arXane › Computer Science › Computer Science
Wednesday, 4 November 2026: 13 new papers, 15 cross-lists and 23 replacements in Computational Complexity (cs.CC), in arXiv's order, with full abstracts. The same papers as arXiv's own listing, set as a newspaper; every title links to arXiv.
Computing a Karush-Kuhn-Tucker (KKT) point of a quadratic polynomial over the unit hypercube is known to be $CLS$-complete, when the coefficients of the polynomial are encoded in binary. We prove that the problem remains $CLS$-complete even when the coefficients are encoded in unary. This is the first problem proven to remain $CLS$-hard when the input is given in unary.
We ask how hard it is to build a neural network when its parameter values are given in advance. An instance supplies a multiset of rational numbers, a target behavior, and a bound on the number of neurons, output included. The question is whether one can choose the graph of a ReLU network within the bound, and place the numbers on its edges and biases so that it meets the target. This separates the numbers from the architecture; permutation training instead fixes the hidden units and only rearranges the output weights. Matching two labeled examples exactly is already $\mathsf{NP}$-complete; we ask what changes when the target must hold on every input or on average. Agreeing with a Boolean circuit on every Boolean input is $\Sigma_2^{\mathsf P}$-complete, already for four neurons. Agreeing with a given ReLU network on every real input lies in $\mathsf{DP}$ for every fixed neuron bound and is $\mathsf{DP}$-complete for three neurons. Approximating a ReLU network on the unit cube within absolute error one everywhere is $\Sigma_2^{\mathsf P}$-complete, even with no hidden neurons and all numbers equal and positive. Deciding whether some network has expected squared loss at most a threshold is $\mathsf{NP}^{\mathsf{PP}}$-complete for uniform Boolean inputs. For uniform inputs on the unit cube and for standard Gaussian inputs with a positive threshold, the same holds for two neurons when a positive risk gap, given in binary and possibly exponentially small, separates yes from no instances. Without a gap, the cube version is complete for guessing a polynomial-length string, then comparing a sum of exponentially many rationals with a threshold, all computable in polynomial time from the input and the guess. Since the graph is chosen too, the reductions must rule out every other network built from the same numbers.
The geometry of constraints in variational inequalities (VIs) plays a central role in applications ranging from resource allocation and equilibrium computation to constrained optimization. Yet, its effect on computational complexity has received surprisingly little attention beyond special cases. We examine this question when the constraint set is an $\ell_p$ ball. This family of problems encompasses two of the most extensively studied classes of problems in algorithmic game theory: symmetric bimatrix games for $p = 1$ and multi-player polymatrix games for $p = \infty$. Both of these problems are known to be PPAD-complete even for affine operators. We establish a comprehensive complexity landscape for VIs over $\ell_p$ balls across the full range $p\in[1,\infty]$. First, for any $p\neq 2$, we show that finding an inverse-polynomial approximate VI solution is PPAD-complete even for affine operators. In contrast, for $p=2$, we give a polynomial-time algorithm for affine VIs, extending a classic result of Ye (1992) beyond symmetric operators. On the other hand, once the operator becomes quadratic, we establish PPAD-completeness even at inverse polynomial precision. From a technical standpoint, we develop a new toolkit for exploiting the geometry of the constraint set and the structure of the operator to characterize the complexity of equilibrium and optimization problems, which is of independent interest.
This note is an exposition of the recent preprints from OpenAI that establish the bound $\omega \leq 9/4$ on the matrix multiplication exponent. We argue that the heart of these preprints is a new technique, which we formulate as the untwisting lemma for semidirect sums of tensors. This gives a generally applicable method for analyzing tensors. We also discuss further consequences of this method, as well as the relation between the OpenAI proof and previous work.
A closed path can return to its starting point while analytic continuation reaches another branch. We construct a finite computational calculus in which retained logarithmic branches supply memory, analytic observations select operations, and descent through a covering exposes deeper memory. Three branch coordinates suffice, and every finite observation uses the same cubic power covering. Each program defines a globally real analytic return map whose integer-lattice execution is realized by actual path lifting. Our first theorem simulates every finite-alphabet, finite-control Post--Turing machine, preserving complete configurations, halting, and divergence. Our second theorem identifies the partial numeric functions of the calculus, under fixed primitive-recursive input and output maps, with exactly the partial recursive functions. Deleting descent leaves a Forward language with a syntax-derived finite observation quotient and a primitive-recursive halting predicate. This yields a strict separation within the calculus and identifies access to deeper memory as its decisive additional operation. The constructions and semantic correspondences are formalized in Lean 4 with Mathlib.
