Henry Fleischmann, Sagnik Mukhopadhyay, Swagato Sanyal
We study $s$--$t$ reachability in an unweighted directed graph on $n$ vertices, accessed through a directed cut oracle: given $S\subseteq V$, the oracle returns the number of edges directed from $S$ to $V\setminus S$. Directed cut functions are submodular, and reachability asks whether the minimum $s$--$t$ cut has positive value. This connects the problem to the query complexity of submodular function minimisation. Whether or not reachability can be solved in truly subquadratic query complexity has been an outstanding open question for a decade. We give a simple algorithmic blueprint with two randomised implementations. The first uses $\widetilde O(n^{9/5})$ outgoing-cut queries; the second improves this bound to $\widetilde O(n^{5/3})$. The common idea is to construct an alternative orientation with the same underlying undirected graph and vertex-wise degree imbalances, thereby preserving reachability. We first approximate the target imbalances using an averaging and rounding procedure, then correct the remaining error with a flow. An earlier manuscript, made available in August 2026 in an online AI repository, reports an $\widetilde O(n^{3/2})$ query bound. Our contribution is a simple and comprehensible alternative approach that already achieves a polynomial saving over quadratic query complexity.
We give a $0.556195$-competitive randomized algorithm for edge-weighted online bipartite matching with free disposal, improving the $0.5368$ guarantee of Blanc and Charikar. The result holds for arbitrary nonnegative edge weights and any input sequence fixed independently of the algorithm's randomness. Our algorithm builds on online correlated selection, introduced by Fahrbach, Huang, Tao, and Zadimoghaddam, and its multiway extension due to Blanc and Charikar. Our main contribution is a state-dependent primal--dual analysis of correlated selection. At each weight threshold, we track the state of the selection process jointly with an auxiliary failure event. The analysis accounts for arrivals below the threshold, which change the state without improving the matching value at that threshold. A potential function bounds the dual increase in both cases. We instantiate the analysis with an explicit win map that assigns a common score to an initial range of states. Its selection inequalities are verified by a finite computer-assisted proof covering every required candidate multiplicity and every real threshold; the verifiers are available in an accompanying repository. Triangular unit-weight inputs asymptotically attain the guarantee of the underlying ideal algorithm, proving that its analysis is tight.
In nonuniform buy-at-bulk network design, we are given an undirected graph and $h$ terminal pairs with positive demands, and we must route every demand at minimum total cost when each network resource has two independent parameters: a fixed buying price paid once if the resource is used, and a per-unit rental cost (or physical length) paid proportionally to the traffic it carries. Unlike the uniform-cost or single-sink variants, the general multicommodity problem allows unrelated buying and rental metrics across arbitrary source--sink pairs, arbitrary demands, and both edge and node resources. We give a deterministic polynomial-time $O(\log^2 h / (\log\log h)^2)$-approximation algorithm for undirected nonuniform buy-at-bulk network design. The guarantee holds for edge costs, node costs, and their combination, and applies to arbitrary positive rational demands with no dependence on the number of vertices or the numerical demand ratio. This improves upon the $O(\log^4 h)$ bound of Chekuri, Hajiaghayi, Kortsarz, and Salavatipour (2010) for unrestricted instances. Our result also establishes the same $O(\log^2 h / (\log\log h)^2)$ upper bound on the integrality gap of the natural affine multicommodity flow LP, bounding integral buying and rental costs separately against their fractional counterparts and preserving zero rental budgets exactly.
The matroid prophet-secretary problem asks an online algorithm to select an independent set of elements whose nonnegative values are drawn independently and revealed in uniformly random order. Its combinatorial-auction counterpart asks the algorithm to allocate disjoint bundles to arriving buyers with independently drawn XOS valuation functions. In both problems the distributions are known, and the benchmark is the expected offline optimum. Ehsani, Hajiaghayi, Kesselheim, and Singla (SODA 2018) give $(1-1/e)$-competitive algorithms for both settings. In this work we show that these two guarantees have different limits. For every matroid, we give a randomized $(1-1/e+\varepsilon_0)$-competitive algorithm, where $\varepsilon_0=10^{-6}$. The guarantee holds for independent nonnegative values with finite expected optimum. For explicitly given finite distributions with rational values and probabilities, the algorithm has polynomial worst-case running time in the matroid independence-oracle model. The oracle tests whether a queried set is independent. The algorithm makes $O(N^4)$ such queries in expectation and $O(N^4\log(N+2))$ in the worst case, where $N$ is the number of elements. For combinatorial auctions with XOS valuations, however, we show that $1-1/e$ is the optimal competitive ratio. We give an explicit family of finite instances showing that no online algorithm can guarantee a larger competitive ratio than $1-1/e$, even when the buyers' valuations are independent and identically distributed, every additive clause has coefficients in $\{0,1\}$, and computation is unrestricted. Thus the matroid improvement exploits structure absent from general allocation with XOS valuations.
