Papers

The maximum-entropy sampling problem (MESP) seeks, for an order-$n$ covariance matrix $C$, a principal submatrix of order $s$ with maximum log-determinant. Mostly for convenience, we assume that $C$ is nonsingular. Al-Thani and Lee (2023) solved MESP in $O(n^5)$ time when $C$ or $C^{-1}$ is tridiagonal. We show that the inner maximization of their recursion depends only on a prefix of the index set and that no piece of a solution is longer than $s$; this gives an $O(ns^2)$-time algorithm that returns the optimal value for every budget $t\le s$. When $C^{-1}$ is tridiagonal, $C$ is, up to scaling, the covariance matrix of an Ornstein–Uhlenbeck process observed at unevenly spaced times, and MESP becomes choosing points on a line under a concave gap function with the Monge property; this gives an $O(ns)$-time algorithm and, in the first-order autoregressive case, a closed-form solution. Given only $C$, we solve MESP in $O(n^2)$ time whenever $C$ or $C^{-1}$ is tridiagonal, up to a symmetric permutation, and we recognize these cases within the same bound. For spiders, we make explicit, and sharpen, the dependence on the number of legs, and, drawing on a hardness result of Ohsaka for stars, we observe that, unless $\mathrm{P}=\mathrm{NP}$, the exponent of the running time must grow with the number of legs.

We study isomorphism decision and basis construction for finite Abelian groups of known order $n$. We measure complexity by the number of additions performed in the groups and by the total running time. We give a randomized isomorphism decision algorithm that performs $\tilde O(n^{1/4})$ group additions and runs in $\tilde O(n^{1/4})$ time. This improves the $\tilde O(\sqrt n)$ upper bound of Chen and Fu and matches Bshouty $\Omega(n^{1/4})$ lower bound up to polylogarithmic factors. We also prove that finding a basis with success probability at least $2/3$ requires $\Omega(\sqrt n)$ group additions in the worst case. This matches the $\tilde O(\sqrt n)$ upper bound of Chen and Fu up to polylogarithmic factors. These results separate isomorphism decision from basis construction in finite Abelian groups: their optimal worst-case complexities are $\tilde\Theta(n^{1/4})$ and $\tilde\Theta(\sqrt n)$, respectivel

The constraint satisfaction problem over a set of relations $\Gamma$ (CSP($\Gamma$)) is the computational problem of deciding if a set of constraints admits at least one solution. The classical complexity for finite-domain CSP($\Gamma$) is settled by the CSP dichotomy theorem: it is tractable if $\Gamma$ satisfies a non-trivial algebraic invariant and is NP-complete otherwise. However, not all these algebraic invariants result in efficient algorithms despite being theoretically tractable. A notable case that generalizes linear equations is that of Maltsev CSPs: an $n$-variable instance with $m$ constraints is solvable in roughly $O(n^8 \cdot m)$ time by Bulatov and Dalmau (SIAM J. Comput. 2006) or $O(n^4 \cdot m)$ time by Dyer and Richerby (SIAM J. Comput. 2013). At the same time, arguably, most "natural" and efficiently usable polynomial-time algorithms rarely exceed a quadratic or cubic time bound. In this paper we revisit Maltsev constraints with this question in mind and find a $O(n^2 \cdot m)$ algorithm (for finite languages, for infinite languages we in addition need to take the total size of the instance into account). The main novel idea is to not attempt to improve the bottleneck in Bulatov and Dalmau (the Fix-Values procedure) but to avoid it altogether with a slightly more refined approach that allows us to search through a smaller space.

Consider the $\mathbb Z_2$ synchronization problem \[ \boldsymbol{Y} = \frac{\lambda}{\sqrt n} \theta\theta^{\top} + \boldsymbol{Z}, \] where $\theta$ is uniform on $\{-1,1\}^n$ and $\boldsymbol{Z}$ is an independent Gaussian Wigner matrix with off-diagonal variance one. We give a polynomial-time posterior sampling algorithm for every fixed $\lambda>1$, for which the conditional output law converges to the posterior in total variation, in expectation over the observation. The construction combines sequential TAP proposals with an independence Metropolis correction. The key is to control signed overlaps after logarithmic pinning and conditional TAP approximations along a random revealing path, which give an efficiently evaluable proposal with a polynomial density-ratio bound outside a set of vanishing posterior mass. To the best of our knowledge, this is the first polynomial-time posterior sampler for $\mathbb Z_2$ synchronization with a total-variation guarantee throughout the supercritical regime. For comparison, the diffusion-based sampler of \cite{montanari2023posterior} gives normalized Wasserstein guarantees at sufficiently large fixed signal-to-noise ratio.