The vertices of a Boolean birthday graph are partial assignments on $\ell$-subsets of $n$ variables. Although a satisfying assignment induces a $K$-clique for $K=\binom n\ell$, an arbitrary selection of $K$ vertices may repeat some views and omit others. This makes soundness substantially harder than under uniform view sampling. For bounded-occurrence, constant-gap Boolean ternary CSPs whose constraints each involve three distinct variables (allowing arbitrary Boolean predicates), we prove that the graph's maximum induced $K$-vertex density satisfies \[ \rho^*(K)\le 2\exp\!\left(-c\, \frac{\ell^3}{n^2\log(en/\ell)}\right) \] for $2\le\ell\le x_0n$ and $\ell\log(en/\ell)\ge C_0\log(n+1)$, with positive constants depending only on the source parameters. The argument refines the biclique counting method by charging the entropy of an adversarially chosen flattening to mutual information and a combinatorial obstruction energy. In the overlapping parameter range it strengthens the explicit soundness exponent previously established for this graph. We also establish an unconditional stability theorem for arbitrary sets $H$ of exactly $K$ distinct local assignments, with missing views and multiple answers: $f_*(H)\le34(n/\ell)q_{\mathrm{con}}(H)$ for every $1\le\ell\le n$. It gives a complementary additive near-clique soundness bound. We do not derive a new fine-grained hardness theorem.
We study the following family of search problems: given a system of equations $x+y+z\equiv 0\pmod p$ promised to admit a solution that is nonconstant on every equation, find a $c$-colouring of the variables that makes no equation monochromatic. For $p\in\{2,3\}$, the problem recovers known tractable Promise Constraint Satisfaction Problems (PCSPs). For $p > 3$, we obtain a complete complexity classification: the problem is polynomial-time solvable for $c \geq 4$, yet even its decision version is NP-hard for $c = 3$, and it is not well-defined for $c = 2$. Our main conceptual contribution is a general tractability framework for PCSPs based on Abelian embeddings, a notion introduced by Bhangale, Khot, and Minzer [STOC 2022] in the study of quantitative approximation of satisfiable CSPs. We show that suitable Abelian embeddings yield polynomial-time PCSP reductions realised by singleton algorithms in the sense of Zhuk [FOCS 2026]. On the hardness side, the promise relation is far removed from the reconfigurable regime covered by the conditional hardness framework of Banakh, Barsukov, and Nakajima [LICS 2026]; our proof instead relies on a new additive-combinatorial rigidity theorem classifying special $3$-colourings of $\mathbb{Z}_p$.
We prove a nearly optimal lower bound on the blocklength of general relaxed locally decodable codes (RLDCs), extending the result of [GGS26] to nonlinear encodings and adaptive decoders with two-sided error. Namely, we show that any $q$-query RLDC $C\colon \{0,1\}^k \to \{0,1\}^n$ must satisfy $n = k^{1+\Omega(1/q)}$, closely matching the upper bound of $n = k^{1+O(1/q)}$ by Ben-Sasson, Goldreich, Harsha, Sudan, and Vadhan (STOC 2004). Our proof reduces showing RLDC lower bounds to estimating the output distribution of adaptive decoders. To this end, we introduce a capped Horvitz--Thompson estimator with uniformly bounded discrete derivatives on the $p$-biased cube. We establish its concentration of measure via self-bounding inequalities and control the downward bias induced by capping through a spreadness argument.
