kolmogorov complexity). The $1$-expander is used to obtain an algorithm that on input $x$ computes in time $\poly(|x|)$ a list with $\widetilde O(|x|^3)$ programs such that at least 1 program prints $x$ and has length $\C(x) + O(1)$. This improves on the $O(|x|^{6+\eps})$ upper bound in~\cite{zim:c:shortlistshortproof} and is close to the $\Omega(|x|^2)$ lower bound from~\cite[theorem 4]{bmvz:j:shortlist}. In the companion paper “Online matching games in bipartite expanders: applications to data structures," the $1$-expander is used to obtain dynamic dictionaries in which the query operation has non-adaptive memory access.

The tensor rank of $n\times 2$-with-$2\times m$ matrix multiplication is at least $\left\lceil \frac{24m+2}{5} \right\rceil$ if $m\ge n=3$ and at least $(n+2)m+n-3$ if $m\ge n\ge 4$, over arbitrary fields. This result matches the Hopcroft-Kerr upper bound for $n=3$ and $m\le 6$.

Single-machine asymmetric weighted earliness–tardiness (AWET) scheduling asks how to sequence jobs around a common synchronization date when early and late completion incur unrelated job-dependent penalties. At the boundary nonrestrictive date $d=\sum_jp_j$, a compact V-shaped schedule reduces the continuous-time problem to a quadratic choice of a nonempty early set. We establish four complementary results for this model. First, the positive-integer problem is strongly NP-complete by a unary-polynomial reduction from Restricted Exact Cover by 3-Sets. Second, unrestricted AWET admits a polynomial-time $(3+2\sqrt2+\varepsilon)$-approximation based on an anchored semidefinite relaxation and deterministic marginal thresholding. Third, when the earliness and tardiness ratio orders are strict reversals, the problem is weakly NP-complete but has an exact two-resource pseudopolynomial dynamic program. Fourth, for fixed total refinements whose ratio permutation is separable, an exact separating-tree recurrence and coordinated geometric trimming yield an FPTAS. The proofs use different manifestations of the same canonical objective: scale-separated prefix penalties, positive-semidefinite minimum-kernel covariance, a dominant completed load square, and a bounded four-coordinate decomposition interface. Together, the results show that the decisive issue is not merely whether the two ratio orders agree, but whether their interaction can be controlled by a global certificate or compressed into a bounded constructive interface.

Quantum algorithms for topological data analysis compute Betti numbers, the ranks of the homology groups of a simplicial complex, which can be read off from the kernel of a combinatorial Laplacian. Deciding whether a Betti number of a clique complex is nonzero is $QMA_1$-hard, and remains so under a spectral gap promise on vertex-weighted graphs. Integral homology, however, contains information inaccessible to the Laplacian spectrum. A new part that appears in integral homology is torsion: cycles that become boundaries only after being traversed several times, as in a projective plane or a Klein bottle. We provide a simple reduction that allows us to establish the first hardness results for the problem of detecting torsion. We attach to an arbitrary clique complex a fixed $31$-vertex triangulation of the projective plane. The $k$th mod-$2$ Betti number of the input then reappears as $2$-torsion two degrees up, while all rational homology disappears and every combinatorial Laplacian acquires a constant spectral gap. We conclude that detecting torsion in clique complexes of unweighted graphs is $NP$-hard, even under a constant gap promise, and that it is $QMA_1$-hard if mod-$2$ clique homology is.

In this paper, we study the computational complexity of finding fixed points for set-valued contractions. We first formulate a computational problem for Nadler's fixed-point theorem: Projected-Nadler, and prove that it is $\mathsf{CLS}$-complete by showing its equivalence to Continuous-LocalOpt. We then establish a stronger converse for Nadler's fixed point theorem that can be applied as a tool to analyze the convergence rate of set-valued basic iteration procedure. Finally, we reduce large-margin triplet stationarity problem to Projected-Nadler. Together with its $\mathsf{CLS}$-hardness introduced in [arXiv:2509.16898], this yields $\mathsf{CLS}$-completeness of large-margin triplet stationarity.

In the best-arm identification problem, we are given $n$ stochastic arms with unknown means and wish to identify the arm with the largest mean with probability at least $1-\delta$, using as few samples as possible. We consider independent Gaussian rewards with unit variance and means in $[0,1]$. Chen and Li [2016] conjectured that the instance-wise sample complexity of this problem is characterized by the gap entropy, up to an additive term arising from the two-arm problem. In this paper, we resolve their gap-entropy and almost instance-wise optimality conjectures. For an instance $I$, let $\Delta_{[i]}$ be the gap between the largest and the $i$-th largest mean, let $H(I)=\sum_{i=2}^{n}\Delta_{[i]}^{-2}$, and let Ent$(I)$ denote the entropy of the normalized complexities of its dyadic gap groups. For every $0<\delta<0.1$, we show that the order-oblivious instance-wise lower bound is $ {\Theta} (H(I)[\log(1/\delta)+Ent(I)]). $ We also give a single $\delta$-correct algorithm with expected sample complexity $ O ( H(I)[\log(1/\delta)+Ent(I)] +D\log(e+\log(e+D))),D=\Delta_{[2]}^{-2}, $ without prior knowledge of the gaps. Our lower bound removes the dyadic-gap and monotonicity restrictions of previous work, and our upper bound removes the additional polylogarithmic factor multiplying the two-arm term. Thus, a single algorithm attains the instance-wise lower bound up to an additive two-arm term. The main theorems have been formalized and proved in Lean 4.

We study replacements in fixed bilinear tensor decompositions, counting changes to complete rank-one summands, including output factors. The shortening frontier records the maximum rank defect of a fixed-size subset and determines the minimum length attainable within a change budget. For the rational 48-term Li–Wang–Hu decomposition \(D(2)\) of \(4\times4\) matrix multiplication over \(\mathbb{C}\), we prove rank radius at least 12, strong radius exactly 11, and border radius at least 8. Every shorter complex decomposition therefore changes at least thirteen original summands. An exact rational twelve-term replacement attains the equal-length barrier. The proofs combine exhaustive support reductions with saturated projected kernels and zero-corner completion arguments controlling arbitrary minimal competitors. A reduced-incidence argument transfers kernel certificates to tensor-space neighborhoods. A Laurent normal form gives strong radius exactly 11 for the sixteen-term core at every nonzero complex parameter. On a nonempty Zariski-open subset of the actual parameter curve, the rank radius is at least 12, the strong radius exactly 11, and the border radius at least 8. We also prove incomparability of the full Kothari–Moitra–Wein sufficient criterion and the Sylvester-equipped kernel criterion. These results describe local decomposition structure rather than a new rank bound for full matrix multiplication.