Papers

We study the number of exact matrix-vector queries needed to sample a centered $d$-dimensional Gaussian with condition number at most $\kappa$ to total variation error $\varepsilon$. The best known algorithm uses $O(\min\{d,\sqrt{\kappa}\log(\frac{d}{\varepsilon})\})$ queries, while the known lower bounds do not grow as $\varepsilon\to0$ and match this bound only for $\kappa\le d^{1/5-\delta}$ at constant accuracy. We introduce a construction based on a hidden determinant sign and prove a lower bound of $\Omega\!\left(\min\left\{d,\sqrt{\kappa}\log\left(1+\frac{d}{\sqrt{\kappa}\,\varepsilon}\right)\right\}\right)$ uniformly for all $d\ge1$, $\kappa\ge4$, and $0<\varepsilon\le\tfrac{1}{16}$. This bound matches the known upper bound up to constant factors whenever $\kappa\le d^{2-\delta}$ for fixed $\delta\in(0,2)$, or $\varepsilon\le d^{-c}$ for fixed $c>0$. We also extend a recent fixed-spectrum construction with a hidden eigenbasis from $\kappa=2$ to $2\le\kappa\le d^{1-\delta}$, for any fixed $\delta\in(0,1)$. This gives an $\Omega_\delta(\sqrt{\kappa}\log d)$ lower bound at constant accuracy that holds even when the spectrum of the precision matrix is known to the algorithm.

While sensitivity analysis for linear programs with parametric right- hand side vectors is well established, cases involving parametric constraint matrices remain less explored. To address this gap, we investigate linear programs where the constraint matrix depends linearly on a scalar parameter. Our goal is to compute the minimum and maximum optimal values of these parametric linear programs within a fixed parameter range. This corresponds to solving two bilevel programs. We show that both decision versions of these optimization problems are NP-complete by proving both NP-hardness and membership in NP. We additionally identify three sufficient conditions on the parametric matrix under which the bilevel programs are no longer NP-hard. We introduce a reformulation for both problems, together with a Farkas-based certificate for the lower-level feasibility condition, and compare the execution time for solving them on standard benchmark instances. The experiments highlight a practical dichotomy. Although both bilevel optimizations are NP-hard, computing the parameter value corresponding to the minimum of the linear program is fast, whereas computing the value corresponding to the maximum is not. We relate this asymmetry to the parametric equality constraints, whose free dual variables leave the relaxation of the maximum reformulation unbounded.

Symmetry of Information (SoI) [ZL70] is a fundamental theorem of classical Kolmogorov complexity whose status for time-bounded computation is closely related to central open problems in complexity theory and cryptography. In a recent development, [HMO26] showed that SoI fails for $rKt$ complexity, namely in the setting of randomized computation. By contrast, many basic properties of Kolmogorov complexity are known to remain valid in the presence of nondeterminism [HLO26], suggesting that SoI may provide a fruitful structural principle for exploring the relative power of randomness and nondeterminism in computation. Our results reveal that the validity of SoI for nondeterministic computation, namely for $nKt$ complexity, is a subtle question connected to several frontiers. We show that symmetry of information holds on average for randomized extensions of $nKt$, while its worst case validity would imply breakthrough complexity separations. Although the standard cryptographic argument for asymmetry does not extend to the nondeterministic setting, we provide evidence that information remains asymmetric in nondeterministic computations. Among other results, we show that SoI would imply a collapse of exponential time classes to low levels of the polynomial hierarchy. Then, building on the work of [Hir22b, GK22, KK26, HMO26], we develop a general framework connecting symmetry of information, meta-complexity, and explicit constructions in the nondeterministic setting. More precisely, we prove that suitable formulations of these three notions are quantitatively equivalent. We apply different aspects of this correspondence to make progress on complexity lower bounds and on the failure of SoI for nondeterministic computation. These consequences are unconditional and provide some of the strongest known lower bounds in meta-complexity.

Network coding allows intermediate nodes to combine messages, whereas routing only forwards them. The Li-Li conjecture asserts that network coding and fractional routing have the same capacity region in undirected multiple-unicast networks. We give a counterexample to this conjecture and apply it to external-memory integer sorting. Since Aggarwal and Vitter posed the question in 1988, whether integer sorting can beat the external-memory comparison bound has remained a central and extensively studied open problem. We give a deterministic sorting algorithm: for $N$ integers of $\Theta(\log N)$ bits, a memory holding $M$ integers, and block transfers of $B$ integers, where $2B\le M\le N$, the number of block transfers is $O\left(\frac{N}{B}\left[1+\frac{\log N}{(\log(M/B))^{1-\varepsilon}(\log M)^\varepsilon}\right]\right)$, where $\varepsilon>0$ is an absolute constant. In a range of parameter regimes, this bound is asymptotically smaller than the classical comparison-sorting bound. We also use Lean 4 to verify the sorting algorithm's correctness and complexity bounds, and the strict advantage of coding over routing.

