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cs.CC: explore 66 source-linked works published from 2023 to 2026, with original documents and citations.
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Sources: arxiv. Collection updated 2026-09-15. Counts describe this index, not the complete source archives.
We formulate the generalized hidden shift problem over arbitrary finite groups and obtain upper and lower bounds for the sample complexity in terms of group theoretic data.
We study the approximability of rank aggregation under the Ulam metric. In the \emph{Ulam median} problem, the goal is to find a ranking (permutation) minimizing the sum of its Ulam distances to the input rankings, while in the \emph{Ulam center} problem, the objective is to minimize the maximum such distance. We prove that, for every $0<\varepsilon< 1/34$, it is $\mathrm{NP}$-hard to approximate either Ulam median or Ulam center within a factor of $35/34-\varepsilon$, even when the input consists of only four rankings. We further show that unless P = NP, neither problem admits a polynomial-time additive approximation scheme. Prior to our work, only the exact versions of both problems were known to be $\mathrm{NP}$-hard, and that too only when the number of input rankings is unbounded [Fischer et al., ESA'25 and Bachmaier et al., J. of Discrete Algorithms'15]. Furthermore, our inapproximability results are optimal in terms of the number of input rankings since for three inputs it is already known to be polynomial-time solvable [Chakraborty, Das, Krauthgamer, SODA'21]. En route, we introduce a new general framework for reducing Boolean constraint satisfaction problems (CSP) to the Ulam median with only four inputs. As a specific instantiation of the reduction framework, we obtain our hardness-of-approximation results. The corresponding hardness for the Ulam center follows from a reduction from the Ulam median.
A central task in knowledge compilation is to compile a CNF-SAT instance into a succinct representation format that allows efficient operations such as testing satisfiability, counting, or enumerating all solutions. Useful representation formats studied in this area range from ordered binary decision diagrams (OBDDs) to circuits in decomposable negation normal form (DNNFs). While it is known that there exist CNF formulas that require exponential size representations, the situation is less well studied for other types of constraints than Boolean disjunctive clauses. The constraint satisfaction problem (CSP) is a powerful framework that generalizes CNF-SAT by allowing arbitrary sets of constraints over any finite domain. The main goal of our work is to understand for which type of constraints (also called the constraint language) it is possible to efficiently compute representations of polynomial size. We answer this question completely and prove two tight characterizations of efficiently compilable constraint languages, depending on whether target format is structured. We first identify the combinatorial property of ``strong blockwise decomposability'' and show that if a constraint language has this property, we can compute DNNF representations of linear size. For all other constraint languages we construct families of CSP-instances that provably require DNNFs of exponential size. For a subclass of ``strong uniformly blockwise decomposable'' constraint languages we obtain a similar dichotomy for structured DNNFs. In fact, strong (uniform) blockwise decomposability even allows efficient compilation into multi-valued analogs of OBDDs and FBDDs, respectively. Thus, we get complete characterizations for all knowledge compilation classes between O(B)DDs and DNNFs.
We describe a Lean~4 formalization revisiting NYU Courant Technical Report TR1995-711 on the average-case complexity of Multilevel Syllogistic (MLS). The development encodes Reischuk--Schindelhauer average-case classes, an axiomatic MLS/EMLS semantics layer, a partial Ferro--Omodeo--Schwartz decision procedure with proved soundness and partial completeness on a membership-free fragment, serialization and step budgets, and conditional NP-average completeness and non-AvP hardness corollaries modulo explicitly documented structural axioms. Full Lean sources are inlined in the appendix modules.