We design an almost instance optimal algorithm for the sum estimation problem using weighted sampling. We show that the sample complexity for sum estimation is closely related to the $\ell_2$ norm of the weighted sampling distribution. We show an almost instance optimal lower bound for this problem as well. We also study the moment estimation problem and design an algorithm that has better instance-wise sample complexity bounds.

Byzantine quorum safety relies on correct replicas refusing to sign conflicting values. A replica that loses its protocol state during recovery but retains its identity and signing key may forget an earlier vote. We study certificates formed by matching signed votes from at least $q$ of $n$ replicas, assuming that each correct replica avoids conflicting votes between recoveries. If two conflicting certificates form, their overlap has size between $2q-n$ and $b+c$, where $b$ counts Byzantine replicas and $c$ counts correct identities that recovered during the execution considered. Our main result decomposes the slack $b+c-(2q-n)$ into four nonnegative counts: extra signers in the first certificate, extra signers in the second, identities in neither certificate, and Byzantine or recovering identities outside their overlap. Zero slack forces an exact signer partition. With $n=3f+1$ replicas, threshold $q=2f+1$, at most $f$ Byzantine replicas, and exactly one correct recovery event, any conflicting pair forces exactly $f$ Byzantine replicas, all in the overlap together with the recovered replica; each certificate has a disjoint side of $f$ correct replicas. A minimal protocol attains this form. We distinguish certificate formation from acceptance, give a sufficient check using configured fault and recovery caps, and explain why durable vote records written before signature release prevent the conflict.

Given a complete graph whose edge weights represent dissimilarities, Metric Violation Distance asks whether at most $k$ weights can be changed to form a metric. Motivated by metric repair for noisy data, the problem admits a polynomial-time $O(\log n)$-approximation due to Cohen-Addad, Fan, Lee and de Mesmay [SIAM J. Comput., 2025]. In the context of parameterized complexity, Fan, Gilbert, Raichel, Sonthalia and Van Buskirk [SWAT 2020] gave a $k^{O(k)}n^{O(1)}$-time algorithm. Fomin, Golovach and More [IPEC 2026] obtained an $O(k^2)$ kernel and a single-exponential algorithm for the ultrametric case. They asked whether general metrics admit a single-exponential algorithm and a polynomial kernel, and whether the tree-metric analogue is fixed-parameter tractable. We answer all three questions. We give a $2^{O(k)}n^{O(1)}$-time algorithm and prove that, unless ETH fails, no $2^{o(k)}n^{O(1)}$-time algorithm exists, even when all input distances lie in $\{1,2,3\}$. We also give a kernel with at most $6k$ vertices. The kernel supports solution lifting and can precede any approximation algorithm. Combined with the $O(\log n)$-approximation of Cohen-Addad, Fan, Lee and de Mesmay, it yields an $O(\log \mathrm{OPT})$-approximation at no asymptotic cost in running time. Both results extend to an interval generalization in which each edge $e$ has an observed value $M_e$ and an admissible range $[A_e,B_e]$ within which it may be reassigned; the kernel then has $7k$ vertices. Finally, Tree Metric Violation Distance is fixed-parameter tractable and solvable in $k^{O(k)}n^{O(1)}$ time.

Probability forecasts are calibrated when predicted probabilities match empirical outcome frequencies: among events assigned a probability $p$, we'd hope that the fraction of positive outcomes is close to $p$. We study the problem of sequential forecasting of binary outcomes. The classical $O(T^{2/3})$ bound on expected cumulative $\ell_1$-calibration error established by Foster and Vohra stood for over two decades until Dagan et al. reduced the exponent $2/3$ by an unspecified constant. We establish a new two-phase recursive labeling strategy for the sign-preservation-with-reuse game that yields the bound $O(n^{\alpha}t^\beta)$ for all choices of space and time. We then sharpen the reduction from upper bounds on sign preservation to calibration by modifying the equivalence of Dagan et al. to use only $O(\log T)$ instances of the sign-preservation-with-reuse game. This lets us establish an explicit bound of $O(T^{0.662942288})$, the first explicit exponent below $2/3$ for sequential calibration, by combining both improvements and choosing explicit feasible parameters.