Greibach and Ginsburg (1969) studied abstract families of languages or AFLs. A full AFL is a class of languages closed under the following operations: homomorphism, inverse homomorphism, intersection with regular languages, union, concatenation, and Kleene star. Greibach (1970) used a restricted substitution, called nested iterated substitution, to introduce the notion of super AFLs, which are full AFLs that are further closed under nested iterated substitution. The smallest super AFL was shown to be the class of all context-free languages. From this fact, we focus on Greibach's limited, univariable application of nested iterated substitution and further generalize it into k-variate. This introduces the notion of nested iterated $k$-variate substitution whose definition does not rely on "machines". Recently, a "two-tape" version of Hibbard's (1967) rewriting systems, which were later reformulated by Pighizzini and Pisoni (2014) as k-limited automata with "single" input/work tapes, has been studied under the name of depth-k storage automata. This work unearths a few characteristics of the languages recognized by those automata under a restriction known as frozen blank sensitivity in terms of full AFLs and nested iterated k-variate substitution. This will promote a further abstract treatment of those languages.

Quantifying the diversity of a set of objects is a fundamental problem in machine learning and statistics. Mironov and Prokhorenkova (2025) recently showed that three axioms-monotonicity, uniqueness, and continuity-are crucial for a reliable diversity measure and that none of the standard metrics in the literature satisfy all three. They also constructed two measures that do satisfy the axioms, but both are NP-hard to compute, leaving open the question of whether efficient axiomatic diversity measures can exist. We study a Euclidean-restricted version of this question. We introduce Geometric Kernel Diversity (GKD), which associates a radial kernel solid with each point and measures the volume of their union. We show that canonical GKD satisfies all three Mironov-Prokhorenkova axioms on Euclidean-realizable inputs. For canonical GKD, we give exact polynomial-time algorithms in dimensions $d \le 3$. In arbitrary dimensions, we give a randomized algorithm that returns a $(1 \pm \varepsilon)$-approximation with probability at least $1-\delta$ in time $O \left(\frac{nd}{\varepsilon^2}\log\frac{2}{\delta}\right)$ for $n$ points. The axiomatic guarantee applies to exact canonical GKD; the randomized algorithm approximates this deterministic quantity. We further show that the two diversity measures of Mironov and Prokhorenkova (2025) remain NP-hard on Euclidean-realizable inputs and, unless $\mathrm{NP} = \mathrm{RP}$, admit no FPRAS even on this restricted domain.

Parikh's famous theorem relates to the semi-linearity of context-free languages. We seek for expanding the scope of the semi-linear property from context-free languages to languages beyond them. Of those languages, this work focuses on the languages that are particularly recognized by one-way nondeterministic depth-k storage automata under a natural restriction of frozen blank sensitivity. Those languages (which are called $k$-storable languages) have been already proven to satisfy a natural generalization of the well-known pumping lemmas for regular and context-free languages. The semi-linearity of k-storable languages generalizes Parikh's theorem on the semi-linearity of context-free languages. As for bounded languages, Ginsburg and Spanier's well-known theorem characterizes context-free languages in terms of stratified semi-linearity. This notion is further generalized and is used to characterize the k-storable languages as well. Such a generalization is interesting in its own right. This helps us take an abstract treatment of k-storable languages without relying on "machines". Our results strengthen the belief that those languages are "natural" generalizations of context-free languages.

Poset games form a fundamental class of combinatorial games that includes classical games such as Nim and Chomp. Determining the outcome of a poset game is known to be PSPACE-complete even for posets of height at most three. However, the exact computational complexity of Two-Level Poset Game, played on posets of height at most two, has remained an open problem for over a decade. We resolve this problem by proving that Two-Level Poset Game is PSPACE-complete via a polynomial-time reduction from Avoid True on positive CNF formulas.