We study variants of the secretary problem, where $N$, the number of candidates, is a random variable with distribution $\mathbf p$. The decision maker, given prior distributional information about $\mathbf p$, wants to maximize the probability of success -- picking the largest number among the $N$ candidates -- using only the sequentially revealed relative ranks of the candidates. When $\mathbf p$ is fully known (full information setting), we show that single-threshold type of strategies can achieve $1/e$-approximation to the optimal probability of success among all possible strategies. In the sampling setting, we give an algorithm that finds a strategy with additive suboptimality at most $ε$ and provide its sample-complexity guarantee. We also give a sampling lower bound; for small $ε$, the upper and lower bounds match up to a logarithmic factor in $1/ε$.
A fault-tolerant (FT) connectivity oracle for an $n$-vertex graph $G=(V,E)$ determines whether $G- F$ is connected for any query set $F\subseteq E$. When the number of failures is unrestricted, prior constructions exhibited a sharp contrast between randomization and determinism: randomized oracles use only $\widetilde{O}(n)$ space [Duan and Pettie, SICOMP 2020], whereas no deterministic construction was known to improve upon storing $G$ explicitly, requiring $Θ(n^2)$ bits in the worst case. The promise that $F\subseteq E$ is essential for subquadratic space: allowing arbitrary queries $F\subseteq V\times V$ forces quadratic space. For queries containing at most $f$ failures, Khanna, Konrad, and Putterman [SODA 2027] recently constructed deterministic oracles using $\widetilde{O}(\sqrt{f}n)$ space that also report approximate distances with polylogarithmic stretch. Their result improves upon the classical $O(fn)$ space bound for deterministic connectivity oracles [Pătraşcu and Thorup, FOCS 2007], but offers no space savings over storing $G$ when $f=Θ(n^2)$. We present the first deterministic connectivity oracles with subquadratic space for failure sets of unrestricted size, using $\widetilde{O}(n^{7/4})$ space. We also establish the first polynomial separation between deterministic and randomized connectivity oracles, proving that every deterministic oracle requires $\widetildeΩ(n^{3/2})$ bits in the worst case. For queries containing at most $f$ failures, we obtain deterministic connectivity oracles using $\widetilde{O}(f^{2/5}n)$ space, improving the dependence on $f$ in the bound of Khanna et al., and prove a lower bound of $\widetildeΩ(f^{1/4}n)$ bits.
Timothy M. Chan, Reina Itakura, Zhiyuan Ma, Yuancheng Yu
We present truly subquartic algorithms for detecting or counting 4-hypercliques in a 3-uniform hypergraph with $n$ vertices, disproving a popular hypothesis in fine-grained complexity. The result immediately implies a faster algorithm for finding a $k$-hyperclique in a 3-uniform hypergraph for all constant $k\ge 4$, and a faster exponential-time algorithm for Max-3-SAT.
Jan Dreier, Nikolas Mählmann, Michał Pilipczuk, Szymon Toruńczyk
A graph class $\mathcal{C}$ is monadically dependent if one cannot interpret all graphs in vertex-colored graphs from $\mathcal{C}$ using a fixed formula of first-order logic. We prove that every monadically dependent graph class has almost bounded merge-width, thereby confirming a conjecture of the first and the last author [STOC '25]. Our proof is algorithmic and the witnessing decompositions in the form of merge sequences can be computed in nearly-cubic time. This is a step towards the proof of fixed-parameter tractability of model checking first-order logic on monadically dependent graph classes, which is achieved in a companion paper.
In the cut-query model, we have access to a (directed) graph via an oracle and we can query the size of the (directed) cut of a given subset of the vertices. One of the most elementary tasks in this model is to decide if there is a path between two fixed vertices $s$ and $t$. While many results are known for undirected graphs, much less is understood for directed graphs in the cut-query model. Even for the basic task of $s$-$t$ reachability, the best known randomized algorithm is to reconstruct the entire graph with a technique by Grebinski and Kucherov using $O(n^2 / \log n)$ queries. On directed acyclic graphs (DAGs), we obtain a deterministic single-source reachability algorithm using $O(n \sqrt{n \log n})$ queries. The result is based on a topological sort algorithm and can also be adapted to compute single-source shortest paths in DAGs. On general directed graphs, we obtain deterministic algorithms for single-source reachability, strongly connected components and minimum $s$-$t$ cut using $O(n^{9/5}\log n)$ queries. The algorithm is simple and relies on two natural greedy subroutines and a modified version of Dinic's algorithm.