We give a deterministic reduction for maintaining simultaneous approximate single-source distance estimates in online decremental directed graphs. For positive integer weights in $[1,W]$, the algorithm maintains an explicit array of integer $(1+\varepsilon)$ upper estimates, identifies unreachable vertices exactly, and answers numerical queries in constant time. Let $N=m+n$, $S=N+\lceil1/\varepsilon\rceil$, and assume $\log W=\operatorname{polylog}(S)$. Initialization and all updates take $O(\Delta)+(N+\Delta_{\mathrm{eff}}+\mathcal{L}/\varepsilon)S^{o(1)}$ time, where $\Delta$ counts raw updates, $\Delta_{\mathrm{eff}}=O(m(1+\log W)/\varepsilon)$ counts filtered updates, and $\mathcal{L}$ measures finite distance growth weighted by the current indegrees of the reachable subgraph. Removed edges and vertices that become unreachable incur no subsequent growth charge. Consequently, the worst-case bound is $O(\Delta)+(N/\varepsilon)S^{o(1)}$, which is almost linear for subpolynomial inverse accuracy. The reduction uses the dynamic minimum-ratio cut and exact reachability algorithms of van den Brand et al. (FOCS 2024). A degree-weighted clipped logarithmic potential turns constant relative cut balance into accuracy at every target; a successor cut then restores strict feasibility. A distance warm start and the energy released by decreasing return coefficients yield the refined growth bound. Return mass also amplifies the accuracy of the existing change detector. The public estimates can be monotone, with $O(n+\mathcal{J}/\varepsilon)$ array writes for an unweighted finite-growth measure $\mathcal{J}\le\mathcal{L}$. Execution uses rational arithmetic. A separate reduction extends the worst-case guarantee to nonnegative integer weights. The algorithm maintains numerical estimates and does not provide fast path reporting.

We give an exact algorithm for continuous separable convex quadratic programming with an integer equality matrix and nonnegative variables. The number of rational arithmetic operations and comparisons is polynomial in the dimensions and the encoding length of the constraint matrix, independently of the right-hand side, linear costs, and all nonnegative quadratic weights. Intermediate rational encodings are polynomial in the complete input. This yields a strongly polynomial algorithm for every integer matrix class with polynomially bounded entry encodings, including arbitrary matrices with entries in $\{0,\pm1\}$. The local solver compresses a quadratic objective on an integer box and solves only the compressed continuous problem. Two-sided proximity on boxes with rational bounds transfers its solution to a nearby original optimum; integer optima are used only in the proof. A single linear program supplies an error bound and a dual potential, while tight-coordinate revelation and fixed feasible anchors ensure termination and control encoding lengths. An explicit lifting extends the result to convex piecewise quadratic functions of signed integer affine features, with operation counts independent of breakpoints and objective coefficients. Applications include continuous multicommodity flow with shared capacities and piecewise quadratic costs of aggregate congestion.

We study the one-pass streaming complexity of CSPs over a fixed finite relation $R$ under three natural objectives: $\textsf{SAT}(R)$, deciding whether all constraints can be satisfied; $\textsf{Min}$-$\textsf{CSP}(R)$, approximately minimizing the number of unsatisfied constraints; and $\textsf{Exact-CSP}(R)$, exactly computing the maximum number of satisfied constraints. Sharma and Velusamy (ESA 2026) characterized the streaming complexity of satisfiability for CSPs with literals using the non-redundancy parameter $\textsf{NRD}_n(R)$, which roughly measures the largest instance in which every constraint is independently necessary. For the most general setting without literals, they obtained the corresponding characterization for Boolean relations and showed obstacles to extending their techniques to larger domains. Kol, Paramonov, Saxena, and Yu (ITCS 2023) similarly characterized the streaming complexity of $\textsf{Exact-CSP}(R)$ for Boolean CSPs with literals in terms of the degree deg$(R)$ of the relation when written as a polynomial, while without literals, the corresponding result was known only for $\textsf{Max-Cut}$. For all three of the objectives we study, we give tight characterizations for unweighted CSP instances over arbitrary finite domains. The streaming algorithms we provide are deterministic, while the corresponding bounds are tight up to polylogarithmic factors, even against randomized algorithms. The central idea in all three results is a reduction of $R$ to its core, a canonical subrelation of $R$ which admits gadgets that allow for the hard-pinning (or fixing) of variables.