We study the possibility of basing fine-grained cryptography on the hardness of fundamental problems in linear algebra. Our main technical contribution is a hardness self-amplification method based on random low-rank updates: we show how to turn an algorithm that solves an inverse-polynomial fraction of random linear systems into a solver which succeeds for every right-hand side of almost every matrix. Our reduction increases the runtime only linearly in the inverse of the original success probability. In particular, this yields fine-grained one-way functions from the almost worst-case hardness of linear systems over large enough fields. Using the same approach, we also obtain tighter worst-case to average-case reductions for matrix multiplication and online matrix-vector multiplication, as well as variants of our reductions that recover exact solutions from approximate algorithms, whose expected coordinate agreement is only $1/q+\varepsilon$, with inverse-quadratic dependence on $\varepsilon$ up to logarithmic factors, over every finite field. Finally, we also use it to connect the existence of fine-grained key agreement to the problem of computing the rank of a matrix: over a suitable sequence of growing characteristic-two fields, the almost-everywhere average-case hardness of rank computation (for any of a large class of distributions) implies infinitely-often fine-grained key agreement.
Can deterministic information loss yield post-quantum pseudorandomness with $TC^0$ evaluation? We show that deterministic rounding achieves this under quantum worst-case lattice hardness, with fixed polynomial parameters and no predetermined sample bound. We introduce learning with algebraic rounding (LAR), which applies an explicit integer rounding map to blocks of secret-dependent linear measurements and encodes the result in a lattice quotient. For one fixed family with polynomially bounded modulus and dimensions, deterministic labels on exactly uniform, secret-independent public matrices are computationally indistinguishable from independent uniform labels for every polynomial number of classical random examples. Our classical reduction from Gaussian LWE, combined with the quantum worst-case reduction to LWE, bases LAR on GapSVP and SIVP over general lattices at approximation factor $\widetilde{O}(q)$, where $q$ is the modulus. The proof uses source-independent affine transport: public affine maps sampled independently of the hidden errors transport the Gaussian error law close to uniform on the rounding map's zero-label fiber while preserving exactly uniform public matrices. For every fixed inverse-polynomial accuracy, we give a polynomial-time sampler for the same public parameter family. This yields the first post-quantum weak pseudorandom function with joint key-and-input evaluation in $TC^0$, secure against uniform quantum polynomial-time distinguishers given any polynomial number of classical uniform random examples. We also prove hardness of learning polynomial-size two-hidden-layer ReLU networks from noiseless uniform random examples, even for quantum polynomial-time improper learners.
We study the degree needed for sum-of-squares (SoS) proofs of several basic principles of real linear algebra. For any $n\times n$ matrix $A$ of rank greater than $r$, the Weak Rank Principle states that the equations $XY^\top=A$ with $n\times r$ matrix variables $X, Y$ have no solution. We show that this principle requires degree exactly $2r+2$ to prove. On the other hand, for $A=I_n$, setting $X = Y$ gives the Tight Frame Principle, which has a degree-$4$ proof. For $n\times n$ matrices, proving that a right inverse is also a left inverse requires degree exactly $2n+2$, while proving that orthonormal rows imply orthonormal columns again has a degree-4 proof. Using our lower bounds, we also derive related SoS degree lower bounds for polynomial systems encoding sign rank, matrix rigidity, small-width homogeneous non-commutative algebraic branching programs, and tensor CP rank. Lastly, we show that quantitative bounds on the Frobenius norms of the factors in the Weak Rank Principle can lower the degree required to refute $XY^\top=I_n$, but sufficiently large bounds still preserve the exact degree $2r+2$ from the unbounded case, giving a quantitative version of the Archimedean property arising in classical results on SoS proofs.
The problem of decomposing an algebraic variety into more structured components (such as equidimensional, irreducible, or connected components) is a fundamental problem in computer algebra and algebraic geometry. In this work, we study the computational complexity of computing equidimensional components of algebraic varieties, represented via Chow forms. We improve the state-of-the-art complexity bounds from PSPACE to CH, the counting hierarchy, for a number of problems related to equidimensional decomposition, including deciding (1) whether the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety vanishes at a given point, (2) whether a given monomial is in the support of the Chow form of the $k^{\text{th}}$ equidimensional component of a given projective variety, (3) membership in a given equidimensional component of a given algebraic variety, and (4) membership in the closure of the image of an irreducible variety under a polynomial map. As an application of our algorithm to decide membership in the closure of the image of a variety under a polynomial map, we show that the approximate polynomial satisfiability problem can be decided in CH. We also show that the degree of the $k^{\text{th}}$ equidimensional component of a given projective variety can be computed in $\text{FP}^{\text{CH}}$. Our main technical result is the construction of a polylogtime-uniform family of constant-depth, exponential-size threshold circuits that compute the Chow forms of the equidimensional components of a given projective variety. We construct these circuits by combining the machinery of geometric resolutions together with the uniform constant-depth, exponential-size threshold circuits of Andrews, Garg, and Schost (FOCS 2026) for the multivariate resultant.
