跳到正文
研踪PaperTrail

YOUR RESEARCH, IN SIGHT

让下一篇重要论文，不再错过。

覆盖计算机全学科，以 AI 为快捷入口。关注方向、作者和关键词，把发现变成持续的研究积累。

部分采集异常
正在读取账号与研究记录…

全站更新概况

+0新增论文
+0其中 AI 相关
+1已有论文修订
+100新增 AI 导读

arXiv 最近完成扫描：2026-11-09 04:05 · 全站最近入库：2026-11-08 06:02各信源进度 →

统计口径与数据新鲜度

统计 2026-11-08 04:15 — 2026-11-09 04:15（北京时间） · 点击数字查看对应简报；AI 相关包含在新增论文内。

最近完成检查：2026-11-09 04:05最近发现新增版本：2026-11-08 06:02库内论文最新更新时间：2026-11-06 02:59

每分钟刷新已保存的数据。空检查不代表采集异常；新收录可能包含历史论文。AI 导读按新增版本计数。

17,100 已收录论文14,661 AI 相关4,057 有修订112 HF 社区发现

浏览论文库

研究日报 / 周报 →按信源浏览 →

时间范围跟随排序：版本更新、本站收录或首次发表。最新收录可能包含历史论文。部分信源异常，查看状态 →

找到 447 篇论文 · 按版本更新时间排序上轮扫描起点 2026-11-09 04:03（北京时间）
新论文 · v1arXiv:2611.06717

Combinatorial Boolean Matrix Multiplication in $n^3 / 2^{Ω(\sqrt{\log n})}$ Time

Piaoyang Cui, Yiming Liu, David P. Woodruff

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06684

The Online $k$-Taxi Problem Is Competitive on All Metrics

Christian Coester, Yichen Huang, Tze-Yang Poon

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2303.04199

Diversity Embeddings and the Hypergraph Sparsest Cut

Adam D. Jozefiak, F. Bruce Shepherd

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06670

Polynomial-Time Differentially Private Synthetic Graph Release with Nearly-Linear Additive Error

Marek Eliáš, Gautam Kamath, Janardhan Kulkarni, George Z. Li, Jingcheng Liu, Sepideh Mahabadi, Pan Peng, Jakub Tarnawski 等 10 位作者

原文摘要

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.

cs.DS · math.ST

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06657

Approximating #BIS in $\exp(\widetilde O(n^{1/3}))$ Time

Jingcheng Liu, Yitong Yin

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2607.07934

Homomorphism Indistinguishability Beyond Graphs: Relational Weisfeiler--Leman and Hypertree Width

Panagiotis Aivasiliotis, Hubie Chen, Andreas Göbel, Matthias Lanzinger, Marc Roth, Markus Schneider

原文摘要

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 $\mathcal A$.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06626

Deterministic Connectivity Sketches Under Edge Deletions

Sanjeev Khanna, Aaron Putterman, Madhu Sudan

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06615

Dimension-Free Private Mean Estimation is Impossible Without the Covariance

Renaud Gaucher, Ephraim Linder, Adam Smith, Jonathan Ullman, Lydia Zakynthinou

原文摘要

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.

cs.LG · cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2601.22183

COL-Trees: Efficient Hierarchical Object Search in Spatial Networks

Tenindra Abeywickrama, Muhammad Aamir Cheema, Sabine Storandt

原文摘要

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.

cs.DB · cs.AI · cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06611

Constant-Time Dynamic Edge Coloring via Dynamic Splitting and LLL

Mohsen Ghaffari, Sijin Peng

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06604

A Sub-$3^n$ Upper Bound on the Number of Stable Matchings

Ken-ichi Kawarabayashi, Hikaru Wada, Masataka Yoneda

原文摘要

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 $2.999^n$.

cs.DS · cs.DM · math.CO

查看论文
arXiv ↗PDF ↗
版本更新 · v5arXiv:2505.07681

Verified Purely Functional Catenable Real-Time Deques

Jules Viennot, Arthur Wendling, Armaël Guéneau, François Pottier

原文摘要

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.

cs.PL · cs.DS · cs.LO

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06494

A $2.81^n$ upper bound on the number of stable matchings

Ebrahim Ghorbani, Matthias Mnich

原文摘要

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.