Maximizing a submodular function subject to a matroid constraint is a cornerstone problem in combinatorial optimization. The state-of-the-art algorithm for this problem obtains $0.401$-approximation~\cite{buchbinder2024constrained}, and the state-of-the-art hardness result shows that no polynomial time algorithm can obtain better than $0.478$-approximation for this problem~\cite{oveisgharan2011submodular}. In this work, we present the first improvement in $15$ years for the hardness result, showing that no polynomial time algorithm in the value-oracle model can obtain better than $8/17 \approx 0.471$-approximation, even for the special case of a cardinality or (simplified) partition matroid constraint.

Causal message ordering provides essential semantics for distributed applications, yet ensuring it within an asynchronous system subject to Byzantine failures presents fundamental theoretical and practical challenges. Prior research establishes that algorithms cannot guarantee both strong safety and liveness without using cryptography under these conditions. Existing Byzantine-tolerant solutions make synchrony assumptions or suffer from $O(n)$ message space overheads and $O(n^2)$ message complexity, where $n$ is the number of processes in the system, or use cryptography, but may not guarantee strong safety. In this paper, we present a novel Byzantine-tolerant causal ordering algorithm that achieves an optimal $O(1)$ application message space overhead for point-to-point messages. Our approach uses a Sender Permission to Send (SPS) invariant and an Isolated-Buffer Optimistic Model for the memory management architecture. To circumvent Byzantine state-pinning and head-of-line blocking attacks, instead of unified causal buffers, we use isolated, per-peer queues equipped with event-driven Cascading Space Evictions. We formally prove that our algorithm guarantees system-wide liveness and satisfies a weakened causal safety abstraction called Congestion-Relaxed Causal Delivery. Under this model, weak causal safety is strictly preserved for all honest-to-honest communications unless extreme network latency or Byzantine withholding attacks exceed quantifiable local buffer capacities, forcing optimistic queue evictions. This formal guarantee successfully balances causal ordering with high throughput, constant message space overhead, low computational overhead, and strictly bounded local space.

We study an online trading problem where a trader, given a sequence of i.i.d. prices drawn from a known distribution $F$ on $[0,1]$, must make irrevocable buy, sell, or hold decisions subject to storage constraints. We investigate achievable algorithmic performance measured in terms of regret, the difference between the expected profit of the hindsight optimal policy that knows the entire price sequence and an online algorithm. We analyze finite atomic and continuous distributions characterized by their local behavior around the median which we capture using a parameter $\beta$. The parameter $\beta$ quantifies how the mass of prices accumulates around the distribution median. We identify a new driver of algorithmic performance, demonstrating that median gaps coupled with an initial inventory level of zero can force regret scaling of $\Omega(T^{(\beta + 1)/(2\beta+4)})$ — establishing a novel spectrum of fundamental limits on algorithmic performance. We then study STARS, short for Storage Trading by Averaging Repeatedly across multiple Scenarios, which simulates possible future price scenarios to approximate the value-to-go function and make buy/sell/hold decisions. We show that STARS obtain near-optimal algorithmic performance (upto poly-logarithmic factors) across a broad range of distributions. In particular, it achieves $O(\log T)$ regret for finite atomic prices, $\widetilde{O}(T^{\beta/(2\beta+2)})$ for continuous distributions without a median gap and $\widetilde{O}(T^{(\beta+1)/(2\beta+4)})$ for continuous distributions with a median gap for $\beta \geq 0$.

The Fréchet Inception Distance (FID) is widely used to evaluate generative models, but its empirical plug-in estimator suffers from finite-sample bias [BSAG18, CF20]. We study the sample complexity $n$ of estimating FID to error $\epsilon$ between $d$-dimensional Gaussians with bounded mean distance and covariances, when one distribution is known. Our contributions are threefold. (1) We establish tight finite-sample $\Theta(\frac{d^2}{n})$ bias and $\Theta(\frac{d}{n} + \frac {d^2} {n^2})$ variance bounds for the empirical plug-in estimator, establishing a $\gtrsim d^2$ sample complexity. (2) To debias the empirical plug-in estimator, we generalize the ${\rm FID}_\infty$ estimator of [CF20] to extrapolation methods of arbitrary order $k$. We further prove tight bias and variance bounds of $\Theta(\frac{d^{k + 2}}{n^{k + 1}})$ and $\Theta(\frac d n + \frac{d^2}{n^2})$ for any order-$k$ extrapolation under our framework. (3) We introduce Relative Taylor Debiasing (RTD), a new, computationally efficient FID estimation algorithm using debiasing techniques inspired by U-statistics. We show that RTD achieves an $O(\frac d {\epsilon^2})$ sample complexity, and prove that this is optimal. We provide a complementary empirical evaluation of our new estimators. Our experiments on synthetic Gaussians validate the predicted residual bias and support the tightness of our bounds. On ImageNet with Inception embeddings, RTD achieves the lowest mean estimation error at the standard 50K sample budget, while our second-order variance-aware extrapolation estimator (VALE$_2$) uses only 10K samples to achieve accuracy comparable to FID$_\infty$ at 50K samples.

