Papers

For a matrix $A$ over a field of characteristic two, let $\Phi(A)$ be the sum of its permutation monomials indexed by odd permutations. Although determinant and permanent coincide in this characteristic, this parity sub-sum retains information that neither gives separately. We show that $\Phi(A)$ and all its first partial derivatives can be computed deterministically in $O(n^\tau)$ field operations for every $n\times n$ matrix, where $2<\tau\le3$ is any fixed admissible matrix multiplication exponent. The result includes singular matrices and the binary field. An inversion-count identity expresses $\Phi$ through complementary minors. For invertible matrices, a decomposition along two binary interval trees aggregates these minors by matrix multiplication; a Boolean border of constant size handles the remaining ranks. We also give an explicit matrix formula for the full gradient. Applied to $\Phi(I+zW)$ for a randomly weighted adjacency matrix $W$, the evaluator computes the shortest even directed-cycle length in $\widetilde{O}(n^{\tau+1})$ bit operations, improving the $\widetilde{O}(n^{\tau+3})$ bound of Björklund, Husfeldt, and Kaski with the same multiplication exponent. The same time bound recovers the union of the arcs of all shortest even cycles with high probability, and deterministically outputs the cycle under a unique-shortest-cycle promise. For general graphs, exact maintenance of a nonzero coefficient gives an $\widetilde{O}(n^4)$ algorithm that outputs a shortest even cycle with high probability. Complementary extraction methods improve this bound for short cycles and for cycles that omit few vertices.

Let $A_1,\ldots,A_n$ be independent $d \times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\sum_{i=1}^n x_i A_i$, $x\in \mathbb{R}^n$. 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 $d\rightarrow \infty, 2n/d^2 \rightarrow \alpha$ 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 $2\sigma(\alpha)$, for an explicit expression of the standard deviation $\sigma(\alpha)<1$ for all $0<\alpha<\alpha_* \simeq 5.74$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025) in this interval.

Robust optimization under interval uncertainty aims to compute solutions that perform well on a range of scenarios that are described by interval-constrained costs. In this paper, we revisit a framework introduced by Ganesh, Maggs and Panigrahi in 2020 to study the robust optimization of NP-hard problems under interval uncertainty. We start by generalizing a result in the $\ell=0$ case, which transforms a category of approximation algorithms into a robust approximation algorithm. Furthermore, in the general case, we provide a theorem that turns any local search-based approximation algorithm into a robust approximation algorithm under three newly formalized conditions over the moves of the local search algorithm. We then use this result to present the first robust approximation algorithm for Weighted $k$-Set Cover, the third NP-hard problem known to admit a robust approximation, and the first since the publication of Ganesh, Maggs and Panigrahi's paper.

We show that there is a fixed planar graph $H$, namely the $5 \times 5$ grid, such that Max Independent Set remains NP-hard in $H$-induced-minor-free graphs. This refutes the Dallard–Milanič–Štorgel conjecture and a weakening of it by Gartland and Lokshtanov, and by Korhonen.

Convex Optimization with Nested Evolving Feasible Sets (CONES)} was introduced in \cite{CONESVaze} where the objective function \(f\) remains fixed but the feasible region evolves over time as a nested sequence \(S_1 \supseteq S_2 \supseteq \cdots \supseteq S_T\). The goal of an online algorithm is to simultaneously minimize the regret with respect to hindsight static optimal benchmark and the total movement cost $M_\cA(T)$ while ensuring feasibility at all times. CONES is an optimization-oriented generalization of the well-known nested convex body chasing (NCBC). In this paper, we extend CONES to allow for loss functions $f_t'$s to also change over time. When all loss functions are convex, we show that the projected proximal algorithm achieves $O(T^{1-\beta}), O(T^\beta)$ simultaneous regret and movement cost, respectively, for any $\beta \in [0,1)$, over a time horizon of $T$. We also show that any weakly adaptive online algorithm with $O(T^\beta)$ regret has a movement cost of $\Omega\left(T^{\frac{1-\beta}{2}}\right)$ for any $\beta \in [0,1)$. When all loss functions are strongly convex, we show that the projected proximal algorithm simultaneously achieves $O(1)$ regret and a movement cost of $O(\log T)$. To complement this, we show that any online algorithm with sublinear anytime regret has a movement cost of $\Omega\left(\log T\right)$.