We determine which major $\TFNP$ classes are hard in the random oracle model (ROM). Prior work establishes average-case hardness of highly structured subclasses of $\TFNP$ from strong cryptographic assumptions such as LWE and indistinguishability obfuscation. It is not known whether weaker assumptions such as one-way functions imply hardness of any of the standard $\TFNP$ subclasses, even in the worst case. The ROM, where all parties have query access to a random function, lies, morally, between these two regimes: a random function is one-way, but public-key cryptography cannot be built from it in a black-box way. We refer to this setting as \emph{Romantica}. Our first result settles a long-standing open problem: $\PPAD$, the class of total search problems reducible to finding a Nash equilibrium, is hard on average in Romantica. Combined with the known hardness of PLS (local search), this gives average-case hardness of $\EoPL = \PPAD \cap \PLS$. In contrast, we show that $\UEoPL$ can be solved in Romantica with polynomially many oracle queries by a computationally unbounded solver, a sharp distinction between $\EoPL$ and $\UEoPL$. This resolves the black-box hardness landscape of all major $\TFNP$ classes in Romantica. Finally, we initiate the study of separations between $\TFNP$ subclasses in Romantica: $\PWPP$ and $\PPA$ do not reduce to $\PLS$ under \emph{public-coin} reductions.
A tournament is an orientation of a complete graph. A vertex $v$ controls a vertex $u\neq v$ if there is a directed path of length at most two from $v$ to $u$, and a $\delta$-king is one that controls at least $\delta(n-1)$ vertices. In the query complexity model, each query reveals the orientation of an edge. Shen, Sheng and Wu [SICOMP'03] established an $\Omega(n^{4/3})$ lower bound on the deterministic query complexity of finding a $1$-king, and a non-matching upper bound of $O(n^{3/2})$. Neither bound has been improved in over two decades, despite substantial progress on variants of the problem. For $\delta$-kings with $\delta < 1$, Biswas, Jayapaul, Raman and Satti [Discrete Applied Mathematics'22] showed that $O(n^{4/3})$ queries suffice to find a $\frac12$-king. Lachish, Reidl and Trehan [FSTTCS'22] improved this guarantee to $\frac12+\frac{2}{17}$ using $\widetilde{O}(n^{4/3})$ queries, and asked whether a larger fraction could be achieved within the same query bound. We answer this question affirmatively for every fixed fraction below one: for every fixed $\varepsilon\in(0,1)$, we give a deterministic algorithm that finds a $(1-\varepsilon)$-king using $O(n^{4/3})$ queries. The design of our algorithm is informed by a structural analysis combining a sparse regularity lemma due to Allen, Kohayakawa, Mota and Parente [Electronic Journal of Combinatorics'14] with a new sampling argument based on a reachability theorem of Afek, Gafni and Linial [arXiv'19].
We study statistical inference and function evaluation problems in which the input binary vector is available only through a binary linear sketch, i.e., a fixed number of parity measurements. We show that under a certain monotonicity condition, simply truncating the input by retaining a subset of its coordinates is optimal among all linear sketches of the same size. In particular, this applies to distributed testing of correlations of binary vectors when they are of opposite signs and evaluating unate Boolean functions on uniformly random inputs. Additionally, given that truncation is optimal, we immediately have lower bounds for arbitrary binary linear sketches. To illustrate this application, we consider the one-way communication complexity of a distributional graph connectivity problem on $n$ vertices and show that truncation, and hence all binary linear sketches, requires $\Theta(n \log n)$ bits to achieve constant error, while a nonlinear protocol only requires $O(\log n)$ bits.