cs.DS · cs.DM · math.CO

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06482

Computing Forest Degree Realizations with Minimum Cardinality Domination

Amotz Bar-Noy, Omri Kosary, David Peleg, Dror Rawitz

原文摘要

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.

cs.DS · cs.DM

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06478

Beyond Gomory-Hu: All-Pairs $\times$ All-Pairs Mincuts in $O(n)$ Space

Koustav Bhanja, Merav Parter, Asaf Petruschka

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06410

Polynomial $(2α+\varepsilon)$- and Quadratic $4$-Approximation Algorithms for the Biased Traveling Salesman Problem in Bipartitioned Networks

Jingyang Zhao, Trung Q. Duong, Een-Kee Hong, Hyundong Shin

原文摘要

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.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06387

Partially Dynamic Algorithms for Single-Source Temporal Paths

Davide Bilò, Bartłomiej Bosek, Jadwiga Czyżewska, Evangelos Kipouridis, Karol Węgrzycki, Anna Zych-Pawlewicz

原文摘要

We present efficient partially dynamic algorithms for maintaining earliest arrival times from a fixed source in temporal digraphs with $n$ vertices. For the incremental scenario where updates are arc insertions, our algorithm has a constant query time and requires a total update time of $O(n + m \log m)$, where $m$ is the total number of arcs added to the arc-less graph. For the decremental scenario where updates are arc removals, our algorithm has a constant query time and requires a total update time of $O(n + m \log m)$, where $m$ is the number of arcs in the graph upon its initialization. We also consider earliest-arrival queries with an arbitrary departure-time threshold and develop an incremental algorithm with $O(n+Δ_s m\log m)$ total update time, $O(\log m)$ query time, and $O(n+m)$ space, where $Δ_s$ is the number of distinct departure times of arcs leaving the source. All of our algorithms are also capable of reporting an earliest arrival path from the source to the queried vertex $v$ in constant time per edge once the earliest arrival time at $v$ is known. We complement our positive results with conditional lower bounds based on the Online Boolean Matrix-Vector Multiplication conjecture (a.k.a. OMv conjecture). For both the incremental and the decremental scenarios, we show the non-existence of an algorithm with $O(n^{1-\varepsilon})$ amortized update time and $O(n^{2-\varepsilon})$ amortized query time, even if the algorithm is randomized and latencies on the arcs are uniform. For arbitrary departure-time queries, we also establish an OMv-based conditional lower bound showing that our incremental algorithm is optimal up to logarithmic factors on dense temporal graphs. Thus, efficient total running time is essentially the best outcome we can get. Our conditional lower bounds hold also for the case of undirected temporal graphs.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06364

Estimating the Volume of a Union of Convex Bodies

Eeshan Banerjee, Aranya Kumar Bal, Debabrota Basu, Sourav Chakraborty, Debarshi Chanda, Uddalok Sarkar

原文摘要

The \textit{Union Volume Estimation problem}, i.e. yielding an $(\varepsilon, δ)$-approximation of the volume of the union of convex bodies, is a classical problem in computational geometry and formal methods. In this paper, we first detect where and why the efficient union volume estimators with approximate sampling and PAC volume estimation oracles fail. Specifically, we show that having only TV-distance guarantees on the approximate sampler does not suffice, and demonstrate this bottleneck by constructing the sunflower families of convex bodies. In this context, we propose the concept of \textit{tail-controlled samplers} that can yield with high-probability almost-$\ell_{\infty}$ closeness. Secondly, we leverage this concept to propose a generic algorithmic framework \texttt{Alpona} that properly combines a PAC volume computation and approximate point sampling oracles with tail-controls to yield an $(\varepsilon, δ)$-estimate of the union volume for any collection of convex bodies. We show that \texttt{Alpona} needs only linear (in number of convex bodies) oracle calls to be $(\varepsilon, δ)$-correct. Finally, we provide a tail-controlled analysis of the Hit-and-Run sampler~\citep{lovasz2006fast}. We further provide a novel geometric analysis to improve the required walk length of Hit-and-Run. We further instantiate \texttt{Alpona}'s correctness for the sparse frequency streams of convex bodies: a special case of the union volume estimation problem where no convex body intersects more than $τ$ many other bodies.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2604.02285

The Computational Complexity of Avoiding Strict Saddle Points in Constrained Optimization