Strong Connectivity Augmentation (SCA) asks whether a directed acyclic graph can be made strongly connected by adding at most $k$ prescribed links whose total weight is within a given budget. Klinkby, Misra, and Saurabh (SODA 2021) gave an $O^*(2^{O(k\log k)})$-time algorithm and asked whether the problem admits a single-exponential parameterized algorithm and a polynomial kernel. We answer both questions affirmatively: SCA can be solved in $O^*(9^k)$ time and admits a polynomial kernel with $O(k^4)$ vertices and $O(k^{16})$ bits. For unweighted SCA, we obtain $O^*(4^k)$ time and a kernel with $O(k^3)$ vertices. Our algorithms are based on a particularly simple reduction to Strongly Connected Spanning Subgraph with two edge costs.

For every fixed $0<\varepsilon\le1$, we give a deterministic strongly polynomial algorithm that, given a nonnegative matrix $A\in\mathbb{R}_{\ge0}^{n\times n}$, returns $Q$ satisfying $\operatorname{per} A\le Q\le(1+\varepsilon)^n\operatorname{per} A$.

Numerical integration studies how well one can estimate the integral of a function $f$ over $[0,1)^d$ using $n$ sample points. The two classical methods, Monte Carlo (MC) and quasi-Monte Carlo (QMC), have complementary strengths and weaknesses, and a fundamental question is to design an approach that combines the benefits of both. Recently, building on the transference principle in discrepancy theory, Bansal and Jiang~\cite{BJ25a} gave a randomized QMC method that bridges MC and QMC guarantees using only i.i.d.\ samples. Their method also goes beyond the classical Koksma–Hlawka inequality: it achieves integration error $\widetilde{O}_d(\sigma_{\mathsf{SO}}(f)/n)$, where the smoothed-out variation $\sigma_{\mathsf{SO}}(f)$ can be substantially smaller than the Hardy–Krause variation that governs the classical bound. However, their algorithm requires $n^2$ i.i.d.\ samples as input, and this quadratic blowup is inherent to any method based on the transference principle. In this work, we bypass the quadratic blowup: for any constant $\varepsilon > 0$, we show that $(1+\varepsilon)n$ i.i.d.\ samples suffice to both obtain the beyond-Hardy–Krause guarantee of~\cite{BJ25a}, resolving an open problem posed there, and to produce low-discrepancy point sequences. Our algorithms are variants of the online Haar-thinning method of Dwivedi, Feldheim, Gurel-Gurevich, and Ramdas~\cite{DFG+19}.

The small-world phenomenon was given an algorithmic foundation by Kleinberg, who showed that in an augmented $k$-dimensional lattice a decentralized greedy algorithm delivers a message in $O(\log^2 n)$ expected steps. We study predicted-greedy routing, in which a mobile agent forwarding the message moves at each step to the neighbor minimizing a noisy $(\varepsilon,\delta)$-prediction of its distance to the target, redrawn at every step from an oracle conditioned on the full routing history. Two cases arise from what this agent can observe. An agent with the coordinate awareness can still compute lattice distance exactly, but not graph distance in the shortcut-augmented network, since that depends on the shortcuts of nodes it has not yet visited; given an $(\varepsilon,\delta)$-prediction of graph distance, information the classical model never supplies, it achieves expected delivery time $O(\log n/(1-4k\varepsilon\delta))$, an asymptotic improvement over $\Theta(\log^2 n)$. An agent with no coordinate awareness at all, the natural model for a privacy-preserving network whose nodes never disclose their coordinates, cannot compute even lattice distance; given an $(\varepsilon,\delta)$-prediction of lattice distance instead, it still reaches the target in $O(n/(1-4k\varepsilon\delta))$ expected steps. Together these results show that a modest amount of predicted information, of the right kind, is enough to accelerate decentralized routing well below Kleinberg's classical bound, and that even when nodes reveal no coordinates at all, reliable delivery remains achievable.