We prove lower bounds for $k$-OV, $k$-XOR, and $k$-SUM in nonuniform $\mathrm{AC}^0$, tracking how the circuit-size exponent scales with $k$ and using no running-time hypothesis. Our framework gives depth-zero projections from colored subgraph isomorphism to the three targets at dimension, row count, or bit width $O(k \log n)$, without increasing depth or size, and preserving gate orientation. For every fixed depth and sufficiently large fixed $k$, we obtain unconditional bounds $n^{Ω(k)}$ for $k$-OV and $(n/k)^{Ω(k)}$ for $k$-XOR and $k$-SUM, with an absolute exponent-rate constant independent of both $k$ and the depth. For growing $k = n^{o(1)}$ and every fixed depth $d$, we obtain the unconditional floor $n^{Ω_d(\min\{\sqrt{k},\log n\})}$. This strengthens to $n^{Ω(k)}$ at depth two for both top-gate orientations, and at depth three for top-disjunction (OR-AND-OR) circuits, with no restriction on fan-in or polarity. The depth-three argument rests on a minterm bound for a single CNF: a fixed CNF is very unlikely to become true for the first time exactly when a randomly planted copy is completed. Assuming a pattern-uniform strengthening of the Li-Razborov-Rossman source lower bound, the same projections complete the $k = n^{o(1)}$ frontier with $n^{Ω_d(k)}$ for the missing top-conjunction depth-three orientation and for every fixed depth $d \geq 4$. The framework is modular in the source bound, so improved source bounds pass directly to all three targets. All direct $k$-XOR bounds concern odd $k$; a black-box lift covers even $k$, and the $k$-SUM projection works for both parities. At the bit width $m = Θ(k \log(\mathrm{e}n/k))$ used by our projection, a block-carry $Σ_3$ upper bound of size $(n/k)^{O(k)}$ matches the depth-three lower bound up to constants in the exponent. Gaps remain at depth two and for top-conjunction depth three.
The $(\min,+)$ convolution is a central problem in fine-grained complexity, and it remains open whether it can be computed in truly subquadratic time. We study it through tropical polynomials, where $(\min,+)$ convolution is exactly tropical polynomial multiplication. We introduce the tropical decomposition width, $\operatorname{tdw}(A)$, which measures how finely a tropical polynomial can be decomposed into factors of small degree. We prove two modular convexity theorems showing that bounded tropical decomposition width forces convexity on arithmetic progression subpolynomials. This yields deterministic algorithms for computing $a\otimes b$ in $$O\left(n\max(\operatorname{tdw}(a),\operatorname{tdw}(b))^2\right)$$ when $\max(\operatorname{tdw}(a),\operatorname{tdw}(b))$ is given, and in $$O\left(ne^{\min(\operatorname{tdw}(a),\operatorname{tdw}(b))(1+o(1))}\right)$$ without prior knowledge of the width. Neither algorithm requires a decomposition of the input sequences. The same structural ideas give a randomized algorithm for Multiple-Sequence $(\min,+)$ Convolution: given $k$ sequences of length at most $n$, their convolution can be computed in $$O\left(kn^2\sqrt{\min(k,n)}\log^{1.5}(kn)\right)$$ time, improving the natural $O(k^2n^2)$ bound. Finally, we introduce interpolation algebras for tropical polynomials and show that classes with bounded tropical decomposition width admit interpolation algebras of finite generating rank, whereas distinguishing all tropical polynomials of degree at most $n$ requires generating rank $\lfloor n/2\rfloor+1$. We also prove that tropical decomposition width cannot decrease under any flat $\mathbb T$-algebra extension. Together, these results connect the tractability of $(\min,+)$ convolution with structural rigidity in tropical polynomial multiplication.
In this work, we present the first analysis of low-degree polynomial threshold functions for the natural hypothesis testing problem of detecting the noisy random lift of a base $d$-regular graph from a uniformly random $d$-regular graph. Along the way, we obtain a new result for the distribution of short cycle counts in noisy random lift up to logarithmic lengths, which generalizes results by McKay, Wormald, and Wysocka and by Johnson in the case of random regular graphs, and results by Greenhill, Janson, and Ruciński and by Fortin and Rudinsky in the case of random lifts.