Information-theoretic MPC among $n$ parties with $t$ corruptions can securely evaluate a circuit $C$ with $O(|C| \cdot t/(n-t))$ online communication in the preprocessing model and $O(|C| \cdot t/(n-2t))$ communication in the honest majority setting. Damg{\aa}rd, Larsen and Nielsen (CRYPTO 2019) showed that both communication bounds are tight for inner products. Fully homomorphic encryption and other assumptions known to imply public-key cryptography beat both communication bounds, but the communication complexity with symmetric-key cryptography alone is less understood. For MPC with preprocessing, we prove that the same $\Omega(|C| \cdot t/(n-t))$ lower bound holds in the random oracle model, so a random oracle cannot reduce the online communication. For honest majority MPC relative to a random oracle, we present an $\Omega(|C|)$ communication lower bound that is matched by a new construction for degree-2 polynomials (including inner products) when $n = O(\log \lambda)$, beating the information-theoretic bound. We also study quantum MPC. First, we prove that information-theoretic protocols with quantum communication (and preprocessing) cannot reduce communication compared to their classical counterparts. Furthermore, we extend our lower bound for MPC with preprocessing to protocols with a quantum-accessible random oracle, but use only classical communication. In contrast, with both quantum communication and a post-quantum one-way function, we present a simple two-party protocol that evaluates the two-party instance of these circuits with $\mathrm{poly}(\lambda, \log |C|)$ communication, beating these lower bounds. Finally, such communication savings do not come for free. We prove that there exist circuits $C$ such that every blackbox protocol with $o(|C|)$ quantum communication must perform $\Omega(|C|)$ cryptographic operations.
The binary perceptron is a random constraint satisfaction problem that asks for a sign vector satisfying a system of random linear inequalities. Its solution space has a strikingly extreme geometry: at every positive constraint density, all but a vanishing fraction of the solutions are isolated. Yet polynomial-time algorithms find solutions at small enough constraint density, which raises the question of how efficient algorithms navigate such a landscape. Linear programming is a natural algorithm to study here, but its analysis has so far been limited to a single round. Li, Schramm, and Zhou (STOC 2025) analyzed one linear program followed by iterative rounding. In this work, we overcome this limitation and give a precise analysis of iterated linear programming for the binary perceptron. (1) For Gaussian matrices and any fixed number of rounds, we characterize the iterated linear programs exactly through a deterministic state-evolution recursion. (2) Adding a final rounding step, we obtain a randomized polynomial-time algorithm for the binary perceptron. Numerical evaluations of the recursion provide evidence that it succeeds at constraint density $\alpha=0.60$ with zero margin, well beyond the range $\alpha\le0.1$ guaranteed by the single-LP analysis and $\alpha\approx0.327$ (Jo and Lee, 2026), the best guarantee so far. (3) We establish universality, showing that the same characterization holds for matrices with i.i.d. entries drawn from any centered, unit-variance subgaussian law. The key idea of our proof is that, although conditioning on earlier rounds makes the matrix non-Gaussian, its conditional law can be computed explicitly and remains close to a Gaussian law, so the convex Gaussian min-max theorem applies in every round. For universality, we compare Gaussian and subgaussian matrices through the Lindeberg principle, using a regularized version of each linear program.
We characterize the optimal space for one-pass spanning-forest computation in dynamic streams of $v$-vertex simple graphs with failure probability at most $\delta$. Nelson and Yu proved an $\Omega(v\log^2v\log(v/\delta))$-bit lower bound, establishing the AGM algorithm's optimality for each fixed $0<\epsilon<1$ and $2^{-v^{1-\epsilon}}<\delta\le1/4$. We resolve the remaining high-confidence regime: for every $0<\delta\le1/4$, matching lower bounds and an algorithm give \[ \Theta\!\left(\min\left\{v^2,\;v\log(v/\delta)\log^2\!\left(2+\frac{v}{\log(v/\delta)}\right)\right\}\right) \] bits, improving space by up to a factor of $\Theta((\log v/\log\log v)^2)$ in this regime. We also study streams with at most $Z$ previously inserted edges deleted and not yet reinserted at report time, where the integer $Z\ge0$ is given in advance. Total deletions, restorations, and intermediate absence counts are unrestricted. For a single report and every $0<\delta\le1/4$, matching lower bounds and a one-pass randomized algorithm yield optimal space $\Theta(\mathcal L(v,Z,\log(1/\delta)))$ bits, where $x_Z(t)=\min\{Z+1,v,\sqrt{vt}\}$ and \[ \mathcal L(v,Z,t)=v\log v\min\{\log^2v,\log^2(Z+2)\} +vt\log\!\left(1+\frac{x_Z(t)}t\right)\log\!\left(1+\frac v{x_Z(t)}\right). \] At zero error, a lower bound for all zero-error algorithms and a matching deterministic algorithm give optimal space \[ \Theta\!\left(\min\{v^2,v(Z+1)\log(1+v/(Z+1))\}\right) \] bits. Together, these results characterize the memory required across all confidence levels and report-time edge-loss bounds.