Hadas Barabash, Jim Geelen, Rose McCarty, Caleb McFarland, Paul Wollan
We give an $O_H(n^5)$-time algorithm to test whether an $n$-vertex labeled tour graph has a full immersion isomorphic to a labeled tour graph $H$. A labeled tour graph is a connected $4$-regular graph whose edges are labeled by binary vectors of fixed length. To obtain a full immersion, a vertex may be removed only by pairing up its four incident edges and "splitting off" both pairs, summing the labels within each pair. Even for one-bit labels, no polynomial-time algorithm was previously known for arbitrary fixed $H$. Our main tool is of independent interest: we give an algorithmic min-max theorem that, in particular, characterizes the maximum number of odd-length circuits in a rooted circuit decomposition of an Eulerian graph. Our results have implications for vertex-minors, whose study is motivated by applications in quantum networking, connections with graph minor theory, and the goal of developing an irrelevant vertex technique for dense graphs. For vertex-minors, circle graphs play the same role as planar graphs do for graph minors. As a corollary of our main theorem, we obtain a fixed-parameter tractable algorithm for apex-vertex-minor testing in $k$-apex circle graphs. We also prove an irrelevant vertex theorem for such graphs of large rank-width. More broadly, our results are essential to forthcoming work of three of the authors proving a structure theorem for graphs with a forbidden vertex-minor. There, our min-max theorem plays the role of both the algorithmic $A$-paths and Two Paths Theorems.
Finnah, Gönsch and Otto [Eur. J. Oper. Res. 2025] recently introduced the Data Validation Problem (DVP): Given a transitive directed acyclic graph $G$ on $n$ vertices (resp. tasks) and $m$ edges (resp. precedences) and a probing budget, a policy can successively pay a probing fee and probe an edge in $G$ as long as the total probing fees do not exceed the budget. If a probe succeeds, the targeted edge $e$ is removed from $G$ and we receive a reward; if the probe fails, the edge stays in $G$ and cannot be probed again. We seek a policy with maximum expected cumulative reward. Crucially, $G$ must stay transitive throughout the probing process, so a policy can only challenge the precedence between consecutive tasks, that is, only probe edges in the transitive reduction of the current precedence graph. We show that DVP is strongly NP-hard and W[1]-hard with the probing budget $T$ as a parameter, even in the deterministic case (when every probe has probability $0$ or $1$ to succeed) and with unit probing costs and rewards. For the same setting, we give an approximation-preserving reduction from the Densest $k$-Subgraph (D$k$S) Problem, with the consequence that there is no polynomial-time $2^{(\log n)^{1-δ}}$-approximation algorithm and no FPT $\sqrt{T}$-approximation algorithm for DVP, assuming the Almost-Exponential Time Hypothesis and the Strongish Planted Clique Hypothesis, respectively. We also present the first known policies with provable worst-case guarantees for DVP, namely an $m$-approximation algorithm for instances with uniform probabilities and probing costs, a $2T/p_{\min}^T$-approximation algorithm for general instances, where $p_{\min}$ is the minimum probing success probability, and an exact FPT algorithm on tournament graphs, in the deterministic setting.
Triangle finding is a fundamental problem in graph theory that has played a central role in the development of quantum algorithmic techniques. For $N$-vertex graphs given by their adjacency matrices, a long line of work has improved the quantum query upper bound from $O(N^{3/2})$ to $O(N^{5/4})$. However, the best known lower bound has remained the elementary $Ω(N)$ bound obtained from unstructured search. We prove the first superlinear quantum query lower bound for triangle finding, showing that $Ω(N^{13/12})$ queries are necessary. Our result also implies the same lower bound for triangle detection and for detecting every fixed connected non-bipartite graph. The proof builds on the compressed oracle approach of Zhandry and the transfer approaches of Belovs. It compares the computation on graphs with a planted triangle to the computation on random graphs conditioned to be triangle-free, through a map on the input registers that almost commutes with every query. We average over local changes to the input while preserving its distribution, through a procedure known as conditional resampling. Our main technical contribution is a set of estimates that let us control the query error under this averaging, despite the dependence introduced by conditioning a dense graph to be triangle-free.