We study the query complexity of finding a Tarski fixed point over $[n]^k$. Previous work has left a large gap between $\smash{{\Omega}(\log^2 n)}$ and $\smash{\log^{O(k)}n}$. We show that both of the previous upper and lower bounds were far from tight: for every $k\geq 3$, \[ \Omega\left((\log n)^{\frac{1}{2}\lceil \log k\rceil}\right)\le \operatorname{Tarski}(n,k)\le O\left(5^k(\log n)^{\lceil \log k\rceil}\right). \] Succinctly, up to the fixed-parameter factor of $5^k$, the complexity is settled at $(\log n)^{\Theta(\log k )}$. In particular, we obtain the first super-polynomial query lower bound for this problem.

We study a Continuous-Variable Quantum Approximate Optimization Algorithm (CV-QAOA) for high-dimensional continuous optimization. Our formulation extends an earlier CV-QAOA proposal with a variationally optimized initial state and recovers the convergence guarantees of Quantum Hamiltonian Descent (QHD) in the high-depth limit. We prove rigorous performance guarantees of CV-QAOA on several families of cost functions. First, we show $d$-step CV-QAOA minimizes any $d$-dimensional strictly convex quadratic function with $2d$ quantum queries to the cost function. We then analyze a family of nonconvex "Rotated Double Well" (RDW) functions with $2^d$ local minima introduced by arXiv:2311.00811. While prior work showed QHD reaches its global minimum with $\tilde O(d^3)$ queries, we prove that 1-step CV-QAOA solves RDW with just two quantum queries. Although general-purpose classical solvers need superpolynomial time for RDW and structure-awareness can reduce the cost to polynomial time, we show that the 1-step CV-QAOA protocol can be efficiently dequantized, and that a gradient-aligned line search succeeds with $O(d)$ queries, nearly matching the information-theoretic $\Omega(d/\log d)$ query lower bound. To move beyond the dequantizable regime, we introduce a “Rotated Square Well” (RSW) problem, whose globally flat landscape suppresses useful local gradient information. For this family, we show that an adiabatic evolution simulated by CV-QAOA can reach the global minimum using $d^{o(1)}$ queries. On the other hand, any classical algorithm that learn the hidden rotation in RSW provably requires $\Omega(d^2/\log d)$ queries, a bound we nearly match with an explicit $\Theta(d^2\log d)$-query classical algorithm.Numerical simulations on deflected corrugated spring and Easom functions illustrate the promising performance of CV-QAOA on more general problems.

We study agnostic tomography of pure bosonic Gaussian states: given copies of an arbitrary $n$-mode bosonic state $\rho$, the goal is to output a pure Gaussian state whose infidelity with $\rho$ is at most $\mathrm{opt} + \epsilon$, where $\mathrm{opt}$ is the minimum infidelity achievable by any pure Gaussian state. We give efficient protocols achieving this in both the high and low fidelity regimes. When $\mathrm{opt}$ is below some universal constant, our protocol has runtime and copy complexity which is strongly polynomial in $n, 1/\epsilon$ and $\log \log E$, where $E$ is the energy of the closest pure Gaussian state. For arbitrary $\mathrm{opt}$, our protocol uses $(n+1)^{\mathrm{poly}(1/\epsilon)} \mathrm{poly}\left(1+\log\log(E)\right)$ copies and runtime. As a corollary, we obtain the first truly tolerant Gaussianity testing protocol for distinguishing whether $\mathrm{opt} > c + \epsilon$ or $\mathrm{opt} < c - \epsilon$, for any threshold $c\in(0,1)$. We also prove $\mathrm{poly}(n,1/\epsilon)$ runtime is impossible, unless $\mathrm{NP}\subseteq\mathrm{BQP}$. Our protocols follow a shared paradigm: first, we iteratively use general Gaussian measurements combined with techniques from classical robust statistics to obtain a good warm start estimate, then we leverage non-Gaussian measurements to refine this warm start using convex and non-convex optimization methods. Interestingly, we prove that non-Gaussian measurements are necessary to match the strong agnostic guarantees we obtain, and in fact these guarantees are provably superior to what is possible for robustly estimating classical Gaussians.