Quantum circuit optimization is an indispensable component of the quantum software stack, used to reduce noise, maximize performance, alleviate costs of quantum error correction, or target specific hardware. A related problem is approximate membership: for a fixed set of unitaries $S$, decide whether a given circuit $C$ is within distance $\varepsilon$ of a unitary in $S$ under some metric $m$. Membership is a special case of optimization, where $S$ is a set of unitaries implementable at no cost (e.g., with noise-minimal circuits, or without magic states). Janzing, Wocjan, and Beth (Int. J. Quantum Inf., 2005) showed that, for the operator metric, the case $S=\{I\}$ (approximate identity) is QMA-complete when $C$ is promised to be either close to the identity or far from it. We generalize this setting in three directions: (i) promise-free inputs, (ii) besides $S=\{I\}$, sets $S$ containing only Clifford unitaries, or only generalized permutation matrices (i.e., no superposition), and (iii) the trace-based, spectral, and operator metrics. We show that, in all cases, approximate membership is PP-hard for wide ranges of $\varepsilon$, provided $C$'s gate-set can implement Clifford+T, as the gate-sets of standard hardware do. If $S=\{I\}$, or if $S$ contains only generalized permutation matrices, $I\in S$, and $m$ is a trace-based distance, PP-hardness holds for all non-trivial $\varepsilon$. If $S=\{I\}$ and $m$ is a trace-based distance, PP-hardness holds even for circuits over classical-reversible gate-sets. Our results indicate that approximate membership is harder over promise-free inputs than the QMA-complete promise version, and that this hardness persists across quantum resources (any gates, non-Clifford, superposition) and distance metrics. They directly imply PP-hardness of circuit optimization when minimizing total, non-Clifford, or superposition gate depth or count.
Binary prefix coding assigns prefix-free binary words to symbols to minimize their frequency-weighted cost. Huffman solved the equal-letter-cost case in polynomial time in 1952 [Proc. IRE'52]. For unequal letter costs, Karp gave an integer-programming formulation in 1961 [IRE Trans. Inf. Theory'61]. Exact polynomial-time algorithms for fixed integer costs were given by Golin and Rote [IEEE Trans. Inf. Theory'98] and Bradford, Golin, Larmore, and Rytter [J. Algorithms'02]. For input-dependent costs, Golin, Mathieu, and Young gave a polynomial-time approximation scheme [SICOMP'12]. The exact complexity when letter costs are part of the input had remained unresolved. We resolve this $65$-year-old problem by proving that deciding whether there exists a binary prefix-free code of frequency-weighted cost at most a given threshold is NP-complete, even when the letter costs are consecutive positive integers $(M,M+1)$ supplied as part of the input. All frequencies, letter costs, and thresholds are encoded in binary. Consequently, finding an optimal code is NP-hard. Our proof uses two polynomial-time reductions, starting from Planar Independent Set. The intermediate problem prescribes the number of leaves at each depth of a full ordered binary tree and minimizes a signed linear score on their sorted right-edge counts. The first reduction represents an independent-set verification program by a tree growing level by level. A fixed-profile identity converts narrow coordinate restrictions into linear penalties that enforce the computation. The second reduction adds large dyadic terms to the symbol frequencies to force prescribed depths. These terms contribute a constant to the weighted number of ones, leaving smaller signed perturbations to reproduce the tree score.