This paper studies fundamental graph optimization problems under differential privacy (DP) and shows new, reconstruction-based lower bounds. We consider a graph $G = (V, E, \vec{w})$ where the vertex set $V$ and edges $E$ are public and the weights $\mathbf{w}:E\rightarrow \mathbb{R}$ must be kept differentially private under an $\ell_1$ neighboring relation. For the problems of releasing a minimum-weight spanning tree and a minimum-weight perfect matching, we show new, tight error bounds of $\Omega(n\cdot\log(m/n)/\epsilon)$ on worst-case graphs with $n$ vertices and $m>2n$ edges. The upper bounds are known pure DP algorithms while the new lower bound holds even under approximate $(\varepsilon,\delta)$-DP as long as $\delta \leq (n/m)^{\Omega(1)}$. Our lower bounds improve the $\Omega(n/\epsilon)$ lower bounds of Sealfon (PODS~'16). The fact that approximate DP does not reduce error for MST under the $\ell_1$ neighboring relation contrasts with the recent upper bound of Pagh et al. (PODS~'25) which shows that approximate DP allows much better error under the $\ell_\infty$ neighboring relation. Going beyond worst-case graphs, we give lower bounds for large families of sparse graphs with expansion properties. We show a lower bound of $\Omega(n / \epsilon)$ for the minimum spanning tree for any graph where the minimum cut is at least $\Omega(\log(n))$. Finally, we consider the problem of private hierarchical clustering under Dasgupta's cost function (STOC~'16) and show the first approximate DP lower bound parameterized by the minimum weight of a balanced cut. This extends lower bounds of Deng et al. (ICLR~'25) to general graphs and to approximate DP.

How much memory is needed to retain the quantum speedup for collision finding? For a uniformly random function $f:[N]\to [N]$, the BHT algorithm finds a collision using $O(N^{1/3})$ queries and a quantumly accessible classical table containing $O(N^{1/3})$ input-output pairs, whereas a logarithmic-space Grover search uses $O(\sqrt N)$ queries. Determining the optimal query-space tradeoff between these extremes remains a major open problem. We resolve this equation within the class of label-symmetric algorithms, which treat the function $f$'s output labels as interchangeable. We prove that such algorithm that makes $T$ queries, uses $S$ qubits, and finds a collision in a uniformly random function $f:[M]\to [N]$ with constant probability satisfies $$T=\Omega(N^{1/3}) \qquad\text{and}\qquad T^2S=\Omega(N\log N).$$ For the setting where $M=N$, these bounds are matched by a space-efficient implementation of the BHT algorithm. As a consequence of our tradeoff, any label-symmetric algorithm for the search version of Element Distinctness on $f: [n] \to [n^2]$ must satisfy $$T=\Omega(n^{2/3}) \qquad\text{and}\qquad T^2S=\Omega(n^2\log n),$$ matching Ambainis's quantum walk. Thus, both tradeoffs are optimal within the class of label-symmetric algorithms. To prove these results, we develop a space-sensitive version of the compressed oracle technique. The compressed oracle records the information learned by the algorithm in an evolving superposition of databases. Using label symmetry and representation theory, we show that an algorithm using $S$ qubits can effectively retain information about only $O(S/\log N)$ collision-free database entries. Substituting this estimate into the compressed oracle technique yields the stated tradeoffs.

In Polyamorous Scheduling Problem, we are given an edge-weighted graph and must find a periodic schedule of matchings in this graph which minimizes the maximal weighted waiting time between consecutive occurrences of the same edge. This NP-hard problem generalises Bamboo Garden Trimming and is motivated by the need to find schedules of pairwise meetings in a complex social group. We present a $4 G^*$ algorithm for Polyamorous Scheduling, improving the previously known bound of $3 + \sqrt{5} \approx 5.236$. Our algorithm is inspired by the Deadline-Driven Heuristic which is optimal for Bamboo Garden Trimming (BGT).