Liu introduced generalized Fibonacci gates on domains of size 3 and 4, proved that Holant problems built from them are tractable by a cut-and-merge algorithm, and conjectured that Fibonacci gates with matching polynomial-time algorithms exist on all higher domains. We extend the definition uniformly to every domain size $k$ and show that a family of domain-$k$ generalized Fibonacci gates is the same thing as a $k$-dimensional commutative Frobenius (equivalently, Artinian Gorenstein) algebra $A$ equipped with an orthonormal basis containing $1$. Under this dictionary Liu's parameter constraints are exactly associativity of $A$, a gate is an element $w\in A$, merging two gates is multiplication in $A$, and the Holant value of a connected graph $G$ is $\tau\bigl(h^{\,|E|-|V|+1}\prod_v w_v\bigr)$ for a fixed “handle” element $h\in A$. This proves Liu's conjecture for every $k$, with an algorithm whose cost is polynomial in $k$ as well as in the graph. We also give a closure criterion for linear recurrences of any depth (a balance condition implies closure, and is necessary when the unit is a combination of the colours), and we map the landscape: semisimple algebras are exactly Liu's orthogonal frames, smoothable algebras with nilpotents are collided frames, and families whose algebra is not smoothable (“frameless” families) exist exactly when $k\ge 14$. Finally we show that every non-semisimple part of a family is invisible to the Holant on connected graphs with two or more independent cycles. This note states the results and sketches the proofs; the proofs of the main theorems are short and are given essentially in full.

We consider the problem of counting solutions to integer linear programming problems of the form $\{x \in \mathbb{Z}^n_{\geq 0} : Ax = b\}$ where $A \in \{-\Delta,\ldots,\Delta\}^{m \times n}$ and $b \in \mathbb{Z}^m$. We present an algorithm that counts solutions with $(n\Delta)^{m + o(1)}\! \cdot \log\lVert b\rVert$ arithmetic operations. This simplifies the algorithm of Gribanov and Malyshev, who considered the problem for the more general setting in which $A$ is $\Delta$-modular.

Bhrushundi et al. (2019) introduced torus polynomial approximation as an approach to separating Majority from $ACC^0$. In this model, approximation is measured modulo integers. Krishan and Vishwanathan (2026) developed a duality framework for lower bounds and posed five open problems. We resolve two of these, concerning integer lifts and explicit polynomial approximations. An integer lift records the integer shifts that turn a torus approximation into an ordinary real approximation. We show that a degree-d torus polynomial approximating Majority with error at most 1/8 and lift contained in an interval of width $B$ must satisfy $d(B + 1) = \Omega(n)$. This rules out low-degree approximating polynomials with lifts in {-1, 0, 1}, answering their Open Problem 2. We also answer Open Problem 5 by giving, for every Boolean function on $n \geq 1$ variables, an explicit torus polynomial approximation of degree at most $n - 1$. Its error is optimal: zero when the accepting set has even cardinality and $2^{-n-1}$ otherwise. The results presented were discovered and formalized in Lean using GPT-6 Astra.

Understanding the average-case complexity of $\mathsf{NP}$ is a central goal in complexity theory and cryptography. A necessary milestone is to understand the corresponding average-case versus worst-case question for the polynomial hierarchy ($\mathsf{PH}$). In a celebrated result, Hirahara [Hir21] proved that if every distributional problem in $\mathsf{PH}$ is easy on average, then every language in $\mathsf{PH}$ admits worst-case algorithms running in time $2^{O(n/\log n)}$. Remarkably, although the statement of Hirahara's theorem makes no reference to meta-complexity, all known proofs appear to use time-bounded Kolmogorov complexity and its associated meta-computational problems in an essential way. We show that the reliance on techniques and perspectives from meta-complexity is not simply a coincidence. Building on previous work of Hirahara [Hir20] and on a recent result of Hu, Lu, and Oliveira [HLO25], we prove that relating the average-case and worst-case complexities of $\mathsf{PH}$ is in fact equivalent to a purely meta-computational implication: efficient algorithms for a standard gap version of $\mathsf{MINpKT}$ with $\mathsf{PH}$ oracle access imply efficient algorithms for a mild gap version. Crucially, in these promise problems the standard gap separates a YES instance with time bound $t$ from a NO instance with time bound $\mathsf{poly}(t)$, whereas the mild gap keeps the YES threshold $t$ and moves the NO threshold to $t+\mathsf{poly}(n)$. Together with prior work, this equivalence points to the study of gap reductions as a potentially promising route toward understanding the relation between average-case and worst-case complexity. We make progress by proving that the gap in running time can be eliminated in the average-case setting. Finally, we prove that in the worst-case setting any efficient reduction between the two problems must use nonrelativizing techniques.