Autobidding equilibria link advertisers' value multipliers or participation probabilities through aggregate budget and return-on-investment (ROI) constraints. We establish the exact computational complexity of finding these equilibria across several canonical settings: exact equilibrium computation is $\mathsf{FIXP}$-complete for second-price ROI pacing, budget- or ROI-constrained generalized second-price (GSP) sponsored shopping, and second-price throttling. Furthermore, approximating the equilibrium control vector in distance to a true equilibrium is $\mathsf{FIXP}_{a}$-complete when the target accuracy is provided as input.
In this paper, we study exact online integer priority queues under differential obliviousness in the balls-and-bins model. Given a public upper bound $T$ on the number of operations and assuming that the priorities are bounded by a polynomial in $T$, we construct an $(\eps,\delta)$-differentially oblivious priority queue supporting insertion and minimum extraction, which requires $\polylog(T)$ private memory. For constant $\varepsilon>0$ and $\delta=\exp(-\Theta(\log^2 T))$, our construction incurs $O(\log T/\log\log T)$ expected amortized total work per operation, which breaks the $\Omega(\log T)$ barrier for fully oblivious priority queues~[Jacob et al., SODA'2019]. Further, we prove a matching lower bound for $(\eps,\delta)$-differentially oblivious priority queues with $\varepsilon=O(1)$ and $\delta=o(1)$, which indicates that our construction is asymptotically optimal in terms of total work.
A \emph{rainbow cycle} in a list-edge-colored graph is a cycle where no color appears more than once. We give a short and simple proof of a result of [ABSZZ] showing that a properly edge-colored $n$-vertex graph with average degree $\Omega(\log n\log\log n)$ always contains a rainbow cycle, and generalize it to weighted graphs with edges list colored with list size at most $\ell$ and where the total weight of all edges containing a given color $i$ incident to a vertex is at most $\Delta$, showing that $\Omega_\ell(\Delta\log n\log\log n)$ average degree guarantees a rainbow cycle. Using the reduction of [AG26] and the new generalized rainbow cycle theorem, we prove that for $r\geq 3$, every binary linear $r$-query locally correctable code $E:\mathbb{F}_2^k\rightarrow \mathbb{F}_2^n$ satisfies $k\leq \tilde{O}(n^{1-2/(r-1)})$ by showing a rainbow analogue of the odd $r$-LDC lower bounds of [BHKL25,JM25], closing the gap between the best known lower bounds for odd- and even-query binary linear $r$-LCCs.
We show that for any fixed $k$, there exists $\varepsilon_k > 0$ such that, given a $(k-1)$-uniform hypergraph $H$ on $n$ vertices, one can count the number of $k$-hypercliques in $H$ in $O(n^{k-\varepsilon_k})$ time. This refutes the hyperclique conjecture of Lincoln, Vassilevska Williams, and Williams (SODA 2018). Our result follows from the uniform construction of a family of arithmetic circuits of size $O(n^{k-\varepsilon_k})$ for the $k$-hyperclique tensor, a natural higher-order generalization of the matrix multiplication tensor. We obtain two corollaries. First, for every $k$, there exists $\varepsilon > 0$ such that Max-$k$SAT can be solved in $2^{(1-\varepsilon)n} \cdot \mathrm{poly}(m,n)$ time. Second, for every $k$, there exists $\varepsilon > 0$ such that arithmetic progressions of length $k$ in subsets of $\{0,1,\ldots,n\}$ can be detected in $n^{2-\varepsilon}$ time. Both corollaries refute other prominent conjectures in the literature.
We prove a broad learning-theoretic generalization of the celebrated hardcore lemma concerning the average-case complexity of computing a Boolean function (Impagliazzo, 1995). Our starting point is the observation that the classic notion of a hardcore measure, which reweights a function's input distribution, is equivalent to an input-dependent masking channel that injects randomness into the function's output distribution until it appears fully pseudorandom. The analogue of density of the measure is the fidelity of the channel. From this viewpoint, we develop a general framework for certifying hardness that allows for non-Boolean functions $f$ and a wide range of notions of hardness besides the classic 0-1 loss, by choosing the right notion of fidelity via convex duality. We show the difficulty of learning $f$ using a class $\mathcal{C}$ to minimize a convex loss is tightly characterized by the maximum fidelity possible for a masking channel, which measures the least amount of noise the channel must add to hide the structure of $f$ and make it $\mathcal{C}$-indistinguishable from random noise. Using our framework, we derive a number of multiclass generalizations of the hardcore lemma, and reveal some unexpected subtleties in going to larger alphabets. We construct a universal masking channel that applies to many natural notions of hardness simultaneously. Such channels are dual to omnipredictors, which optimize multiple losses at once (Gopalan, Kalai, Reingold, Sharan, and Wieder, 2022). We observe a symmetry between masking channels and generalizations of the hardcore lemma using pseudoentropy. We present a tight characterization of pseudoentropy using the MaxEnt algorithm from machine learning. A long line of research has shown that learning and hardness are two sides of the same coin; our framework gives a crisp and general formulation of this principle via convex duality.
