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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 $Ω(n^{2/3})$ 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 $Ω(n^{2/3})$ lower bound for such problems with large alphabet sizes. We complement these algorithms with an $Ω(\sqrt{n\log n})$ quantum query lower bound for graph collision, exceeding the $O(\sqrt n)$ certificate complexity barrier for the positive adversary method.

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.

We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ Gaussian random matrices has growth $ρ$ is at least inverse quasi-polynomial: $Ω\left(\exp\left(-c \log^2 \left(\fracρ{\sqrt{n}}\right)\log(n)\right)\right)$ for some constant $c > 0$. In particular, the probability of growth $ρ\ge n^{1/2 + ε}$ is at least $\exp(- O (ε^2 \log^3 (n) ))$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.

We study approximation algorithms for All-Pairs Shortest Paths (APSP) in directed graphs under the roundtrip metric. In directed graphs, even finite-factor approximation of ordinary one-way distances must distinguish reachable from unreachable vertex pairs, and therefore computes transitive closure, a problem equivalent to Boolean matrix multiplication (BMM) under standard reductions. The roundtrip distance, defined as $\text{dist}_G(u \leftrightarrows v) = \text{dist}_G(u \to v) + \text{dist}_G(v \to u)$, offers a natural and robust relaxation. It measures the cost of traveling from $u$ to $v$ and back, satisfies the triangle inequality, and is inherently symmetric. While this metric has become central to the study of directed roundtrip spanners and compact routing, the fundamental problem of all-pairs roundtrip distance approximation has remained largely unexplored. In this work, we present a comprehensive study of roundtrip APSP and distance oracles, presenting several new lower and upper bounds. On the algorithmic side, our main result is a combinatorial algorithm that computes a stretch-$3$ approximation to all-pairs roundtrip distances in $\widetilde{O}(n^2 + \min\{m\sqrt{n},\, m^{1/3}n^{5/3}\})$ time. For edge weights in $[1,W]$, we also obtain a $(3+\varepsilon)$-approximation in $\widetilde O((n^{2.037}+nm^{2/3})\log(nW)/\varepsilon)$ time and a $(4+\varepsilon)$-approximation in $\widetilde O(n^{2.032}\log(nW)/\varepsilon)$ time using fast matrix multiplication. On the hardness side, we show that any stretch below $3$ would imply a breakthrough for BMM. We also obtain additional tradeoffs between stretch and running time.

We introduce Gaussian partitioning, an algorithmic primitive for distributions smooth with respect to a Gaussian. It partitions space into polyhedral regions so that almost all mass lies in thin strips restricting a few orthogonal directions. Within each strip, the conditional distribution approximately matches low-degree Gaussian moments in the unrestricted directions. Our construction builds on logconcave partitioning of Diakonikolas, Kane, and Ma (FOCS 2026). Under input marginals smooth with respect to the standard Gaussian, we obtain agnostic learners with error $(1+c)\mathrm{opt}+ε$ for broad classes of Boolean multi-index models, including functions of halfspaces, low-dimensional convex concepts, and low-dimensional polynomial threshold functions. For functions of $k$ halfspaces, both agnostic and testable agnostic learners achieve this error with dimension dependence $d^{\mathrm{poly}(k/c)}$, improving the previous exponential dependence on $k/c$ in the agnostic exponent. For learning from positive examples with imperfect unlabeled data (PUI), we give algorithms and Statistical Query (SQ) lower bounds that qualitatively characterize complexity through Gaussian moment matching. For fixed smoothness and positive-mass parameters, we learn intersections of $k$ halfspaces with dimension dependence $d^{O(\log(k/ε))}$ and matching SQ lower bounds, improving the previous $d^{\mathrm{poly}(1/ε)}$ upper bound even for one halfspace. A general reduction from agnostic reliable learning to PUI yields negative reliable learners for these intersections with the same dimension dependence, for fixed smoothness. Finally, for PQ learning and testable learning with distribution shift, we prove qualitatively optimal SQ lower bounds of $d^{Ω(1/ε^2)}$, even for intersections of two halfspaces with Gaussian training data and arbitrary test distributions.

Given the full truth table of a Boolean function whose input is split between two players, we study the complexity of determining the minimum worst-case number of bits they must communicate to compute it. This is a basic meta-complexity question, posed by Yao in 1979. We prove NP-hardness of approximation within every fixed additive error for deterministic communication complexity, under deterministic polynomial-time reductions. The result holds both in the standard model and in Yao's original model, where the players alternate after every bit and only the receiver of the last bit must determine the answer. In particular, it resolves the exact computation question under Yao's original conventions. Earlier versions of this same paper established exact NP-hardness for the standard model, independently and concurrently with Hirahara, Ilango, and Loff. In both models, a polynomial-time estimator with any fixed additive error would yield a polynomial-time $3$-SAT algorithm, deterministic or bounded-error randomized according to the estimator. For the standard model, we also give quantitative implications for errors that grow with the number $T$ of table entries. A bounded-error randomized polynomial-time estimator with additive error $c\log T/\log\log T$, for any fixed $0\le c<1/8$, yields a bounded-error randomized $3$-SAT algorithm on $N$ variables and $O(N)$ clauses running in time $2^{O(N^α)}$ for every fixed $8c<α<1$. A deterministic polynomial-time estimator with additive error $c_*\log\log T$, for an effective absolute constant $c_*>0$, yields a deterministic algorithm running in time $2^{O((\log N)^2)}$ on such formulas.