Andreas Kontogiannis, Ioannis Panageas, Vasilis Pollatos

原文摘要

The complexity of finding second-order stationary points (SOSPs) in constrained settings has remained unclear and has been highlighted as an open question by the works of Hollender and Zampetakis [2023] and Kontogiannis et al. [2024]. In particular, under one established definition (SOSP1) where nonnegative curvature is required only along the null space of the active constraints [Bertsekas, 1997], the problem lies in TFNP, and algorithms with $O(poly(1/ε))$ running times have been proposed. Under another established notion of stationarity (SOSP2) checking is coNP-hard in high dimensions. We investigate the computational landscape of finding SOSPs under polytope constraints. We prove that, even if promised that all KKT points lie a constant distance away from the domain's boundary--where SOSP1 and SOSP2 reduce to the standard unconstrained conditions--finding an approximate SOSP under either definition is PLS-hard, and the hardness continues to hold even in $[0,1]^2$. As a direct consequence, our result establishes a fundamental barrier on the continuity of SOSP-computing algorithms: unless PLS=CLS, no deterministic, iterative algorithm with a computationally efficient, continuous update rule can guarantee that every $ε$-approximate fixed point is an $(ε,\sqrtε)$-SOSP, even under the above interior promise. We establish PLS-completeness by proving that the problem of finding approximate SOSP1 belongs to PLS in high dimensions. In addition, we prove that, for every fixed $d\ge 2$, finding an approximate SOSP2 is also PLS-complete, escaping the coNP-hardness which appears in high dimensions. We also establish PLS-hardness and a polynomial-time search reduction to LOCALOPT for finding $(r^{2+γ},r)$-local pure Nash equilibria, for every fixed rational $γ\in(0,1]$, in two-player nonconvex common-interest games with actions in $[0,1]$.

cs.CC · cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v3arXiv:2610.31513

Closing All-Exp Gaps in Local Search for the Traveling Salesman Problem and Maximum Constraint Assignment

Sophia Heimann, Hung P. Hoang, Stefan Hougardy

原文摘要

We prove the all-exp property for the Traveling Salesman Problem with the $2$-opt neighborhood, and for $(3,2,2)$- and $(2,3,2)$-Maximum Constraint Assignment with the flip neighborhood. The all-exp property states that infinitely many instances have an initial solution from which every sequence of improving moves to a local optimum, and hence every run of local search, has exponential length. For the Traveling Salesman Problem, the all-exp property was previously known for every $k$-opt neighborhood with $k \ge 3$, so our result for $2$-opt completes the picture for all $k$-kopt neighborhoods. In $(p,q,r)$-Maximum Constraint Assignment, each constraint involves at most $p$ variables, each variable occurs in at most $q$ constraints, and each domain has size $r$. Decreasing any of the parameters of $(3,2,2)$ or $(2,3,2)$ yields a problem in which every sequence of improving flips has polynomial length, so both Maximum Constraint Assignment results are tight. All three results settle long-standing open problems. Our proofs build on a common construction idea, which reveals a close connection between the local-search dynamics of the three problems.

cs.DS · math.CO

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06250

A Polynomial Erdős-Pósa bound for Eulerian digraphs

Esther Galby, Roohani Sharma, Bart van der Steenhoven

原文摘要

In [Combinatorica 1996], Reed, Robertson, Seymour and Thomas proved Younger's conjecture dating back to 1973 that states that directed cycles have the Erdős-Pósa property. Specifically, they showed that every directed graph $D$ contains either $k$ arc-disjoint cycles or a set of at most $f(k)$ arcs that intersect every directed cycle. The function $f$, depending only on $k$, is however notoriously large--in fact, not even elementary. The first improvement on this function was only very recently provided by Hatzel, Kreutzer, Milani, and Muzi [FOCS 2024] who showed that $f$ can be taken to be an exponential tower of height $8$. In this work, we improve this function to $O(k^4)$ in the case where the digraph is Eulerian. This also answers an open question by Wahlström [Springer Monographs in Mathematics 2018] who asked explicitly when Eulerian digraphs have a polynomial Erdős-Pósa property. The only other known result about Erdős-Pósa on Eulerian digraphs is a result from Seymour [Combinatorica 1996] who showed that $f(k)=k$ when the Eulerian digraph admits a linkless embedding in 3-space, and such an equality is not possible for general Eulerian digraphs. We also complement our positive result, by showing that the result of Seymour cannot be extended to give a linear bound ($O(k)$) on general Eulerian digraphs, by showing that even on Eulerian digraphs the lower bound for the Erdős-Pósa property is $Ω(k \log k)$.