We study the generalized sorting problem where only a subset of the pairs can be compared. Equivalently, we are given an $n$-vertex $m$-edge graph $G=(V, E)$ where the vertex set $V$ is to be sorted, and the edge set $E$ indicates possible comparisons. The goal is to sort $V$ with as few (adaptive) queries to $E$ as possible. When $G$ is the complete graph, the problem is equivalent to standard sorting for which $Θ(n \log n)$ queries are sufficient and necessary. All classical sorting algorithms crucially rely on the ability to compare any pair and do not imply an $o(n^2)$-query algorithm in the general case. This was improved 15 years ago by Huang, Kannan, and Khanna [FOCS'11] who achieved a bound of $\widetilde{O}(n\sqrt{n})$. Kuszmaul and Narayanan [FOCS'21] later gave a bound of $\widetilde{O}(\sqrt{m n})$, which is more efficient for sparse graphs, but matches the $\widetilde{O}(n\sqrt{n})$ barrier for dense graphs. In this paper, we present an algorithm for generalized sorting that makes at most $\widetilde{O}(n^{4/3})$ queries, therefore breaking the longstanding $\widetilde{O}(n\sqrt{n})$ barrier for all graphs. The key to our result is to query edges that belong to many (specific types of) four-cycles. We show that querying such edges allows the algorithm to verify the direction of many other edges (specifically, their opposing edges in the four-cycles) and thus achieve a much better amortized query cost.
Adam R. Klivans, Konstantinos Stavropoulos, Sergei Tikhonov, Arsen Vasilyan
In an important recent work, Blanc (2026) gave an algorithm for robustly learning Boolean concept classes with respect to a fixed distribution that outputs a (randomized) classifier achieving the optimal error of $η+ \varepsilon$ where $η$ is the noise rate. In contrast, it is well known that deterministic hypotheses cannot achieve error less than $2η+ \varepsilon$. Blanc's algorithm is computationally inefficient, and the main problem left open in his work is to find a polynomial-time algorithm given access to an oracle for empirical risk minimization (ERM). Here we resolve this problem and give such an algorithm. Additionally, we give an efficient algorithm (no ERM oracle required) for robustly learning any function class that admits low-degree polynomial approximators with respect to hypercontractive distributions. This captures basic settings including halfspaces with respect to Gaussian marginals. Our techniques make new connections to various types of online convex optimization algorithms. As a surprising consequence, we show that efficient SQ learnability in the agnostic setting implies efficient learning with contamination (with respect to Gaussians or uniform over the hypercube).
For over fifty years, combinatorial algorithms for Boolean matrix multiplication (BMM) and triangle detection saved only polylogarithmic factors over cubic time. At STOC 2024, Abboud, Fischer, Kelley, Lovett, and Meka broke this barrier with a deterministic $O(n^3 / 2^{Ω((\log n)^{1/7})})$-time algorithm, and asked whether $n^3 / 2^{Ω(\sqrt{\log n})}$ time is achievable. We answer this question in the affirmative, giving deterministic combinatorial algorithms for both BMM and triangle detection running in $O(n^3 / 2^{Ω(\sqrt{\log n})})$ time. Following Abboud et al., we decompose $(A,B)$ into pairs of submatrices $(A_k,B_k)$, called \emph{pieces}, each of which is either sparse or pseudorandom in a sense that forces $A_k\star B_k$ to be nearly all-ones on its vertex sets. Whereas their decomposition produces $2^{O(d^7)}$ pieces at regularity parameter $d$, we obtain $2^{O(d^2)}$ pieces, directly improving the runtime exponent from $(\log n)^{1/7}$ to $\sqrt{\log n}$. Prior work found pseudorandom pieces by repeatedly restricting to denser submatrices, paying worst-case vertex losses at each step that compounded across nested recursion levels, together with an expensive derandomization step. We eliminate the extra powers of $d$ by matching the vertex loss at each step to the size of its density jump so losses telescope across scales, derandomizing witness search via $L_d$-norm estimation so the sampler length stays $2^{O(d)}$, and bounding overlap across the recursion with a fractional density-weighted volume budget rather than raw submatrix volume.