We consider the complexity implications of proofs of quantumness and (quantum verifier) oblivious state preparation. We show that both of these objects can coexist with the easiness of nondeterministic computation. That is, we give evidence that proofs of quantumness can exist when $\mathsf{P} = \mathsf{NP}$ and that (quantum verifier) oblivious state preparation can exist when $\mathsf{BQP} = \mathsf{QCMA}$. In particular, we give a classical oracle relative to which (1) $\mathsf{P}^\mathcal{O} = \mathsf{NP}^\mathcal{O}$ (2) $\mathsf{BQP}^\mathcal{O} = \mathsf{PSPACE}^\mathcal{O}$ (3) Proofs of quantumness exist. In particular, there exists a language in $\mathsf{IP}^\mathcal{O} \subseteq \mathsf{BQP}^\mathcal{O}$ which is hard on average against $\mathsf{P}^\mathcal{O}/poly$. We give a unitary oracle such that (1) $\mathsf{BQP}^\mathcal{O} = \mathsf{QCMA}^\mathcal{O}$ (2) Quantum verifier oblivious state preparation exists. These oracle separations directly imply a number of related consequences. For example, our first result implies that proofs of quantumness cannot be used to construct classical or quantum cryptography with a relativizing proof. Our second result implies that quantum computation/classical communication oblivious transfer cannot be used to construct post-quantum one-way functions with a relativizing proof.
Given a class of machine learning models and sample access to a data distribution with binary labels, which computational task is harder: training a discriminative model that distinguishes positive and negative labels as well as the best member of the class, or training a generative model whose outputs are indistinguishable from real data by all members of the class? We prove that these two fundamental tasks in computational learning theory are not just closely related, but actually equivalent to each other in complexity. One direction of this equivalence was previously known. Indeed, the powerful regularity lemma in complexity theory gives an efficient oracle reduction from the indistinguishability task (a.k.a. multiaccuracy) to the distinguishing task (a.k.a. agnostic learning) in the style of boosting algorithms. Our main contribution is an efficient oracle reduction in the other direction. Taking this idea further, we show that a multiaccuracy oracle is a remarkably powerful primitive, achieving not just agnostic learning but also a wide range of seemingly more demanding learning objectives via similarly efficient and direct constructions. Our results crucially use the power of making adaptive queries to the oracle, and stand in stark contrast to prior work, which showed that post-processing a single multiaccurate predictor is insufficient for even weak forms of learning. On a technical level, we draw a connection between our (offline) problem and a version of no-regret learning in the online setting, which allows us to use follow-the-leader type algorithms to modify the label distribution and aggregate the predictions of the multiaccuracy oracle.
In the graph collision problem, the task is to decide whether a known graph on $n$ vertices contains an edge with both endpoints marked, given query access to the input bits specifying the marked vertices. We first refute the conjectured $\Omega(n^{2/3})$ query lower bound with a bounded error quantum algorithm using $\widetilde O(n^{92/139})$ queries. We then refine this approach to obtain a $\widetilde O(n^{11/17})$ query algorithm. We also obtain a $\widetilde O((nq)^{11/17})$ query bound for all (partial) Boolean output functions with positive certificate complexity at most 2 and alphabet size $q$, contrasting a known $\Omega(n^{2/3})$ lower bound for such problems with large alphabet sizes. We complement these algorithms with an $\Omega(\sqrt{n\log n})$ quantum query lower bound for graph collision, exceeding the $O(\sqrt n)$ certificate complexity barrier for the positive adversary method.