We give the first deterministic polynomial time algorithm that approximates the permanent of arbitrary nonnegative rational matrices within a subexponential factor. For a matrix of order $n$, the approximation factor is \[ \exp\!\left(O\!\left(\frac{n(\log\log n)^2}{\log n}\right)\right)=\exp(o(n)). \] All previously known deterministic polynomial time guarantees for unrestricted inputs had approximation factors $\exp(\Omega(n))$. Our proof uses convex optimization to tighten an upper bound on the permanent. The bound is based on weighted sums over all matchings in a bipartite graph representing the matrix, and correlations between unmatched vertices control its error. We approximate these sums deterministically using correlation decay and a bound on the effect of vertex deletion.

We determine the optimal sample complexity of low-rank quantum state tomography when each measurement may act jointly on at most $t$ samples. For sufficiently small $\varepsilon$, estimating an unknown state on $\mathbb{C}^d$ of rank at most $r$ to trace norm error $\varepsilon$ with constant success probability requires, and is achievable with, $$\Theta\left(\frac{dr}{\varepsilon^2}\mathop{\mathrm{max}}\left\{1,\frac{r}{\sqrt{t}}\right\}\right)$$ samples. The lower bound allows the protocol to choose each joint measurement adaptively using all previous classical outcomes; the matching upper bound is nonadaptive. Thus joint measurements on at most $t$ samples improve the complexity of algorithms making single-sample measurements by at most a factor $\sqrt{t}$. Further, measuring order $r^2$ samples jointly is necessary and sufficient to attain the unrestricted collective rate. For the lower bound, we vary the support of a state with fixed uniform spectrum and bound the Fisher information trace of every joint measurement on $t$ samples. The adaptive Fisher chain rule and the van Trees inequality then give the trace norm lower bound. For the upper bound, we construct and analyze a nonadaptive tomography protocol based on a Gaussian joint measurement. An explicit second moment identity and a conditional Gaussian law outside the state's support give a rank-dependent error analysis, yielding the matching rate.

We design an adaptive two-sided error testing algorithm for the binary rank of a $0,1$ matrix $M$ with query complexity $O(d^3\log(d+1)/\epsilon^2)$, where $d$ is the tested binary rank bound and $\epsilon$ is the distance parameter. This answers an open question posed by Parnas, Ron and Shraibman~\cite{parnas2021property}, who asked if the binary rank can be tested with query complexity polynomial in $d$ and $1/\epsilon$. Furthermore, our testing algorithm can be used to find an approximate binary decomposition of $M$ with an additional $d(n+m)$ queries. That is, under the promise that the binary rank of $M$ is at most $d$, we show how to find, with probability at least $5/6$, two $0,1$ matrices $A',B'$ such that $M' = A' \cdot B'$ is a $0,1$ matrix which differs from $M$ on at most an $O(\epsilon)$ fraction of its entries. Our results also imply a testing algorithm with polynomial query complexity for the equivalent problem of testing if the edges of a bipartite graph can be partitioned into at most $d$ bicliques.

We study the vantage point selection problem, introduced by Ashvinkumar, Chowdhury, Gao, Goswami, Mitchell, and Polishchuk [WADS'25] to model the problem of estimating bottleneck capacities on the Internet. The input is a weighted undirected graph with unique shortest paths where every edge has a distinct unknown capacity. When the algorithm queries a vertex $v$, it reveals the minimum-capacity edge on the shortest path from $v$ to every other vertex reachable from $v$. The goal is to maximize the total number of revealed edges. The quality of an algorithm is measured by its competitive ratio against an optimal algorithm that knows all edge capacities a priori. We first consider the foundational single-query setting, where both the algorithm and the optimal algorithm are restricted to a single query. There is a trivial upper bound of $O(n)$ on the competitive ratio and the best known lower bound was $\tilde{\Omega}(\sqrt{n})$. We provide an algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible competitive ratio is $\tilde{\Theta}(n^{2/3})$. Furthermore, we extend our results to the general setting where the optimal algorithm is allowed $k$ queries and our algorithm is allowed $\alpha k$ queries for $\alpha\geq 1$. We present a randomized non-adaptive algorithm and matching lower bound (up to polylogarithmic factors) showing that the best possible expected competitive ratio for non-adaptive algorithms is the following surprisingly complex bound: $$ \tilde{\Theta}\left( \min\left\{ \frac{n}{\alpha k}, \max\left( \sqrt{\frac{n}{\alpha}}, \frac{n^{2/3}}{\alpha k^{1/3}} \right) \right\} \right). $$