The online $k$-taxi problem is a natural generalization of the $k$-server problem: $k$ taxis must serve a sequence of requests in a metric space. Each request is a pair of points $(r,s)$, and the algorithm serves the request by first moving a taxi to $r$ (to pick up a passenger) and then to $s$ (to drop off the passenger). The cost is the total distance traveled by the taxis without a passenger. Previous bounds for general $k$ all depended on the number of points $n$ or the aspect ratio $Δ$ of the metric space, both of which can be infinite on general metrics, and competitive algorithms were previously known on general metrics only for $k \le 3$. In this work, we give a $9^{k-1}$-competitive randomized algorithm against adaptive online adversaries and, as a corollary, an $81^{k-1}$-competitive deterministic algorithm for the online $k$-taxi problem.
Good approximations have been attained for the sparsest cut problem by rounding solutions to convex relaxations via low-distortion metric embeddings. Recently, Bryant and Tupper showed that this approach extends to the hypergraph setting by formulating a linear program whose solutions are so-called diversities which are rounded via diversity embeddings into $\ell_1$. Diversities are a generalization of metric spaces in which the nonnegative function is defined on all subsets as opposed to only on pairs of elements. We show that this approach yields a polytime $O(\log{n})$-approximation when either the supply or demands are given by a graph. This result improves upon Plotkin et al.'s $O(\log{(kn)}\log{n})$-approximation, where $k$ is the number of demands, for the setting where the supply is given by a graph and the demands are given by a hypergraph. Additionally, we provide a polytime $O(\min{\{r_G,r_H\}}\log{r_H}\log{n})$-approximation for when the supply and demands are given by hypergraphs whose hyperedges are bounded in cardinality by $r_G$ and $r_H$ respectively. To establish these results we provide an $O(\log{n})$-distortion $\ell_1$ embedding for the class of diversities known as diameter diversities. This improves upon Bryant and Tupper's $O(\log\^2{n})$-distortion embedding. The smallest known distortion with which an arbitrary diversity can be embedded into $\ell_1$ is $O(n)$. We show that for any $ε> 0$ and any $p>0$, there is a family of diversities which cannot be embedded into $\ell_1$ in polynomial time with distortion smaller than $O(n^{1-ε})$ based on querying the diversities on sets of cardinality at most $O(\log^p{n})$, unless $P=NP$. This disproves (an algorithmic refinement of) Bryant and Tupper's conjecture that there exists an $O(\sqrt{n})$-distortion $\ell_1$ embedding based off a diversity's induced metric.
Marek Eliáš, Gautam Kamath, Janardhan Kulkarni, George Z. Li, Jingcheng Liu, Sepideh Mahabadi, Pan Peng, Jakub Tarnawski 等 10 位作者
We give a polynomial-time $(\varepsilon, δ)$-edge-differentially private algorithm that releases a synthetic graph approximately preserving all cut values of a nonnegative weighted input graph $G$. Our algorithm outputs a nonnegative weighted graph $H$ such that, with probability at least $1-β$, simultaneously for all $S\subseteq V$, \[ \left|w_G(S,V\setminus S)-w_H(S,V\setminus S)\right| \leq γw_G(S,V\setminus S) + O\left(\frac{n}{γ\varepsilon} \log^7 \frac{n}{δβγ\varepsilon}\right). \] For fixed $γ, \varepsilon$ and $δ= β= n^{-10}$, the additive error is $n \operatorname{polylog} n$, improving upon the previous efficient bound of $n^{13/12 + o(1)}$. Furthermore, we establish an $Ω(n/γ\varepsilon)$ lower bound for the relevant parameter regime, showing that our result is tight up to polylogarithmic factors. Our algorithm uses a log-det-regularized SDP to generate fractional queries for multiplicative-weights updates and certify accuracy for all cuts. The privacy analysis charges privacy loss only fractionally per mechanism call, based on the amount of mass on edge coordinates retired from subsequent iterations. Our synthetic graph release yields improved error bounds for several downstream tasks in private graph analysis including correlation clustering, Max-Cut, Max-$k$-Cut, constrained cut problems and hierarchical clustering. Most notably, for unweighted complete-graph correlation clustering, we achieve the optimal $\widetilde O(n)$ additive error while reducing the best previously known approximation ratio for this additive error from several thousand to $1.387$. We also provide a formalization of our results in Lean.
We give randomized algorithms for approximately counting and nearly uniformly sampling independent sets in arbitrary bipartite graphs (#BIS). At fixed accuracy, both algorithms run in $\exp(O(n^{1/3}\log^{4/3}n))$ time on $n$-vertex graphs. More generally, the same running-time dependence on the number of variables holds for approximate counting and sampling with efficiently evaluable positive log-supermodular weights on Boolean variables. A weighted Sinclair-Jerrum walk on the self-reduction tree reduces these tasks to coarse lower estimates of residual log-partition functions. We obtain these estimates by optimizing a concave relaxation of an exact variational principle on the tree; threshold smoothing certifies that the low-degree search suffices.