We consider the problem of quantum 1-PCA: given copies of an unknown $n$-qubit mixed state, recover a classical description of its leading eigenvector. Our goal is to do so using non-adaptive and single-qubit measurements. For an $n$-qubit state with top eigenvalue $\lambda$ and spectral gap at least $\Delta > 0$, we give an algorithm that recovers the leading eigenvector to fidelity at least $1 - \varepsilon$ with high probability using $\tilde{O}\left( {2^n \cdot \eta^2} / {\Delta^3 \varepsilon^3}\right)$ copies and $\tilde{O}((2^n/\Delta \varepsilon) \cdot \operatorname{poly}(\eta/\Delta\varepsilon))$ time, where $\eta = \max (1 - \lambda, \varepsilon)$. All of our measurements are non-adaptively chosen, and performed in single-qubit Pauli bases. When the spectral gap is constant and the desired accuracy is comparable to the noise level, i.e. $\varepsilon = \Omega (\eta)$, our runtime and copy complexity become $\tilde{O} (2^n / \varepsilon)$. This generalizes the guarantees of Grewal et al. [arXiv:2601.04444], who achieved similar rates, but under the assumption $\eta = 0$, i.e., that the state was pure. Our results show that the same rates hold in the presence of state misspecification, up to polylogarithmic factors. From a technical perspective, our algorithm works by recursively constructing low-dimensional subspaces that approximately preserve the target eigenvector. To achieve nearly linear runtime dependence on the dimension of the Hilbert space, we develop a novel structured Pauli sampling scheme that enables fast batched computation of exponentially many projected Pauli matrices.

The Sachdev-Ye-Kitaev model is a strongly interacting fermionic system that has been well-studied in condensed matter and high energy physics. It is highly quantum: Gaussian states are far from the thermal state (Hastings and O'Donnell, STOC'22) and representing the thermal state requires large polynomial-size quantum circuits (Anschuetz et al., QIP'25). Very recently, it was nonetheless proven that classical algorithms can estimate local thermal expectations at sufficiently high temperature in quasipolynomial time (Zlokapa, FOCS'26). We show that classical algorithms can in fact estimate local observables at all constant temperatures in polynomial time. Our techniques also extend straightforwardly to classical systems: we resolve an open question about computing thermal expectations of a classical spin glass up to its phase transition (Bencs et al., STOC'26). Our proof develops a fully rigorous quantum cavity method. Due to the success of the classical cavity method in optimization, sampling, inference and learning, we expect the quantum cavity method to find further applications of independent interest. As an example, we give a quantum algorithm that learns SYK Hamiltonians from the Gibbs state at any constant temperature with polynomial time and sample complexity.

For $\kappa>1$, a directed graph is $\kappa$-conditioned if it is $\kappa$-mixing and its stationary distribution is approximated by the uniform distribution within a factor of $\kappa$. We present a deterministic algorithm that approximates the stationary distribution of a $\kappa$-conditioned graph to inverse polynomial relative error in $O((\log n + \log^2 \kappa) \log \log n)$ space. In the regime $\kappa = \exp(\Theta(\log^\alpha n))$ for any $\alpha \in (0, 2/3)$, our result improves the best-known $O(\log n \sqrt{\log \kappa} / \sqrt{\log \log n})$ space bound for approximating $\kappa$-step random walks in general directed graphs by [Hoza, RANDOM 2021]. We release this preliminary version due to recent rumors of LLM-based progress on related problems and uncertainty about when those results may appear. Further implementation details will be provided in a subsequent version.