We study the randomised and deterministic demand-query complexity of approximate bipartite matching with few query rounds. The demand-query model, introduced in [Blumrosen and Nisan SICOMP 2009; Nisan SODA 2021], is inspired by combinatorial auctions and welfare maximisation, where an oracle returns the first neighbouring vertex (item) of a vertex u (agent) according to a given permutation pi (preference ordering of the agent over the items). For randomised algorithms, our main result gives a trade-off between the number of rounds k, the per-round query budget q=n^{1+\delta}, and the achievable approximation ratio. Let 1 <= k <= log n, and -1 <= delta <= 1/k. Every k-round randomised algorithm using O(q) demand queries per round has a worst-case expected-size approximation factor of Omega(n^{1/2k-delta/2}/k). Conversely, we give a randomised algorithm achieving approximation factor O(n^{(1-k*delta)/(k+1)}/k) in expectation. For deterministic algorithms, for k*q >= n, we show that the optimal deterministic approximation factor is ~Theta(n^2/(q*k^2)), thereby improving the previous lower bound of [Nisan SODA 2021] by a polynomial factor over a wide range of parameters. In particular, for q=Theta(n), we improve the deterministic round-complexity lower bound for constant-factor approximation from ~Omega(n^{1/3}) to ~Omega(sqrt n). Our algorithms produce matching and vertex-cover certificates; the deterministic trade-off also extends to minimum vertex cover and maximum fractional matching. We extend the randomised upper bounds to the weaker Demand-OR model [Nisan 2021]. We prove a non-adaptive simulation of uniform random-neighbour queries using known techniques, avoiding the O(log n)-round penalty of the trivial binary-search simulation. Consequently, our upper bounds yield the same k-round approximation guarantees using O(q * log^2 n) Demand-OR queries per round.

For Vertex Cover, a minimal blocking set of a graph $G$ is an inclusion-wise minimal vertex set contained in no minimum vertex cover of $G$. Bougeret et al. [SIDMA 2022] showed that, for all minor-closed graph classes $\mathcal{G}$, Vertex Cover has a polynomial kernel parameterized by the distance to $\mathcal{G}$ if and only if $\mathcal{G}$ has bounded minimal blocking sets. In this work, we explore how far this connection between minimal blocking sets and kernelization extends to $\mathcal{F}$-Minor Deletion parameterized by the deletion distance to $\mathcal{G}$. A minimal $\mathcal{F}$-blocking set of $G$ is a minimal vertex set not contained in any optimal $\mathcal{F}$-Minor Deletion solution of $G$. For connected families $\mathcal{F}$, we show that if some graph class $\mathcal{G}$ closed under disjoint union has unbounded minimal $\mathcal{F}$-blocking sets, then $\mathcal{F}$-Minor Deletion parameterized by the deletion distance to $\mathcal{G}$ has no polynomial kernel. This generalizes a result of Hols et al. [SIDMA 2022] for Vertex Cover. We then show that this criterion does not capture all kernelization lower bounds for $\mathcal{F}$-Minor Deletion, even for minor-closed graph classes. Motivated by the work of Dekker and Jansen [Discrete Appl. Math. 2024], we study the more general notion of extended $\mathcal{F}$-blocking sets and the kernelization lower bounds they yield. For every fixed integer $\eta\geq 1$, Pw-$\eta$ Deletion is the problem of deleting at most $k$ vertices to reduce the pathwidth to at most $\eta$. We show that it has no polynomial kernel parameterized by the distance to graphs with pathwidth at most $\eta+1$. Moreover, for connected families $\mathcal{F}$, we show that $\mathcal{F}$-Minor Deletion parameterized by the distance to graphs with pathwidth at most $\max_{F\in\mathcal{F}} pw(F)+1$ has no polynomial kernel.

Non-clashing teaching minimizes teaching complexity among models satisfying the Goldman–Mathias collusion-avoidance criterion. While the Vapnik–Chervonenkis dimension (VCdim) measures the complexity of learning from samples, non-clashing teaching dimension (NCTD) measures the number of examples required when a teacher can select them. Whether NCTD is always bounded above by VCdim remains a fundamental open question in learning theory. In this paper, we investigate this question for the concept classes of closed neighborhoods in planar graphs. We prove a tight upper bound of two on their NCTD, thereby establishing the conjectured inequality for this family. For the positive variant NCTD+, in which teaching sets contain only positive examples, we obtain a tight upper bound of four. Finally, we show that deciding whether either NCTD+ or NCTD is at most one is NP-complete, even for concept classes of closed neighborhoods in planar graphs. In particular, despite the upper bound of two on NCTD for such classes, distinguishing between NCTD values of one and two remains NP-hard.