We establish an $Ω(d)$ lower bound on the competitive ratio of convex body chasing in $\mathbb{R}^d$ with movement costs measured in any norm. Previously, the best known lower bound valid across all norms was $Ω(d^{1/6})$, established by Papazov (2023); stronger lower bounds were known for $\ell_p$ norms, where Bubeck, Klartag, Lee, Li, and Sellke (2019) had established a lower bound of $Ω(\max\{\sqrt{d}, d^{1-1/p}\})$. In particular, a linear lower bound was known only for the $\ell_\infty$ norm, realized by the standard construction of chasing faces of a hypercube. Our lower bound strictly improves on the prior universal and $\ell_p$ lower bounds for $p < \infty$ and shows that the functional Steiner point algorithm of Sellke (2023) - which has competitive ratio $O(d)$ - is asymptotically optimal for every norm. Our approach is based on the recursive construction of Bubeck, Coester, and Rabani (2023) for metrical service systems, which gives a lower bound of $Ω(r^2)$ for requests of size $O(r)$. We show that requests in their construction, each comprising a set of discrete points in a graph metric, can be mapped to the vertices of convex requests in a $(2r+1)$-dimensional $\ell_p$ vector space while approximately preserving movement costs up to a factor of $O(r)$; the general result then follows via Milman's (1985) quotient-of-a-subspace theorem.

A defining feature of the linear-probing hash table is its spatial locality: at load factor $1 - \varepsilon$, the hash table can be implemented so that each operation scans a single contiguous region of expected size $O(1/\varepsilon)$ (this is known as the \emph{scan cost}). This guarantee, however, relies on all items having unit size. In this paper, we ask whether it is possible to achieve a similar guarantee, but in a hash table that stores items of arbitrary lengths $|x| \ge 1$. The main result of the paper is a new hash table called \emph{Catalan hashing}, which supports amortized expected scan cost $O(|x|/\varepsilon)$ for positive queries, $O(1/\varepsilon)$ for negative queries, and $O(|x|\varepsilon^{-1}\log^{1+δ}|x|)$ for insertions/deletions, for any constant $δ> 0$. As a corollary, we obtain the same bounds in the cache-oblivious model, with a scan cost of $S$ translating to $1 + O(S/B)$ I/Os. Along the way, we introduce several high-level techniques that may be of independent interest; this includes a novel way of using erasure codes in order to perform write optimization within a hash table, and a connection to scheduling that allows us to place elements in a way which avoids clustering at high load factors.

Let $μ$ be a probability distribution on a spin system with $n$ sites and possibly infinite spin spaces. We prove local-to-global spectral-gap bounds for block Glauber dynamics. Given a symmetric, entrywise nonnegative matrix $A$ with zero diagonal and $λ_{\max}(A)<1$ encoding local spectral bounds for a spin system, we derive global spectral-gap bounds for block dynamics on its marginal distributions. These bounds are sharp: the Gaussian system with covariance $(I-A)^{-1}$ attains equality for every choice of disjoint blocks. Our proof extends the Bochner method through an abstract inequality for orthogonal projections on Hilbert spaces. As an application, we prove that Glauber dynamics for the $O(N)$ model, with $N\geq 2$, arbitrary external fields, and couplings of either sign, has spectral gap at least $(1-\|J_{\mathrm{abs}}\|_{\mathrm{op}}/N)/n$, where $J$ is the interaction matrix and $J_{\mathrm{abs}}=(|J_{ij}|)$. We also obtain spectral bounds for swap walks and local spectral expansion of partite simplicial complexes. For complexes formed from disjoint groups of ranks in random flags of subspaces over $\mathbb F_q$, we obtain a one-sided local spectral expansion bound of $2\sqrt q/(q+1)$. This bound is independent of the number and sizes of each group, and is nontrivial for every prime power $q$. Finally, for random forests $F$ of any finite graph with arbitrary positive edge weights, we prove that \[ \Pr(e,f\in F)\le \frac{4}{3} \Pr(e\in F)\Pr(f\in F) \] for distinct edges $e,f$, improving the factor 2 supplied by general log-concavity bounds for matroid independent sets. The argument combines our extended Bochner method with the analysis of an auxiliary real stable polynomial.

Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_iA_i$, $x\in \mathbb R^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\to \infty, 2n/d^2 \to α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \in \{-1,+1\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for a standard deviation $σ(α)<1$ for all $α>0$, and $σ(α) = O(1/\sqrt{α\log α})$ for large $α$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025).

Colonel Blotto is a fundamental resource-allocation game in which two players divide their budgets across battlefields, and the higher allocation on each battlefield wins its value. The problem of computing an equilibrium efficiently has remained unresolved for its classical continuous version with arbitrary budgets and with values that may vary across battlefields but are the same for both players. We give a fully polynomial-time approximation scheme that computes an $\varepsilon$-Nash equilibrium in time polynomial in the input length and $1/\varepsilon$, and nearly linear in the input length for fixed $\varepsilon$. For sufficiently balanced budgets and battlefield values, we also compute exact equilibria in linear time.