We give the first polynomial improvements over the textbook algorithms for $3$SUM and All-Pairs Shortest Paths (APSP): we show how to deterministically solve $3$SUM on $n$ integers of polynomial size in $O(n^{1.9992})$ time and APSP on directed $n$-vertex graphs with polynomially bounded integer weights in $O(n^{2.9995})$ time. This refutes the $3$SUM and APSP hypotheses. Using known reductions, we also refute the real-valued versions of the $3$SUM and APSP hypotheses, the Exact Triangle hypothesis, the Zero-Weight $k$-Clique hypotheses, and the three rectangular hinted Online Matrix–Vector conjectures of van den Brand, Nanongkai, and Saranurak, and we give polynomial speedups for a variety of other problems. All of these results follow from a single new algorithm for thin matrix products. Let $X$ be an $N\times D$ integer matrix and $Y$ a $D\times N$ integer matrix with $D\le N^{1/18}$, and let $W$ be any set of at most $N^2/\sqrt D$ positions. We compute the entries $(XY)[I,J]$, $(I,J)\in W$, in $O(N^2/D^{0.063})$ operations, which is polynomially less than the time needed to write down $XY$ or to compute $N^2/\sqrt D$ inner products one by one. We design this algorithm by modifying a variant of Coppersmith's rectangular matrix multiplication algorithm, built from a ten-multiplication identity of Schönhage, to perform only the operations needed for the entries in $W$, and show that few operations are needed. Interpreted as a graph algorithm, this solves the All-Edges Sparse Triangle problem in truly subquadratic time on sparse lopsided tripartite graphs where two parts have $n$ vertices but one part has $n^{\varepsilon}$ vertices for $\varepsilon<0.12$. By known reductions, Exact Triangle, and hence $3$SUM and APSP, reduce to this problem. We also give a data structure version that answers queries for single entries of $XY$, not known in advance.

How can random order change the role of quantum memory in streaming? Later classical input can restore the usefulness of a quantum state consumed by earlier queries. We call this replenishment. We construct an artificial problem based on Hidden Matching, with repeated coordinate data and online matching requests. It admits a one-pass quantum algorithm using polylogarithmic space in uniformly random order, but unconditionally requires polynomial space both classically in random order and quantumly when all updates precede the requests. To prove the quantum lower bound, we strengthen the consumability bounds of Gilboa, Jain, and McClean for Multiple Hidden Matching. Without prior entanglement, any quantum encoding supporting $r$ independent matching requests requires $\Omega(r)$ qubits for $r\le N^{1/2-\delta}$ and every fixed $\delta\in(0,1/2)$, even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through $r=\Theta(\sqrt N)$. We also adapt Kallaugher's triangle-counting algorithm to uniformly random streams in which every edge is repeated equally often. Rebuilding the quantum sketch and resampling the classical estimator improve its expected-space bound in suitable parameter regimes. Finally, we extend the robust Noisy Gap Cycle framework of Assadi and Sundaresan to quantum streaming. A quantum communication lower bound for Block Hidden XOR yields an $\Omega(n)$ space lower bound in random edge order for large enough cycles, with consequences for several graph problems. Thus random order can enable replenishment of small quantum representations, while substantial space requirements persist for other tasks.

We design the first polynomial-time algorithms for approximately counting and almost uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the study of quantum many-body systems, and beyond. Our approach has two key ingredients. $\bullet$ We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "$\textit{soft intersection}$" satisfies a Poincare inequality uniformly over all external fields. $\bullet$ We solve a dual maximum entropy convex program to compute external fields under which the hard constraint is satisfied with high probability under the soft intersection measure. We bound this success probability directly using the uniform Poincare inequality and smallness of the gradient norm. $\textbf{AI Disclosure}$ GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper. A more complete discussion is included in the acknowledgments.

We give optimal bounds, up to polylogarithmic factors, for rounding convex bodies to near-isotropic position. A convex body $B(0,r)\subseteq K\subseteq B(0,R)$ in $\R^n$ can be rounded using $\Ot(n^3)$ membership queries; the upper bound extends to logconcave distributions. We prove a matching $\Omegat(n^3)$ lower bound, even when $R/r=n^{O(1)}$. The key lemma states that if $B(0,1)\subseteq K$ and $\Cov(\Unif(K))\preceq\kappa I_n$, with $\kappa\ge1$, then approximate uniform sampling from an initial density bounded by a constant times the uniform density uses $\Ot(n^2\sqrt\kappa)$ expected membership queries. We prove this by simulating reflected kinetic dynamics using the analysis of Eberle and Lörler~\cite{EL26:journal}. Combining this sampler with the ideas of Jia, Laddha, Lee, and Vempala~\cite{JLLV26:journal} for rounding well-rounded bodies yields our $\Ot(n^3)$ query rounding algorithm. We also give a counterexample to an ellipsoid-growth conjecture from previous papers on this topic.