We study the complexity of computing approximate first-order Nash equilibria in min-max optimization problems with one dimensional min and max players and product constraints. For black-box access to the objective, we prove that any algorithm requires $\Omega(1/\sqrt{\epsilon})$ queries to find an $\epsilon$-first-order Nash point. In the white-box setting, we establish CLS-hardness by reducing from End-of-Potential-Line. These results show that even the simplest min-max setting with one dimensional min and max players is at least as hard as CLS and therefore does not admit polynomial-time algorithms unless $\mathsf{CLS}\subseteq \mathsf{P}$.

We construct Clifford+$T$ circuits implementing arbitrary $n$-qubit unitaries with worst-case $T$-count $\widetilde{O}(N^{8/7})$ at constant operator-norm error, where $N=2^n$, improving the previous exponent $5/4$. The construction uses $\widetilde{O}(N^{6/7})$ clean ancillas. The main idea is to reduce the data addressed by each query to the block-encoding circuit for a sparse piece while controlling the additional simulation work needed to combine the pieces. We simulate a Hermitian logarithm of the target by flattening its traceless part with a two-layer random similarity transformation, randomly partitioning the transformed generator into block-sparse Hermitian pieces, and combining their evolutions using qDRIFT. For a target operator-norm error $\epsilon\in(0,1/2)$, let $L=\log_2(N/\epsilon)$, the qubit count plus the requested bits of accuracy. By successively correcting the approximation error, we obtain a circuit using $\widetilde{O}(N^{8/7}L^{3/7}+NL)$ $T$ gates and $\widetilde{O}(N)$ clean ancillas whenever $L\le N$. We also prove that, at constant precision, the exponent $8/7$ is optimal within a sequential block-SELECT model.

The Boolean Satisfiability (SAT) problem is a remarkably expressive computational problem, serving both as a cornerstone of complexity theory and as a unifying language for a wide range of search and reasoning tasks. While $\mathrm{NP}$-complete in general, SAT is well known to be linear-time solvable on instances with bounded treewidth. This motivates the search for "hidden" tree-like structures in formulas—for instance by detecting so-called weak $t$-treewidth backdoors. Such backdoors are sets of $k$ variables that can be assigned in such a way that the remaining instance has treewidth at most $t$ and is satisfiable whenever the original instance is. Ever since Gaspers and Szeider [ICALP 2012] showed that finding a weak $t$-treewidth backdoor for a SAT formula is $\mathrm{W}[2]$-hard with respect to $k$, tractability results have been restricted to structurally sparse formulas—most notably, a linear-time algorithm for $d$-SAT (bounded clause-length SAT) by Fomin, Lokshtanov, Misra, Ramanujan, and Saurabh [SODA 2015] and a non-uniform polynomial-time algorithm when the incidence graph of the input formula belongs to a nowhere-dense graph class (or more generally is $K_{d,d}$-free) by Lokshtanov, Panolan, and Ramanujan [ICALP 2022]. In this paper, we push the tractability boundaries of finding weak $t$-treewidth backdoors beyond sparsity and into the regime of "well-behaved" dense formulas. Aiming towards relaxing structural input restrictions from nowhere-dense to edge-stable graph classes, as featured in the theory of beyond sparsity for FO model checking, or to bounded VC-dimension set systems, we identify signed semi-ladder-freeness as the correct dense structural restriction that establishes the boundary between tractability and intractability along this line.

In the Hunters and Rabbit game, hunters shoot at a set of vertices at each round, while an invisible rabbit must move along an edge. We prove that deciding whether a given number of hunters can guarantee capture is PSPACE-complete, even on simple bipartite graphs. The reduction uses intersection nonemptiness for inverse finite automata: one common word must put all automata in accepting states simultaneously. A clique acting as a safe haven and two families of guards force every winning strategy to simulate common transitions. Existing transformations remove the restriction on the rabbit's initial positions and yield PSPACE-completeness for the directed game with one hunter, even on oriented graphs with positive minimum outdegree. Adjusting the guard sizes also proves PSPACE-completeness of the firefighting game, in which the rabbit may remain at its current vertex.