We introduce a graph decomposition primitive called introvert clustering, which strengthens standard low-diameter clustering by guaranteeing that every clustered vertex keeps at least a $\left(\frac12-\varepsilon\right)$-fraction of its relevant neighbors in its own cluster. Repeatedly applying this primitive yields a layered introvert network decomposition with $O(\log n)$ layers and weak diameter $O(\log n)$. We give two applications in the $\mathsf{LOCAL}$ model. For every constant $\varepsilon>0$, we obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for list $\left(\frac32+\varepsilon\right)\Delta$-edge coloring on graphs of maximum degree $\Delta\geq\Delta_0(\varepsilon)$; for bipartite graphs, the result holds for all $\Delta$. For every constant $0<\varepsilon<1/4$, we also obtain a $\widetilde O(\log^2 n)$-round deterministic algorithm for a $\left(\frac14-\varepsilon\right)$-locally balanced cut, where every vertex has at least a $\left(\frac14-\varepsilon\right)$-fraction of its neighbors on the opposite side. The resulting algorithms are remarkably simple: edge coloring processes the layers in reverse order and colors each cluster, while locally balanced cut processes them forward and computes a locally maximum cut within each cluster. The introvert guarantee enables these procedures beyond the usual greedy regime of network decomposition. We construct the decomposition in $O(\log^2 n)$ randomized rounds using Miller–Peng–Xu low-diameter clustering and a simple trimming procedure, and deterministically in $\widetilde O(\log^2 n)$ rounds via a white-box adaptation of the recursive network decomposition algorithm of Ghaffari and Grunau [FOCS 2024].

The 11/6 worst-case constant of the Wang-Sitters rounding scheme, which a companion note establishes, can naturally be attributed to the freedom in Step 3, where an arbitrary valid slot matching is permitted. We show that eliminating that freedom does not improve the constant. A minimum-makespan completion oracle still has worst-case constant exactly 11/6 against the optimum; both natural 7/4 statements about it are false; and restricting Step 1 to vertices of the relaxation does not help. The loss therefore cannot be attributed solely to the freedom in Step 3. We also record what structure survives: a reduction confining every overload to two shapes, a seven-machine instance defeating the natural two-phase repair, and a strict 7/4 bound on the generalized three-path family.

Consistent submodular maximization studies the tradeoff between solution quality and stability when elements arrive over time. For a monotone submodular objective, which models diminishing returns, an algorithm maintains a set of at most $k$ available elements and changes only $O(1)$ elements after each insertion. Dütting et al. [2025] established a tight $2/3$ approximation with unrestricted computation and a polynomial-time $0.51$ approximation. They left open at STOC 2025 whether efficient algorithms can match the offline $1-1/e$ guarantee. We resolve this problem by proving that the supremum approximation achievable with polynomially many value queries and worst-case constant recourse is \[ \beta=2-\sqrt2\approx0.5858<1-1/e. \] For every $\varepsilon>0$, our randomized algorithm attains $\beta-\varepsilon$ with $O(\varepsilon^{-2})$ changes per insertion. Any fixed improvement requires exponentially many queries before one critical insertion or linear recourse of $\Omega(k)$ changes at that insertion, even with unlimited queries afterwards. This gap quantifies the cost of consistency: the current oracle hides which elements will be needed after an arrival. We also determine the exact curvature-dependent threshold $1-(\sqrt2-1)\vartheta$, attain $1-1/e-\varepsilon$ for weighted coverage with $O(\varepsilon^{-1})$ recourse, and separate the existence of universal future-price certificates from their efficient computation. Our algorithm has a bounded-bit polynomial-time implementation for polynomial-bit rational oracle answers; the lower bound uses only logarithmic-bit rational answers.