We study exact single-source shortest paths when the output is only the materialized labeled distance vector ($\mathrm{DIST}$), rather than a distance order. In the full deterministic comparison-addition model, the minimum worst-case number of additions on every fixed directed topology is exactly the maximum number $ρ_{\mathrm{fwd}}$ of forward nonsource endpoint classes over rooted vertex orders; the lower bound permits adaptive control, literals, and arbitrary mixed sums. This arithmetic law aligns with the comparison optimum on DAGs, where the full resource region is an exact rectangle. Cycles destroy that alignment: a two-spoke shared-hub graph has coordinatewise optima $(4,2)$ but requires five comparisons at the two-addition budget. Its $k$-spoke extension forces $k\log_2 k+O(k)$ comparisons at the addition optimum and has an entropy-tight deterministic tradeoff $C_{k+r}^*(H_k)=Θ(k+Λ_{k,r})$, where $Λ_{k,r}=\log_2(k!/[r!(r+1)^{k-r}])$, with leading constant one when $Λ_{k,r}/k\to\infty$. Because the two coordinatewise minima need not belong to one program, these conflicts lead to the same-program benchmark $\operatorname{OPT}_{\mathrm{DIST}}=\inf_A\sup_w(C_A(w)+P_A(w))$. An exact transcript-cone game yields one uniform interpreter whose charged addition-comparison cost equals $\operatorname{OPT}_{\mathrm{DIST}}$ on every topology; its optimal actions are synthesizable in polynomial space but may require exponential time. Finally, an active-core reduction and the current deterministic directed-SSSP bound give an efficient uniform $O\!\bigl(\operatorname{OPT}_{\mathrm{DIST}}\sqrt{\log(2+\operatorname{OPT}_{\mathrm{DIST}})\log\log(4+\operatorname{OPT}_{\mathrm{DIST}})}\bigr)$ charged-operation bound. Thus optimal numerical policies exist uniformly, while efficient constant-competitive navigation remains open.
The widespread adoption of artificial intelligence (AI) within real-world applications has raised a lot of concerns regarding their trustworthiness, especially in critical applications. The field of eXplainable AI (XAI) has emerged with the objective of providing explanations to the users about the decisions made by AI systems. Several explanations for boolean classifiers have been introduced in the literature, including abductive and contrastive explanations, each giving a different insight on the decision of the classifier. However, computing an explanation for a decision of a boolean classifier is a hard problem in general. One way to deal with this complexity is to rely on a compiled representation of the classifier for which each explanation can be computed efficiently. Unfortunately, we prove in this paper that several classes of abductive explanations, remain hard to compute even for Ordered Binary Decision Diagrams, one of the most tractable subsets of the knowledge compilation map. Included in such classes are shorter abductive explanations or abductive explanations that include the explainee's preferences. To recover the benefits of working with compiled representations, we show that a proper representation of the dual-rail encoding of the classifier can be used to compute efficiently these classes of explanations.
Algebraic combinatorics often seeks bijections that explain identities between distributions object by object. Encoding combinatorial objects as words lets automata theory study such a bijection as a word-to-word computation and measure its memory, input access, and control of output order. This refines existence questions by asking which computational mechanisms a bijection requires. We develop this viewpoint for Dyck paths. Our motivating example is the $q,t$-Catalan polynomial. Let $D_n$ be the set of Dyck paths of semilength $n$, let $D=\bigcup_{n\ge 0}D_n$, and let $area, dinv, bounce \colon D\to\mathbb{N}$ be the standard statistics. Then, \[ C_n(q,t)=\sum_{P\in D_n}q^{area(P)}t^{bounce(P)} =\sum_{P\in D_n}q^{dinv(P)}t^{area(P)}. \] Haglund's zeta map $ζ\colon D\to D$ gives a bijective proof: it preserves semilength and sends $(dinv,area)$ to $(area,bounce)$. By contrast, the full symmetry $C_n(q,t)=C_n(t,q)$ still lacks a direct explanation: no explicit, uniform, semilength-preserving bijection is known that swaps area and dinv on every Dyck path. Polyregular maps from automata theory provide a natural computational starting point, but we prove that neither $ζ$ nor the classical height-sweep bijection witnessing Narayana symmetry is polyregular. The missing mechanism is global ordering by numerical levels whose range grows with the input. We call this a \emph{rank sort} and introduce \emph{weighted-rank polyregular maps} (WRP), extending polyregular maps by one such sort and containing both bijections. Nevertheless, WRP is a proper subclass of deterministic logspace. We prove that $ζ^{-1}$ lies outside WRP and that no WRP map can realise a semilength-preserving area-dinv swap. Thus the rank-sorting strategy behind $ζ$ cannot be extended within WRP to exchange the two statistics.