Panagiotis Aivasiliotis, Hubie Chen, Andreas Göbel, Matthias Lanzinger, Marc Roth, Markus Schneider
The Weisfeiler--Leman (WL) algorithm is one of the most influential heuristics for the graph isomorphism problem. The expressive power of WL has been extensively studied in the contexts of descriptive complexity, logics, graph neural networks, and the theory of homomorphism indistinguishabily. Notably, two graphs are indistinguishable by the $k$-dimensional WL algorithm if and only if they are indistinguishable by homomorphism-counts from graphs of treewidth at most $k$. An intrinsic question is to find a natural version of the WL algorithm for relational structures of higher arity admitting an equivalent characterisation via homomorphism indistinguishability along bounded generalised hypertree width (GHW). Scheidt and Schweikardt solved this for $k=1$ by defining the RCR algorithm and showing indistinguishability from $α$-acyclic structures. In this work, we resolve this for all $k\ge1$: we develop a $k$-dimensional version of RCR ($k$-RCR for short) and show that two structures $\mathcal A$ and $\mathcal B$ are indistinguishable by $k$-RCR if and only if they have the same number of homomorphisms from all structures $\mathcal C$ of generalised hypertreewidth at most $k$. Furthermore, we show that on a structure $\mathcal A$ over a fixed signature, with $\mathsf{ghw}(\mathcal{A})\leq k$, $k$-RCR runs in $O(\|\mathcal A\|^k\cdot\log(\|\mathcal A\|))$, where $\|\mathcal A\|$ is the number of all tuples that appear in the relations of $\mathcal A$.
How much information about a graph is needed to determine its connectivity after arbitrary valid edge deletions? This question is at the heart of designing deterministic dynamic streaming algorithms for graph connectivity. Towards answering this question, we study the deterministic one-way communication problem in which Alice holds a graph $G_A = (V, E_A)$ on $n$ vertices, Bob holds a graph $G_B = (V, E_B)$ with $E_B \subseteq E_A$, and Alice sends a message $M \in \{0,1\}^m$ that allows Bob to decide whether $G_A - G_B$ is connected. Randomized sketches solve this problem using $\tilde O(n)$ bits, whereas deterministic recovery of even one surviving edge requires $Ω(n^2)$ bits. We prove an upper bound of $m = \widetilde{O}(n^{7/4})$ bits and a lower bound of $m = \widetildeΩ(n^{3/2})$ bits for deciding connectivity. Thus, connectivity admits a subquadratic deterministic representation despite the quadratic cost of edge recovery, but remains polynomially harder than its randomized counterpart. The upper bound immediately yields an $\widetilde{O}(n^{7/4})$-bit deterministic $n^2$-fault tolerant connectivity oracle, which reports the connected components of a graph $G$ after any valid set of edge deletions. We also obtain an $\tilde Ω(n f^{1/4})$-bit lower bound for deterministic $f$-fault tolerant connectivity oracles for every $f \in [1,n^2]$. Finally, we show that the separation between connectivity and edge recovery persists in deterministic single-pass dynamic streams with bounded update multiplicity. For every fixed $ε\in (0,1/3)$, our algorithm reports the connected components of the final simple graph using $\widetilde{O}(n^{15/8 + 3ε/8})$ bits (subquadratic space), provided each edge undergoes at most $n^ε$ insertion or deletion updates in total.
Renaud Gaucher, Ephraim Linder, Adam Smith, Jonathan Ullman, Lydia Zakynthinou
We study the limits of differentially private Gaussian mean estimation when the covariance matrix is unknown. Prior work shows that when the covariance $Σ$ is known, the optimal sample complexity can depend only on its spectrum, rather than on the ambient dimension $d$. We establish new lower bounds for Gaussian mean estimation with Euclidean error which show that private estimators cannot achieve similar guarantees when the covariance is unknown. First, when the sample size $n$ is smaller than the effective dimension $\mathrm{trace}(Σ)/\|Σ\|_{\text{op}}$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d/n}$. Second, when the sample size is larger than the effective dimension, even for diagonal covariance matrices with effective dimension bounded by $\log d$, the optimal error when the covariance is unknown can be larger than the optimal error when it is known by a factor of $\sqrt{d}/n^{3/2}$. Together, these results show that in the unknown covariance setting, exploiting anisotropic structure under differential privacy can require a dimension-dependent error or sample complexity.