Coloring Discovery asks whether a possibly improper initial coloring can be made proper within a prescribed number of allowed changes. We study the parameterized complexity of three modification step models that were studied previously in the literature: recoloring one vertex (color flipping), swapping the colors of arbitrary vertices (color swapping), and swapping colors only across an edge (color sliding). For color flipping, we give exact fixed-parameter algorithms for the parameters vertex cover and distance to complete. For color swapping, we obtain fixed-parameter tractability for the parameter vertex cover plus the number of colors. Our lower bounds show W[1]-hardness for treedepth plus feedback vertex set in the color flipping model and for the number of colors plus bandwidth or distance to disjoint paths in the swapping and sliding models. All three variants remain NP-complete with four colors on graphs of diameter two.

In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of $\log T$ between the regret upper bound $O(d \log T)$ and the lower bound $\Omega(d)$ is unresolved (here $T$ is the total number of rounds and $d$ is the dimension). When the action set is M-convex, the regret is known to be bounded by $O(d \log d)$, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake, and applies it to online gradient descent, the online Newton step, and MetaGrad. When the forward problem is an integer linear program with a unique optimal solution, the number of mistakes is bounded, for all three, by a quantity independent of $T$; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes $O(d^2)$, that is, the factor $\log T$ is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.

We study fast algorithms for sparse-PCA certification. Given a positive semidefinite matrix $M$, the problem asks either to rule out a large $k$-sparse quadratic form or to return a high-value (relaxed) witness. The standard semidefinite relaxation provides such certificates, but existing general-purpose solvers require $\Omega(d^4)$ time. We give a bicriteria algorithm running in $O(d^2+d k^{O(\log k)})$ time: if some $k$-sparse unit vector has quadratic form greater than $2$, it returns either an $O(k^2)$-sparse unit vector or an SDP-feasible matrix of value at least $1$. For $k\leq\exp(O(\sqrt{\log d}))$, this running time is $O(d^2)$. We also go below the quadratic barrier in the sample-access model: Given $n=d^{o(1)}$ samples, our algorithm obtains a related one-sided certificate in $d^{2 - \Omega(1)}$ time for $k=\mathrm{polylog}(d)$, without forming the empirical covariance matrix. As an application, these certificate routines yield the first quadratic and subquadratic-time algorithms for robust sparse estimation for broad families of distributions. Our sparse-PCA algorithm reduces a high-value sparse direction to a bounded-radius set in the graph of large correlations and searches the resulting candidate supports. The subquadratic implementation constructs this graph using fast correlation detection.

Byzantine agreement (BA) is a foundational building block in distributed systems, and the security analysis of BA protocols under multi-instance executions has attracted increasing attention. However, most existing adversary models focus solely on party corruption and neglect important threats posed by adversarial manipulations of communication channels in the network. Through channel attacks, messages can be reordered across multiple executions and lead to violations of the protocol's security guarantees, In this work, we present the first adversary model that combines party corruption and channel attacks. Based on this model, we establish new security thresholds for Byzantine agreement under parallel and concurrent compositions, supported by complementary impossibility and possibility results that match each other to form a tight bound. For the impossibility result, we show that even authenticated Byzantine agreement protocols cannot be secure under parallel composition when $n \leq 3t$ or $n \leq 2c + 2t + 1$, where $t$ and $c$ denote the number of corrupted parties and communication channels, respectively, and $n$ is the number of parties. For the possibility result, we prove the existence of secure protocols for unauthenticated Byzantine agreement under parallel and concurrent composition, when $n > \max\{3t, 2c+2t+1\}$. We first provide general black-box compilers that transform any single-instance secure BA protocol into one that is secure under parallel and concurrent executions without additional security assumptions. To optimize performance, we further design refined compilers using erasure-correcting codes. These refined versions significantly reduce communication overhead, particularly for long messages, where they achieve a constant multiplicative overhead compared with the original protocol, thus achieving the same asymptotic communication complexity.