We consider the problem of establishing limitations on the power of quantum query algorithms via the completely bounded polynomial method. In particular, we prove several optimal functional inequalities involving different notions of completely bounded polynomials. These inequalities lead to limiting theorems for the power of quantum query algorithms that improve on prior works. 1. An optimal root-influence bound for block-multilinear polynomials. Prior work showed that block-multilinear polynomials $p$ of degree $t$ satisfy a root-influence bound, $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t^2$, which is stronger than the bound appearing in the Aaronson-Ambainis conjecture. We find the optimal constant in that inequality: $\|p\|_{\text{cb}}\geq \sum_i \sqrt{\mathrm{Inf}_i[p]}/t$. Since the amplitudes of quantum algorithms that query disjoint blocks of inputs-such as $t$-fold forrelation- are block-multilinear polynomials with $\|p\|_{\text{cb}}\leq 1,$ our inequality shows that they satisfy $t\geq \sum_i\sqrt{\mathrm{Inf}_i[p]}$. We prove that this inequality yields both a more efficient classical simulation than prior results based on the Aaronson-Ambainis argument, and a qualitative improvement: all classical queries are nonadaptive. 2. Optimal Fourier growth of the highest level of quantum query algorithms. We show that for every polynomial $p$ defined on $\{-1,1\}^n$ of degree $2t$, the Fourier Growth at the level $2t,$ namely $\|\widehat p_{2t}\|_{\ell_1}$, satisfies $\|\widehat p_{2t}\|_{\ell_1}\leq (en/(2t-1))^{\frac{2t-1}{2}}\|p\|_{\text{cb}}$. This is optimal up to the factor $e$, as witnessed by $2t$-fold forrelation. As quantum query algorithms that make $t$ queries (to the whole input) satisfy $\|p\|_{\text{cb}}\leq 1$, this yields a Fourier growth bound for these algorithms, partially resolving a question by Girish (STOC, 2026).
The \emph{Stanley--Yan} (SY) \emph{inequality} gives the ultra-log-concavity for the numbers of bases of a matroid which have given sizes of intersections with $k$ fixed disjoint sets. The inequality was proved by Stanley (1981) for regular matroids, and by Yan (2023) in full generality. In the original paper, Stanley asked for equality conditions of the SY~inequality, and proved total equality conditions for regular matroids in the case $k=0$. In this paper, we completely resolve Stanley's problem. First, we obtain an explicit description of the equality cases of the SY inequality for $k=0$, extending Stanley's results to general matroids and removing the ``total equality'' assumption. Second, for $k\ge 1$, we prove that the equality cases of the SY inequality cannot be described in a sense that they are not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level.
One of the fundamental open problems in the field of tensors is the border Comon's conjecture: given a symmetric tensor $F\in(\mathbb{C}^n)^{\otimes d}$ for $d\geq 3$, its border and symmetric border ranks are equal. In this paper, we prove the conjecture for large classes of concise tensors in $(\mathbb{C}^n)^{\otimes d}$ of border rank $n$, i.e., tensors of minimal border rank. These families include all tame tensors and all tensors whenever $n\leq d+1$. Our technical tools are border apolarity and border varieties of sums of powers.
Neural networks with ReLU activations are a widely used model in machine learning. It is thus important to have a profound understanding of the properties of the functions computed by such networks. Recently, there has been increasing interest in the (parameterized) computational complexity of determining these properties. In this work, we close several gaps and resolve an open problem posed by Froese et al. [COLT '25] regarding the parameterized complexity of various problems related to network verification. In particular, we prove that, for all $\ell\ge 2$, deciding positivity (and thus surjectivity) of a function $f:\mathbb{R}^d\to\mathbb{R}$ computed by an $\ell$-layer ReLU network is W[$\ell-1$]-hard when parameterized by the input dimension $d$. The case $\ell=2$ implies that zonotope non-containment (a problem that is of independent interest in computational geometry, control theory, and robotics) is W[1]-hard with respect to the ambient dimension $d$. Moreover, we show that approximating the maximum within any multiplicative factor and computing the $L_p$-Lipschitz constant for $p\in(0,\infty]$ in $\ell$-layer networks is NP-hard and W[$\ell-1$]-hard with respect to $d$. For $\ell\ge 3$, approximating the $L_p$-Lipschitz constant is NP- and W[$\ell-2$]-hard. We further show that the above problems are NP- and W[$t$]-hard (for all $t\ge 1$) with respect to $\ell$ for constant $d$. Notably, our hardness results imply that the naive enumeration-based methods for these fundamental problems running in $n^{(\ell-1) d}\cdot\operatorname{poly}(N)$ time are all essentially optimal under the Exponential Time Hypothesis.