We consider the problem of computing the probability of existence of a source-to-target path or walk satisfying a given length constraint in an input probabilistic graph. The length constraints we consider can require the length to be at least $\ell$, at most $\ell$, or exactly $\ell$, for $\ell$ a target length given as input. The probabilistic graphs that we consider are directed graphs where each edge is annotated with an independent probability of existence. As this task is generally intractable (#P-hard), we aim for algorithms applying to input graphs that are assumed to be of bounded treewidth. We classify the complexity landscape of these problems: - We give an XP algorithm parameterised by treewidth to compute the probability that an input probabilistic graph contains a source-to-target path/walk of length $\geq \ell$, and do the same for a path/walk of length $\leq \ell$. - We show that the four problems mentioned in the last item are W[1]-hard parameterised by pathwidth (hence not FPT under the standard assumption that FPT$\neq$W[1]). This is in contrast to the case of non-probabilistic graphs, where the corresponding problems were known to be FPT parameterised by treewidth; and to the connectivity problem (and more generally MSO evaluation) on probabilistic graphs, where the same was known. - Surprisingly, we show that it is #P-hard to compute the probability that an input graph contains a source-to-target path/walk of length exactly $\ell$, even on graphs of pathwidth $2$. By contrast, all our problems are tractable on graphs of treewidth $1$. - Last, we investigate the complexity of our problems when the edges can carry weights. This causes all our problems to become #P-hard already on pathwidth-2 graphs, and causes specifically the “exactly $\ell$” walk problem to be #P-hard even on pathwidth-1 graphs.

For a fixed finite collection of graphs $\mathcal{F}$, the $\mathcal{F}$-Minor-Free Deletion problem asks, given a graph $G$ and an integer $k$, whether there is a set $S\subseteq V(G)$ with $|S|\leq k$ such that $G-S$ contains no graph of $\mathcal{F}$ as a minor. This generalizes Vertex Cover, Feedback Vertex Set, and Vertex Planarization, among many others. Baste, Sau, and Thilikos [JCSS 2020, SICOMP 2023] showed that, assuming the Exponential Time Hypothesis (ETH), there are collections $\mathcal{F}$ for which the problem cannot be solved in time $2^{o(\mathrm{tw}\log\mathrm{tw})}n^{O(1)}$ on $n$-vertex graphs of treewidth $\mathrm{tw}$, and gave a matching algorithm. We ask whether this super-exponential dependence persists under more restrictive structural parameters. First, we strengthen their lower bound. Assuming ETH, for every fixed $h\geq 6$, $\{P_h\}$-Minor-Free Deletion, where $P_h$ is the path on $h$ vertices, cannot be solved in time $2^{o(t\log t)}n^{O(1)}$, where $t=\mathrm{fvs}+\mathrm{td}$ is the sum of the feedback vertex set number and treedepth of the input graph. These two parameters are incomparable, but both individually upper bound the treewidth. Together with the $2^{O(\mathrm{tw}\log\mathrm{tw})}n^{O(1)}$-time algorithm of Baste, Sau, and Thilikos [SICOMP 2023], our lower bound is tight. In contrast, we show that asymptotically faster algorithms are possible when parameterized by the vertex cover number $\mathrm{vc}$. Specifically, for any finite family $\mathcal{F}$, $\mathcal{F}$-Minor-Free Deletion can be solved in time $2^{O(\mathrm{vc})}n^{O(1)}$. Taken together, our results show that the boundary between single-exponential and super-exponential complexity for this problem lies between the vertex cover number and the feedback vertex set number (plus treedepth).

We study the number of uniform quantum examples needed to exactly learn a Boolean function with at most $k$ nonzero Fourier coefficients. We give a learner with near-optimal $\widetilde{O}(k)$ sample complexity independent of the input length $n$. The learner uses only separate measurements of individual examples. This improves the previous $O(k^{3/2}\log^2 k)$ upper bound (Arunachalam, Chakraborty, Lee, Paraashar, and de Wolf, 2021). The main ingredient is a candidate set formed by symmetric differences of subsets from independent groups of Fourier samples. A multilinear majority polynomial combines unbiased random approximations into an accurate witness whose Fourier support lies in this set. Granularity then forces the set to contain the entire Fourier support, including subsets never directly sampled. For comparison, we derive the tight classical sample complexity $\Theta(nk)$ by combining known sparse-recovery and lower-bound results.