Motivated by data produced by generative systems, \cite{LZ26b} formulates similarity-aware statistics via a weighted similarity graph, replacing equality with similarity in classical frequency-based statistics. Although this framework captures semantic relationships between nonidentical items, under general similarity functions even coarse one-pass approximation can require linear space. We therefore ask whether the geometric structure present in natural vector similarities can overcome this barrier. We answer this question affirmatively for the Gaussian kernel. For fixed-dimensional Euclidean vector streams, we study similarity-aware analogues of classical frequency statistics, including the number of distinct elements and frequency moments, through the diversity index and Gaussian density moments. We give one-pass sublinear-space approximation algorithms that exploit the geometric and analytic properties of the Gaussian kernel, and complement them with lower bounds. Our results show that geometric structure can fundamentally change the streaming complexity of similarity-aware statistical analysis.

We study online vector balancing with $N$ random vectors in $\mathbb{R}^M$ revealed sequentially, where each vector must be assigned an irrevocable sign upon arrival. The goal is to minimize the expected $\ell^\infty$ norm of the final signed sum. For i.i.d. entries with mean zero, variance one, and a finite fourth moment, we prove that, as $M/N\to\alpha\in(0,\infty)$, the optimal value divided by $\sqrt N$ converges to a limit $R_\alpha$ independent of the entry distribution. This limit is the stochastic control value identified for Gaussian inputs by Fiedler, Jackson, Lacker, and Niles-Weed. In particular, it determines the exact asymptotic optimum for Rademacher inputs. For every $\kappa>R_\alpha$, we construct a randomized online algorithm whose final signed sum has $\ell^\infty$ norm at most $\kappa\sqrt N$ with high probability; for $\kappa<R_\alpha$, every online algorithm has vanishing success probability. Consequently, the online threshold of the symmetric binary perceptron is universal at every positive margin. The main step is a coupling that transfers Brownian controls to non-Gaussian inputs, while truncation controls rare large entries.

We study the oracle complexity of computing a point with small fixed-point residual $\|T(x)-x\| \leq \epsilon$, for a general norm $\|\cdot\|$ and a self-map $T$ of a compact convex set. We study this problem in the setting where $T$ is nonexpansive with respect to the same norm $\|\cdot\|$ and accessed via an unbiased stochastic oracle with bounded variance $\sigma^2$. We provide an algorithm that solves such instances for any norm with a weak Rademacher type $q > 1$, with high probability. The algorithm is based on a recursive anchoring technique. For type-$2$ spaces, such as $\ell_p$-spaces for $p \in [2, \infty]$, our algorithm attains stochastic oracle complexity $\tilde O(\sigma^2 \epsilon^{-3} + \epsilon^{-1})$. We further prove a near-matching lower bound (i.e., matching up to poly-log factors) for such $\ell_{\infty}$-norm instances in high dimensions. Our lower bound holds against any randomized algorithm that succeeds with constant probability. It further extends to settings with “sparse” noise, where variance measured with respect to any $\ell_p$ norm is of the same order, ruling out the possibility of improving oracle complexity as a function of $\varepsilon$ by measuring variance in a non-matching $\ell_p$ norm.

We study the approximate nearest neighbor problem under $\ell_\infty$ in the ultra-high dimensional setting where the dimension $d$ is significantly larger than the number of points $n$. Thus, we desire data structures with no dependence on $d$ in the query time. [Herold-Nanongkai-Spoerhase-Varma-Wu, SoCG 2025] introduce this problem and give data structures in $\ell_p$: for $p=1,2$, they give $(1+\varepsilon)$-approximation data structures with space $\tilde{O}(n\log d/\text{poly}(\varepsilon))$ and query time $\tilde{O}(n/\text{poly}(\varepsilon))$. Since any data structure must have query time $\Omega(\min \{n,d\})$, this query time is nearly tight. However, their results are inefficient for $\ell_\infty$, with query time $\Omega(nd)$. In order to handle the challenges of $\ell_\infty$, we introduce a notion of subset embeddings, which embed points by simply selecting a subset of dimensions. In particular, we show one may preserve all pairwise distances of an $n$ point dataset up to a factor of $O(c)$ by computing distances on only $n^{1+1/c}$ coordinates. We also show a matching lower bound: for any $c > 1$, there exists a set of $n$ points in $\mathbb{R}^{d}$ such that any subset embedding for the set with approximation $c$ must have at least $n^{1+\Omega(1/c)}$ coordinates. Using our subset embeddings, we give data structures for approximate nearest neighbor in $\ell_\infty$ with space $O(n^2\log d)$, query time $\tilde{O}(n^{1+1/c})$, and approximation $O(c\log\log n)$ for any $c \geq 1$. Finally, we give another data structure for the approximate nearest neighbor under $\ell_\infty$ with the same space and query time as our subset embedding approach, but with approximation $O(c^{\log_2 3}) \approx O(c^{1.58})$. This allows us to achieve $O(1)$-approximation with query time e.g.~$n^{1.01}$