We consider the following data compression problem. Given a string $x \in \{0,1\}^m$ of Hamming weight at most $n$, compress it into a shorter string $y \in \{0,1\}^s$ so that any bit $x_i$ of $x$ can be retrieved without any error using at most $t$ quantum queries to the standard oracle encoding of $y$. If queries are allowed to be adaptive we show how optimal compression up to a logarithmic factor can be achieved. If the queries are required to be made non-adaptively, we show schemes whose space is optimal in its dependence on $m$ except for a logarithmic factor, and is at most quadratically worse when compared to the optimum in its dependence on $n$.

We give a fast algorithm for solving min-cost $k$-commodity flow. The basic idea is to construct an auxiliary linear program that has low rank and whose minimum value is at most $1/k$ times the minimum value of the original problem (thus, solving this auxiliary linear program will make at least $1/k$ fraction of progress in the original problem). Low-rank linear programs can be solved quickly using black-box techniques. Thus, our algorithm runs in time $\tilde{O}(\operatorname{poly}(k)(n^{2.5}+m\sqrt{n}))$ on a directed graph with $n$ vertices and $m$ edges. We do not rely on any fast matrix multiplication.

A classical well-quasi-ordering result guarantees the existence of non-uniform linear-time algorithms for all problems in Strict NP on relational structures of bounded treedepth; however, this provides neither a procedure for constructing these algorithms nor computable bounds on their parameter dependence. We turn this existential result into a uniform algorithmic metatheorem. Given a Strict NP sentence $\varphi$ and a relational structure $\mathcal{R}$, our algorithm decides whether $\mathcal{R}\models\varphi$ in time $f(|\varphi|, td(\mathcal{R})) \cdot |\mathcal{R}|$ for a computable function $f$, where the treedepth of $\mathcal{R}$ is measured on the Gaifman graph. The algorithm also constructs witness relations, with the polynomial exponent depending on their arity, and provides a unified framework for settling hereditary graph problems parameterized by treedepth. We also present several applications - among others, our result resolves open questions on the fixed-parameter tractability of computing the stack number, queue number, track number and twin-width parameterized by treedepth.

Braess paradox originates when latency at Wardrop equilibrium in traffic networks decreases because of removing edges. The graph-theoretic property of networks suffering from the Braess paradox was called vulnerability by Roughgarden in 2006; it was then characterized and algorithmically checked both for undirected and for directed nets. In this paper, we provide a decremental algorithm of linear amortized complexity to check vulnerability for dynamically evolving networks. The basic idea of our dynamic algorithm is to use a static algorithm that marks some edges as irrelevant for the vulnerability of the graph and ignore those edges for all subsequent runs of the decremental procedure. To get a linear amortized cost for every edge remotion, we also provide a new version of such a static algorithm that improves its complexity from O(n m^2) to O(m^2), that is in turn aligned with the cost of the best state-of-the-art static algorithm for vulnerability.

We study deterministic and randomized midpoint discretizations of Langevin dynamics for a target $\pi \propto e^{-V}$, where $0 \prec \alpha I\preceq\nabla^2V\preceq\beta I$ and $\kappa=\beta/\alpha$. To achieve $\sqrt\alpha\,W_2\leqslant\varepsilon$, we show that deterministic Heun uses at most $\widetilde O(\kappa^{4/3}d^{1/3}\varepsilon^{-2/3})$ gradient queries, and underdamped exponential midpoint uses $\widetilde O(\kappa^{5/4}d^{1/4}\varepsilon^{-1/2})$. The proofs exploit cancellation at stationarity and smoothing using techniques from Malliavin calculus, outperforming previous upper bounds based on standard couplings. At bounded condition number, a lower bound matches the $d$ and $\varepsilon$ powers of both deterministic methods. To contrast, for the randomized midpoint methods and Poisson midpoint with at least two grid points (both overdamped and underdamped variants), a simple Gaussian calculation yields a lower bound $d^{1/3}\varepsilon^{-1/3}$ to get an $\varepsilon$-close sample despite starting at a benign initialization. This shows surprisingly that in high dimensions, deterministic discretizations can outperform their random counterparts.