Let $R_1 := \begin{pmatrix}1&1\\1&0\end{pmatrix}$, and write $R_n := R_1^{\otimes n}$ for the $N \times N$ disjointness matrix, where $N = 2^n$. We construct new depth-2 linear circuits for $R_n$: one of size $O(N^{1.2449})$ and one of maximum input-output degree $O(N^{0.3199})$, improving upon the results of Alman and Li [AL25] (FOCS 2025), who achieved size $O(N^{1.2495})$ and degree $O(N^{1/3})$. In this paper, we develop a number of mechanisms for designing depth-2 linear circuits that compute linear transforms represented by Kronecker powers of a general base matrix. For $R_1$ in particular, we obtain our refined bounds by expressing a certain recursive circuit construction as a random matrix product and analyzing its quenched Lyapunov exponent, a quantity studied in the theory of random dynamical systems. Our new upper bounds imply faster deterministic algorithms for #OV, more efficient online data structures for the Subset Query and Partial Match primitives, and smaller low-depth linear circuits for both disjointness and a class of related non-Kronecker-power matrices.
We study the computational complexity of solving promise systems of equations over finite algebras. Given two algebras $\mathbf{A}$ and $\mathbf{B}$ with a homomorphism from $\mathbf{A}$ to $\mathbf{B}$, the promise system of equations problem is to determine if an input system of equations has a solution in $\mathbf{A}$ or not even in $\mathbf{B}$. We generalize the results of Larrauri, Mottet, and Živný [ACM ToCL'26] to obtain a $\mathbf{P}-\mathbf{NP}$-hard dichotomy result for promise systems of equations over a class of algebras which contains all monoids, and a dichotomy result for promise systems of equations over algebras in a congruence modular variety. We then consider the metaproblem for promise systems of equations over algebras in a congruence modular variety: given finite algebras $\mathbf{A}$ and $\mathbf{B}$ such that $\mathbf{A}$ is in a congruence modular variety, we show there is a quasi-polynomial time algorithm for determining whether or not the associated promise system of equations problem is in $\mathbf{P}$.
The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $Ω(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $Ω(k)$ and $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.
What can a single layer of self-attention compute? We study head complexity: the minimum number of attention heads required to compute a function in a one-layer attention-only model. We establish an exact hierarchy under this measure: $k$ heads compute $k$-bit parity but cannot compute $(k+1)$-bit parity. The lower bound is unconditional in the two resources a transformer might otherwise exploit; it holds at unbounded embedding dimension and unbounded numerical precision. The proof rests on an alternating-sum obstruction: after clearing the softmax denominators, every monomial in the resulting decision polynomial omits at least one of the $k+1$ input bits, forcing its correlation with parity to vanish. The same obstruction yields lower bounds for related tasks, including the well-studied multi-hop induction-head task. We also establish compactness bounds for embedding dimension and numerical precision. Specifically, a compactness theorem shows that any function computable at all can be computed with embedding dimension and precision bounded by the discrete data of the task, namely, head count, alphabet size, and length. Thus, potentially unbounded dimension or precision provably cannot substitute for heads. Finally, we derive nearly matching universal bounds for general binary functions: $2^n$ heads suffice to compute every $n$-bit binary function, with one head per monomial in its multilinear expansion, while a counting argument shows almost all such functions require $Ω(2^n/n^2)$ heads. This lower bound matches the upper bound to within a $\operatorname{poly}(n)$ factor, even when dimension and precision are unbounded. Together, these results characterize head requirements for Boolean computation in this model.