Consider a permutation of $[n]$ whose values arrive one at a time. We resolve two questions about the space needed to decide natural properties of such input: First, computing the parity of the permutation requires $\Theta(n)$ bits, even with randomization and constant error, and a constant number of passes. Second, every permutation pattern of length three can be detected deterministically in one pass using $O(\log n)$ bits. Together with the 2026 lower bounds of Berendsohn, this completes the classification of fixed permutation patterns; The optimal space complexity is $\Theta(\log n)$ for monotone patterns and patterns of length at most three, and $\Theta(n)$ for every other pattern. As a consequence, we observe that we can verify BST traversals in streaming with logarithmic memory.

We establish a linear quantum query lower bound for high-accuracy convex optimization over an explicit family of $n$-dimensional ellipsoids. We focus on linear optimization with an explicitly given objective, where the feasible set is accessed through a membership oracle. We show that any algorithm that, for every unit linear objective, returns an exactly feasible point with additive objective error $\Theta(n^{-1/2})$ requires $\Omega(n)$ membership queries, where the ellipsoids are contained in the unit ball. The same lower bound can be shown to hold if the returned point is only required to be approximately feasible, within $\Theta(n^{-3/2})$ distance from the feasible set. This resolves an open question posed by Chakrabarti, Childs, Li, and Wu~(Quantum, 2020) and by van Apeldoorn, Gilyén, Gribling, and de Wolf~(Quantum, 2020). Coupled with the upper bounds in these papers, the query complexity of high-accuracy convex optimization is characterized tightly up to logarithmic factors. The proof is built around a lower bound for determinant computation that is derived via a novel polynomial method based on Fourier-rank. In the continuous matrix phase-query model, computing the determinant of a real $n\times n$ matrix requires at least $n/2$ matrix-vector product queries. The construction also yields an $\Omega(n)$ phase-query lower bound for estimating the minimum eigenvalue of a real symmetric $n\times n$ matrix to additive accuracy $\Theta(n^{-2})$. These results extend the determinant and minimum-eigenvalue lower bounds of Childs, Hung, and Li~(ICALP 2021) from finite fields to the real-valued setting. Based on the same constructions, we also prove an $\Omega(\min\{\sqrt\kappa,n\})$ gradient-query lower bound for constant-accuracy optimization of $1$-strongly convex and $\kappa$-smooth functions, which is optimal up to a logarithmic factor.

For an undirected graph $G = (V,E)$ and a fault bound $f$, an edge-fault-tolerant connectivity labeling scheme assigns short labels to vertices and edges, so that for any vertex pair $(s,t)$ and failed edge set $F\subseteq E$ with $|F|\leq f$, the connectivity between $s$ and $t$ in $G-F$ can be answered by inspecting only the labels of $s$, $t$ and edges in $F$. In this paper, we present a labeling scheme that uses $O(\log^{2}n)$-bit labels that can be computed in deterministic polynomial time. This improves upon the previous $\tilde{O}(\sqrt{f})$ deterministic bound of [Long, Pettie, Saranurak'25], and even slightly improves the $O(\min\{f+\log n,\log^{2}n\log f\})$ randomized bound of [Dory, Parter'21] and [Long, Pettie, Saranurak'25] when $f = \Omega(\log^{2}n)$. Moreover, for a general $f$, this is the first labeling scheme that produces an $\tilde{O}(1)$-size labeling which is simultaneously correct across all queries. Our approach combines the cycle-space-based labeling scheme from Dory and Parter with a recent result by [Knauer'26] on sparse cycle bases.