Tenindra Abeywickrama, Muhammad Aamir Cheema, Sabine Storandt
Many spatial network applications, such as map-based services, rely heavily on efficient methods to search for relevant nearby points-of-interest (POIs). A $k$ Nearest Neighbor ($k$NN) query is one such example that finds the $k$ closest POIs from an agent's location. While most existing techniques for spatial networks focus on retrieving nearby POIs for a single agent, these search heuristics do not translate to many other applications. For example, Aggregate $k$ Nearest Neighbor (A$k$NN) queries require POIs that are close to multiple agents. $k$ Farthest Neighbors ($k$FN) queries require POIs that are the antithesis of nearest. Such problems naturally benefit from a hierarchical approach, but existing methods rely on Euclidean-based heuristics, which have diminished effectiveness in spatial networks such as road network graphs. We propose a novel data structure, COL-Tree (Compacted Object-Landmark Tree), to address this gap by enabling efficient hierarchical graph traversal using a more accurate landmark-based heuristic. We then present query algorithms that utilize COL-Trees to efficiently answer A$k$NN, $k$FN, and other queries. Our experiments evaluate performance on real-world and synthetic datasets including multiple types of spatial networks, such as road networks of varying sizes and edge weights, terrain networks, and 2D physics meshes. The results demonstrate that our techniques significantly outperform existing approaches, achieving up to three orders of magnitude improvement. Moreover, this comes at a small pre-processing overhead in both theory and practice.
We give a dynamic algorithm that maintains a near-optimal edge coloring in constant expected time per update. For every fixed $ε>0$, the algorithm maintains a $(1+ε)Δ$ edge coloring under edge insertions and deletions in $\mathrm{poly}(1/ε)$ worst-case expected update time against an oblivious adversary, provided $Δ\geΔ_ε=\widetildeΘ(1/ε^2)$. The previous constant-time algorithm of Bhattacharya, Costa, Panski, and Solomon (BCPS) required $Δ\ge(\log n/ε^4)^{Θ(\log(1/ε)/ε)}$, where $n$ is the number of vertices. We remove this dependence on $n$, answering their open question. Our approach differs substantially from that of BCPS, which relied on coloring in locally tree-like subgraphs. Instead, we draw on distributed algorithms that organize partial coloring through successive applications of the Lovász Local Lemma (LLL). We develop a dynamic LLL solver with worst-case expected update bounds and a dynamic splitting algorithm that partitions edges into low-degree subgraphs. A key technical contribution is a bottom-up method for iterative splitting, which avoids the logarithmic overhead of maintaining successive levels. It achieves constant expected update time by adaptively choosing which levels to inspect and repair, rather than inspecting every level after each update. We believe these dynamic splitting and LLL tools can be of interest beyond edge coloring.
We study the problem of determining the maximum number of stable matchings. This classical counting problem was posed by Knuth and has been studied for five decades. We obtain a new $2.999^n$ upper bound on the number of stable matchings with $n$ men and $n$ women. This improves the best-known previous result by Palmer and Pálvölgyi. This result gets much closer to the well-known lower bound $\frac{2.28^n}{\text{poly}(n)}$. We refine their entropy-based framework by capturing broader structural relationships among stable matchings. To capture these relationships, we introduce new frameworks, including cones and supports. This establishes the $3.332^n$ bound. We also exploit relationships among local structures and use a charging argument to improve the bound to $2.999^n$.
Jules Viennot, Arthur Wendling, Armaël Guéneau, François Pottier
We present OCaml and Rocq implementations of Kaplan and Tarjan's purely functional, real-time catenable deques. The correctness of our Rocq code is machine-checked.
We study the maximum number ${\rm SM}(n)$ of stable matchings in an instance with $n$ men and $n$ women. Karlin, Oveis Gharan, and Weber (STOC 2018) proved the first single-exponential upper bound; their argument gives ${\rm SM}(n)\le 2^{17n}$ for sufficiently large $n$. Palmer and Pálvölgyi (FOCS 2021) improved this bound to $3.55^n+O(1)$. We prove that ${\rm SM}(n)<2.81^n$. This brings the exponential base closer to $2.28$, the base of the best known lower bound. More precisely, we bound the number of stable matchings in each instance in terms of each person's number of stable partners, where a stable partner is someone matched to that person in at least one stable matching. Our proof assigns each person a weight determined by this number and uses these weights to bound the number of stable matchings by induction.