Rubinstein et al. introduce the study of connectivity and minimum cut problems in the cut query model, where an algorithm can only access the edges of a graph through CUT queries. In this work, we examine the query complexity of densest subgraph problems. Our first contribution is to generalize the uniform sparsification approach for Densest Subgraph and other heavy subgraph problems to weighted graphs, achieving the following results. >A $(1+\varepsilon)$-approximation for weighted Densest Subgraph in $O(n\log^2 n/\varepsilon^2)$ queries, roughly matching the $\Omega(n/\log n)$ lower bound we prove for randomized algorithms. >A $(1+\varepsilon)$-approximation for weighted Max Cut in $O(n\log^2 n/\varepsilon^2)$ queries. This improves upon the bound of $O(n\log^3 n(\log n+1/\varepsilon^2))$ queries achieved by Plevrakis et al., making significant progress towards their lower bound of $\Omega(n/\log n)$ queries. In contrast, we rule out approximation schemes with truly subquadratic query complexity for Densest $k$-Subgraph and settle its exact query complexity in both the unweighted and deterministic settings. For hypergraphs, we show CUT queries alone cannot achieve $o(\sqrt{n})$-approximation for Densest Subhypergraph or $o(n)$-approximation for Densest $k$-Subhypergraph, regardless of the number of queries. Augmenting CUT queries with the EDGE oracle of Chen et al., we overcome this barrier for Densest Subhypergraph and obtain a $(1+\varepsilon)$-approximation on rank-$r$ hypergraphs using $O(nr\log n/\varepsilon^2)$ queries. We also extend our Densest $k$-Subgraph hardness results to hypergraphs and prove exponential query lower bounds for exact Densest $k$-Subhypergraph. Lastly, confirming a conjecture of Chakrabarty and Liao, we show that simulating an ADD query using CUT and IS-EDGE$^?$ queries requires exponentially-many queries.

We improve the shared-leg separation construction in Proposition 3.1 of OpenAI, *An Upper Bound of $9/4$ for the Matrix Multiplication Exponent* (October 2, 2026). If $M$ tensor blocks have a common first space and independent matched sectors on their other two spaces, then $M+n-1$ copies of their matched sum degenerate to the full direct sum of the blocks, each tensored with an $n$-dimensional dot product. For $n=M$, this reduces the number of source copies from $5M$ to $2M-1$. The construction uses polynomial interpolation and truncated local substitution, with no roots of unity or factorial denominators. It works over every field containing enough distinct elements. We verify that the remaining spectral and discrete-growth arguments of the cited preprint are independent of characteristic. Over an algebraic closure, the replacement therefore gives $\omega\leq9/4$ in every characteristic.

We study block sensitivity and certificate complexity of rotation symmetric Boolean functions, with particular emphasis on monomial rotation symmetric Boolean functions. We first show that every odd-degree monomial rotation symmetric Boolean function on $n$ variables satisfies $s=bs=C=n$. We then determine sensitivity, block sensitivity, and certificate complexity for every quadratic monomial rotation symmetric Boolean function. For the quadratic function $Q_{n,k}$ generated by $x_0x_k$, we prove, outside the short-orbit case, that \[ C(Q_{n,k})= \begin{cases} n,&4\mid n/\gcd(n,k),\\ n-\gcd(n,k),&4\nmid n/\gcd(n,k). \end{cases} \] Moreover, \(bs(Q_{n,k})=C(Q_{n,k})\) for every quadratic MRS function, and outside the short-orbit case, strict separation \(s<bs=C\) occurs exactly when \[ \frac{n}{\gcd(n,k)}\equiv2\pmod4. \] Finally, for the consecutive quartic family \[ R_n(x)=\bigoplus_{i\in\mathbb Z_n}x_ix_{i+1}x_{i+2}x_{i+3}, \] we determine all three measures in every residue class modulo $6$, using the sensitivity formulas and, in one residue class, finite-state results established in our companion preprint~\cite{LiZhangRSBFSensitivity}. In particular, for $r=2,3$ and $6q+r\ge5$ we obtain strict separation, \[ \boxed{s(R_{6q+r})=6q<bs(R_{6q+r})=C(R_{6q+r})=6q+1,\qquad r=2,3.} \] Thus this family gives an infinite collection of rotation symmetric Boolean functions for which sensitivity is strictly smaller than block sensitivity, while block sensitivity and certificate complexity coincide.

Rotation symmetric Boolean functions are Boolean functions invariant under cyclic permutation of their variables and can be represented by binary necklaces. We express local sensitivity as a weighted bichromatic degree in the necklace quotient and derive an exact formula for the multiplicities between neighboring necklaces. For an odd prime number of variables, we obtain explicit sensitivity formulas and show that multiplicity corrections occur only on an asymptotically negligible family of necklaces. For monomial rotation symmetric Boolean functions, we prove that every odd-degree function has maximum sensitivity, determine the sensitivity of every quadratic function, and determine the sensitivity of the consecutive quartic family. We also determine exactly the sensitivity of cyclic run-threshold functions for arbitrary length and threshold.