cs.DM · cs.DS · math.CO

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2604.23334

A Lagrangian Approach to Interdiction of Linear Minimization Problems

Yu Cong, Kangyi Tian

原文摘要

We study budgeted interdiction of linear minimization problems. Given a family of feasible sets, element weights, and deletion costs, the objective is to delete elements within a prescribed budget and minimize the remaining weight of a feasible set. We give an alternative, simpler Lagrangian approach that generalizes the connectivity interdiction framework of Huang, Obscura Acosta, and Yingchareonthawornchai (IPCO' 24). When the interdiction optimum is positive, we show that the optimal Lagrange multiplier gives truncated element weights, in which every feasible set supporting an optimal interdiction solution has weight less than twice the minimum under the truncated weights. We apply the reduction to graph cuts, $k$ cuts, and cocircuits of perturbed graphic matroids. We obtain deterministic fully polynomial-time approximation schemes for graph connectivity interdiction and for cogirth interdiction on perturbed graphic matroids of fixed perturbation rank. For graph connectivity, the running time is $\tilde O(n^3m^2/\varepsilon)$. The $k$-cut application gives a fully polynomial-time randomized approximation scheme.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06234

A Dichotomy for Terminal Cyclability

Matthias Bentert, Dario Cavallaro, Sebastian Wiederrecht

原文摘要

Given an undirected graph $G$ and a set $R \subseteq V(G)$ of terminals, Terminal Cyclability asks whether $G$ contains a simple cycle visiting every vertex of $R$. We establish a complexity dichotomy for this problem on red-minor-closed classes of annotated graphs. In particular, we show that Terminal Cyclability is polynomial-time solvable on every such class excluding any annotated planar graph. The problem is NP-complete whenever the class contains all annotated planar graphs. More precisely, we give an $|G|^{2^{\mathbf{poly}(b)}}$-time algorithm, where $b$ is the bidimensionality of $(G,R)$. Our main technical contribution is the introduction of a weak irrelevant vertex rule. Rather than removing irrelevant vertices until the treewidth becomes constant, we only remove irrelevant vertices until the treewidth is logarithmic in the graph size. Such a weakening is necessary as a parameter-only treewidth threshold for an irrelevant vertex rule in this setting is known to be impossible to realise. Combined with a single-exponential algorithm for Terminal Cyclability parameterized by treewidth, this yields the polynomial-time algorithm for our dichotomy.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06223

Consistent Submodular Maximization with Full Foresight

Eric Balkanski, Paul Dütting, Federico Fusco, Silvio Lattanzi, Ashkan Norouzi-Fard, Ola Svensson, Morteza Zadimoghaddam

原文摘要

Online algorithms with bounded recourse face three intertwined obstacles: uncertainty about the future, computational limits, and a limited budget for changes. We ask a fundamental question that separates consistency from the other two: what does solution stability cost when the entire future is known and computation is unlimited? We answer this question for monotone submodular maximization, where an algorithm must maintain a solution of at most $k$ arrived elements, with a constant number of changes per step. We determine the optimal approximation thresholds: $3/4$ for deterministic algorithms and $ξ^\star \approx 0.7942$ for randomized algorithms. For every $\varepsilon>0$, we get within $\varepsilon$ of these thresholds with worst-case recourse polynomial in $1/\varepsilon$, independent of $k$ and the stream length. We prove matching impossibility results even for coverage functions. We attain the sharp bounds by optimally balancing two opposing forces: preparing for future arrivals and protecting the current solution. For randomized algorithms, we use addition robustness against prefixes of the known future, which we analyze tightly via an interval-selection problem. For deterministic algorithms, we introduce certificates, paths in a fractional relaxation that maintain high value, which we round into an update schedule.