Amotz Bar-Noy, Omri Kosary, David Peleg, Dror Rawitz
In the Degree Realization problem with respect to a family $\mathcal{P}$ of graphs the input is a non-increasing sequence $d = (d_1,\ldots,d_n)$ of positive integers, and the goal is to decide whether there exists a simple undirected graph $G \in \mathcal{P}$, whose degrees correspond to $d$, i.e., such that $deg(G) = d$. In this paper we consider the version of Degree Realization in which the realization is required to be a forest (i.e., $\mathcal{P}$ is the family of forests). We consider \emph{optimized} Degree Realization in which the goal is to obtain a realization that minimizes an objective function $f$. That is, the goal is to find a realization $G$ that minimizes $f(G)$ among the realizations of the given input sequence. More specifically, we focus on the following functions: the size of an optimal \emph{vertex cover} and the size of an optimal \emph{dominating set}. We also consider the \emph{total} and \emph{paired} versions of both Min Vertex Cover and Min Dominating Set. We provide characterizations and linear time realization algorithms for all the above-mentioned problems. Since Min Paired Vertex Cover is closely related to Min Edge Dominating Set, our results for Degree Realization with respect to the former extend to the latter.
For over four decades, two classical structures have offered complementary views of mincuts. The Gomory-Hu tree encodes one $(x,y)$-mincut for every vertex pair $x,y$ in $O(n)$ space, but retains only one mincut per pair. In contrast, for a fixed pair $x,y$, the Picard-Queyranne representation admits an $O(n)$ space vertex ordering that determines whether some $(x,y)$-mincut separates any given pair $u,v$ and reports such a cut. Can these complementary guarantees hold simultaneously in $O(n)$ space? We answer affirmatively. For undirected graphs, we construct an $O(n)$ space structure encoding, for every four vertices $x,y,u,v$, an $(x,y)$-mincut separating $u,v$ whenever one exists. More precisely, given any terminal pair $x,y$, the structure recovers the Picard-Queyranne ordering in $O(n)$ time. Thereafter, separability queries take $O(1)$ time, and reporting a separating mincut $C$ takes $O(|C|)$ time. Thus, information for all $Θ(n^4)$ choices of $x,y,u,v$ is encoded in $O(n)$ space. This is essentially the strongest information of this form that fits in $O(n)$ space: if one additionally requires the mincut to place $u$ on side of $x$ and $v$ on side of $y$, then $Ω(n^2)$ space is necessary even for fixed $x,y$. As a consequence, we obtain an $O(n)$ space all-pairs mincut sensitivity oracle for a single edge failure in multigraphs, with optimal $O(n)$ time to report a new mincut. This resolves an open question of Baswana and Pandey [SODA'22] on achieving $o(n^2)$ space and $o(m)$ query time simultaneously. Our construction substantially generalizes the $O(n)$ space nearest-mincut-hierarchy [SODA'27], which reports only the nearest $(x,y)$-mincut. Recovering the entire Picard-Queyranne ordering for an arbitrary pair requires new structural properties of the Dinitz-Vainshtein [STOC'94] connectivity carcass, extending its use beyond the Steiner mincuts it represents.
Jingyang Zhao, Trung Q. Duong, Een-Kee Hong, Hyundong Shin
The traveling salesman problem (TSP) is one of the fundamental problems in combinatorial optimization that asks for a minimum-cost Hamiltonian cycle in a complete undirected graph with $N$ vertices. For the metric TSP, where edge costs satisfy the triangle inequality, a long line of work has produced constant-factor approximation algorithms, culminating in the best-known approximation ratio $α<3/2-10^{-36}$. The biased TSP (BTSP) generalizes the metric TSP to bipartitioned networks: the vertex set is partitioned into two subsets, edge costs satisfy the triangle inequality within each subset, while edges crossing the partition may violate it by a certain bias factor. This model captures heterogeneous routing scenarios in which distances are metric only locally but may be distorted across different regions or vertex types. Previously, the only known result for the BTSP was a $7/2$-approximation algorithm with running time $O(N^3)$. In this paper, we present two new approximation algorithms that significantly improve the state of the art. The first achieves an approximation ratio of $2α+\varepsilon$ (strictly less than $3$) in polynomial time $N^{O(1)}$ for any constant $\varepsilon>0$, where the exponent grows at least linearly with $1/\varepsilon$. The second achieves a $4$-approximation in only $O(N^2)$ running time, reducing the previous cubic-time bound by a full polynomial factor. These advances directly support emerging networked systems---including multi-tier 6G architectures and hybrid quantum-classical networks---where cross-region transitions incur biased and non-metric overheads.