We study the computational complexity of graph distortion distances, a family of graph metrics measuring how vertex distances differ between two graphs. Formally, these distances are defined in terms of standard matrix norms on the difference of distance matrices under an optimal vertex alignment. The resulting metrics have natural interpretations, corresponding to three ways of measuring the distortion of shortest-path distances: the maximum distortion among all vertex pairs (entrywise maximum norm), global aggregation (entrywise $p$-norms), and local aggregation (operator $p$-norms) of distortions. For maximum distortion, we show that approximation below a factor of two is NP-hard even for highly restricted trees. Due to the close connection to the Gromov-Hausdorff distance, we obtain a similar hardness result for the latter. In both cases, we show essentially optimal additive inapproximability. On the algorithmic side, we give a subcubic-time algorithm for optimal embeddings of connected graphs into paths, based on a structural decomposition along a diametral path. Additionally, we give a simple approximation algorithm when one input graph has a "relatively path-like" metric structure, i.e. has bounded pathlength. For local and global distortion, we derive various hardness results even for restricted inputs. Making use of a close connection between the global distortion distances and Gromov-Wasserstein distances, we give the first formal proof of NP-hardness of the latter, which has been claimed in the literature many times but has, to the best of our knowledge, never been correctly proven.

The central object in this work is the Boolean network, a function $f:\{0,1\}^V \to \{0,1\}^V$. Often identified with $f$ is its dynamics $D_f$, the digraph with exactly $\{x \to y : f(x)=y\}$ as its arcs. The interaction graph of $f$ is the digraph $G$ on $V$ which can be thought of as representing the dependencies of the components of $f$ on one another. On the other hand, model-checking is a major field of study in graph algorithms and complexity. We consider problems at the intersection of these paradigms: for a given property $\phi$, we say that a digraph $G$ is $\phi$-dynamic if $G$ is the interaction graph of some Boolean network $f$ such that $D_f \models \phi$. The corresponding decision problem $\phi$-DYNAMIC asks whether $G$ is $\phi$-dynamic. Problems of this form are well-studied in the Boolean networks community. For example, it is a well-known result that, letting $\phi$ be 'No fixed points' (i.e. $\nexists x:x\to x$), $\phi$-DYNAMIC is equivalent to the structural digraph problem NON-ACYCLIC, which is solvable in linear time. We establish, for the first time, properties $\phi$ for which $\phi$-DYNAMIC is computationally hard. These include the 'Unique fixed point' property $\exists!x:x\to x$, and a range of properties concerning the periodic points of $f$ (i.e. vertices of $D_f$ which lie on a cycle). More widely, we exhibit an infinite family of natural properties $\phi$ such that $\phi$-DYNAMIC is $\mathrm{NEXPTIME}$-complete, which we call PERIOD PIECE. As a special case, these include all properties of the form 'Unique cycle of length $k$' and 'Exactly $k$ fixed points' for any $k \ge 1$. On the other hand, we show that if the input digraph has bounded in-degree, then for those same problems $\phi$-DYNAMIC collapses down to $\mathrm{NP}^{\mathrm{NP}}$, the second level of the polynomial hierarchy, though it remains $\mathrm{NP}^{\mathrm{NP}}$-complete.

The idea that learning is closely tied to compression has long served as both a powerful principle and, at times, an informal analogy across machine learning and learning theory. Here we focus on settings in which that intuition can be made mathematically precise, surveying formal connections among learning algorithms, data compression, and computational complexity theory, with an emphasis on recent progress and current frontiers. We describe how techniques from complexity theory can guide the design of both supervised and unsupervised learning algorithms with provable guarantees, and conversely, how learning algorithms can yield new insights and results in complexity theory. A central theme is the role of data compression as a bridge between learning and complexity, especially through an area called meta-complexity, which studies the computational difficulty of compression problems and time-bounded variants of Kolmogorov complexity.

In the context of graph reconstruction studies, this paper addresses the problem defined as follows: given a sample of vertex-coloured graphs, each associated with a number, does there exist a connected graph containing exactly the given number of induced subgraphs isomorphic to each graph in the sample, such that every edge of the solution graph is included in at least one of these subgraphs. This paper first shows that this problem is $\Sigma$______-complete. The associated optimisation problem is to determine the minimum number of vertices in a solution graph, if one exists. For instances consisting of two paths, we prove NP-completeness in general and give a polynomial-time dynamic programming algorithm when one path is monochromatic.