cs.DS · cs.LG

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06209

Sublinear Approximations for Spanning Tree Congestion

Sunny Atalig, Marek Chrobak, Petr Kolman, Hans Raj Tiwary

原文摘要

The spanning tree congestion of a connected graph \(G\) is the minimum, over all spanning trees \(T\) of \(G\), of the maximum number of edges of \(G\) crossing a cut induced by deleting an edge of \(T\). In the spanning tree congestion problem we are asked to compute a spanning tree with minimum congestion. We describe two sublinear-ratio approximation algorithms for the spanning tree congestion problem (STC): a simple $\Ot(n^{8/9})$-approximation, and a more involved $\Ot(n^{2/3})$-approximation, where the notation $\Ot$ hides factors polynomial in $\log n$.. Until now, no algorithm with sublinear ratio was known. We also prove that for any fixed $0<\varepsilon<1$, approximation of the spanning tree congestion within $2-\varepsilon$ is NP-hard. This improves the previous best known lower bound $6/5$.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v3arXiv:2609.06327

Query-Oblivious Coresets for Softmax Attention: Improved Bounds and Efficient Constructions

Ofek I. Cohen

原文摘要

A query-oblivious coreset for a softmax-attention head is a subset of the key--value pairs whose attention output is within $\varepsilon$ of the full one for every query in a ball. With $ρ$ the query radius times the centred key radius, we build such coresets of size $O(e^ρ(\sqrt{d_v}+\sqrt{d_k\log(1+ρ)})/\varepsilon)$ in randomised polynomial time, improving the existential $O(\sqrt d\,e^{ρ+\frac12\logρ+o(\log\logρ)}/\varepsilon)$ of Liberty, Andoni and Kleiner to within $\sqrt{\log(1+ρ)}$ of their $Ω(\sqrt d\,e^ρ/\varepsilon)$ lower bound: a spherical lift of both balls into one exponential-kernel instance admits the Bozzai--Rothvoss chaining bound, and Chevet's inequality splits key from value dimension. A sampling cap $O(e^{2ρ}/\varepsilon^{2})$ completes the envelope, and in fixed dimension Tai's diameter-free bound removes the logarithm. A census of every head of Qwen2.5-7B-Instruct and Llama-3.1-8B-Instruct finds $ρ\ge24.100$, so every whole-ball bound with a factor $e^ρ/\varepsilon$ prescribes a coreset larger than the cache: the algorithmic contribution is asymptotic on these models.

cs.DS · cs.CL

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06178

Efficient Stable Word-RAM Simulation by Population Protocols

Leszek Gąsieniec, Tytus Grodzicki, Grzegorz Stachowiak

原文摘要

Population protocols model computation by indistinguishable agents interacting in uniformly random pairs, with unary inputs and outputs encoded by state counts. Efficiency requires polylogarithmically many states and polylogarithmic stabilisation time, measured as interactions divided by population size $n$. Stability requires eventual stabilisation to the correct output with probability $1$; silence requires that no agent changes state after stabilisation. Fast stable leader election and majority in leaderless populations require growing state spaces. Despite progress on individual problems, the general computational power of efficient stable protocols remains largely unexplored. We show that it extends beyond semilinear predicates, which characterise constant-state stable computation. We give an efficient, stable, and silent simulation of every $\mathcal{P}$-program: a word-RAM program with running time and number of registers polynomial in $w=Θ(\log n)$. Stabilisation takes polylogarithmic time both in expectation and with high probability. The framework supports general word-oriented computations, including multiple outputs. The construction extends MC$+$ protocols, fast Monte Carlo protocols whose failures are eventually detected as anomalies, and the system of weights. MC$+$ majority comparisons against the weights convert unary inputs into binary registers, with one agent per bit. MC$+$ operations on individual bits emulate RAM instructions, after which outputs are converted back to unary. A detected anomaly triggers a new universal rank-based backup that stabilises correctly and silently. These results open a systematic study of computational power and resource trade-offs, including extensions to external or spatial data.

cs.DC · cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2606.30525

Working With Measurement-Based Computations on Qudits

Piotr Mitosek, Miriam Backens

原文摘要

Measurement-based quantum computing is a universal model of quantum computation in which successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure an overall deterministic computation. Flow structures characterise cases in which such an adaptive correction procedure is possible. Recently, flow has been defined in a setting where the resource states are prime-dimensional qudit graph states rather than the usual qubit graph states. Yet, this qudit flow definition is more burdensome to work with than analogous definitions for qubits. Here, we give a simpler definition of qudit flow and consider various useful properties of this flow, drawing on results for the qubit case. In particular, we show how to focus qudit flow and argue that focused flow is canonical. We improve the previous algebraic formulation to capture focused flow and use it to obtain an O(n^3) flow-finding algorithm (where n is the number of qudits), matching the best known complexity for qubit flows and improving on the previous O(n^4) result for qudits. Furthermore, we explore multiple flow-preserving transformations, thus opening a pathway to using flow for optimisation. These transformations include pivoting, removal and insertion of certain types of vertices, and reversibility of flow. Lastly, we propose an algorithmic approach to generating large qudit computations with flow, for testing or machine learning.

quant-ph · cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06092

Constant-Round Massively Parallel Algorithms for Approximate MSTs in Metric Spaces

Eunjin Oh, Hyeonjun Shin

原文摘要

We develop a unified algorithmic framework for computing approximate minimum spanning trees (MSTs) in metric spaces in the fully scalable Massively Parallel Computation (MPC) model. Using this framework, we obtain the following results for any fixed constant $σ\in(0,1)$: - General metrics: Given all pairwise distances, we compute a $(1+\varepsilon)$-approximate MST in $O(\log(1/\varepsilon))$ rounds using $n^σ$ local space per machine and $\widetilde{O}(n^2)$ total space. - Euclidean spaces: Given $n$ points in $\mathbb{R}^d$, we compute an $O(1/σ^4)$-approximate MST in $O(1)$ rounds using $(nd)^σ$ local space per machine and $O(nd+n^{1+σ})$ total space. - Doubling metrics: Given $n$ points in a metric of doubling dimension $\mathrm{ddim}=O(σ\log n)$, we compute an $O(1/σ)$-approximate MST in $O(\mathrm{ddim}/σ^4)$ rounds using $n^σ$ local space per machine and $\widetilde{O}(n^{1+σ})\cdot 2^{O(\mathrm{ddim})}$ total space. For any fixed constant $\varepsilon\in(0,1)$, these results give the first constant-round MPC algorithms achieving $(1+\varepsilon)$-approximation in general metrics and $O(1)$-approximation in a high-dimensional Euclidean space, respectively. They improve upon the $O(\log\log n+\log(1/\varepsilon))$-round algorithm for general metrics [SODA'25] and the $O(\log\log n\cdot\log\log\log n)$-round algorithm for Euclidean spaces [SODA'24], respectively. Moreover, our round complexity for general metrics is asymptotically optimal in its dependence on $\varepsilon$, matching the $Ω(\log(1/\varepsilon))$ lower bound of [SODA'25] under the 1-vs-2-cycles conjecture. Our result also answers an open question raised by [STOC'23].

cs.DS · cs.CG · cs.DC

查看论文
arXiv ↗PDF ↗
版本更新 · v3arXiv:2602.09497

Beyond Vizing Chains: Improved Recourse in Dynamic Edge Coloring

Yaniv Sadeh, Haim Kaplan

原文摘要

We study the maintenance of a $(Δ+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $Δ$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a new technique based on an object which we call a \emph{shift-tree}. This object tracks multiple possible recolorings of $G$ and enables us to maintain a proper coloring with small recourse in polynomial time. We shift colors over a path of edges, but unlike many other algorithms, we do not use \emph{fans} and \emph{alternating bicolored paths}. We combine the shift-tree with additional techniques to obtain an algorithm with a \emph{tight} recourse of $O\big( \frac{\log n}{\log \frac{Δ+C}{Δ-C}}\big)$ for $C \ge 0.62Δ$ where $Δ-C = O(n^{1-δ})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $Δ-C=o(Δ)$. This result settles the theoretical complexity of the recourse for large palettes. Furthermore, we believe that viewing the possible shifts as a tree can lead to similar tree-based techniques that extend to lower values of $C$, and to improved update times. A second application is to graphs with low arboricity $α$. Previous works [BCPS24, CRV24] achieve $O(ε^{-1}\log n)$ recourse per update with $C\ge (4+ε)α$, and we improve by achieving the same recourse while only requiring $C \ge (2+ε)α- 1$. This result is $Δ$-adaptive, i.e., it uses $Δ_t+C$ colors where $Δ_t$ is the current maximum degree. Trying to understand the limitations of our technique, and shift-based algorithms in general, we show a separation between the recourse achievable by algorithms that only shift colors along a path, and more general algorithms such as ones using the Nibbling Method [BGW21, BCPS24].

cs.DS

查看论文
arXiv ↗PDF ↗