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

+367新增论文
+288其中 AI 相关
+33已有论文修订
+100新增 AI 导读

arXiv 最近完成扫描：2026-11-09 10:18 · 全站最近入库：2026-11-09 10:17各信源进度 →

统计口径与数据新鲜度

统计 2026-11-08 11:11 — 2026-11-09 11:11（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 10:18最近发现新增版本：2026-11-09 10:17库内论文最新更新时间：2026-11-06 12:44

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。

17,467 已收录论文14,949 AI 相关4,222 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 462 篇论文 · 按版本更新时间排序上轮扫描起点 2026-11-09 10:16（北京时间）
新论文 · v1arXiv:2611.04736

Smallest Multiplicative Stretch for an $α$-APASP Below $\widetilde O \left(n^ω\right)$ Runtime

Lakshay Saggi, Ariel Sapir

原文摘要

Dor, Halperin and Zwick [FOCS'96; SICOMP'00] showed that $(2-\varepsilon)$-APASP is at least as hard as BMM for every fixed $0<\varepsilon<1$. Zwick [FOCS'98; JACM'02] gave a $2$-APASP algorithm in $\widetilde O(n^{ω+o(1)}\log W_{\max})$ runtime, where $W_{\max}$ is the maximum edge weight. Bringmann, K{ü}nnemann and W{ę}grzycki [STOC'19] removed this logarithmic factor. Using OpenAI's recent bound $ω\leq\frac{9}{4}$ gives $\widetilde O\left(n^{\frac{9}{4}+o(1)}\right)$ time. We present two weighted undirected $2$-APASP algorithms. The first uses $(\min,\max)$-products. Earlier rectangular FMM bounds of Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'24] and Alman, Duan, Vassilevska Williams, Yinzhan Xu, Zixuan Xu and Zhou [SODA'25] give $\widetilde O(n^{2.36999465})$ runtime, below the previous $ω\leq2.371177$ of Dupont, Eisenberger, Kozlovskii, Mehrabian, Ruiz, See, Zhou, Alman, Vassilevska Williams and Balog [arXiv'26]. Current rectangular FMM bounds give $\widetilde O(n^{2.34549151})$, independent of $W_{\max}$. For integer weights, our second algorithm has $\widetilde O(n^{ω-\frac{(ω-2)^2}{ω-1}+o(1)}\log^{O(1)}W_{\max})$ runtime, with exponent below $ω$ whenever $ω>2$. The rectangular bounds of Uzna{ń}ski [arXiv'26] or of Amanov, van den Berg and Zuiddam [arXiv'26] give $\widetilde O(n^{2.17157288}\log^{O(1)}W_{\max})$.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04731

Tree Covers for Directed Graphs: Distance Oracles, Routing Schemes and Spanners

Shiri Chechik, Gur Lifshitz

原文摘要

In a seminal work [STOC'01; JACM'05], Thorup and Zwick showed that every weighted undirected graph admits a tree cover where each vertex belongs to $O(kn^{1/k})$ trees in expectation, and every pair has a path of stretch at most $2k-1$ in some tree. In this paper, we extend this structural property to directed graphs with nonnegative edge weights. Let $\mathrm{dist}_G(u \leftrightarrows v) = \mathrm{dist}_G(u \to v) + \mathrm{dist}_G(v \to u)$ denote the roundtrip distance. For every integer $k$, we construct a collection $\mathcal{T}$ of tree-pairs, each consisting of an in-tree and an out-tree rooted at the same vertex. Each vertex belongs to $O(kn^{1/k})$ tree-pairs in expectation, and for every ordered pair $(u, v)$ some tree-pair $T_{in}\cup T_{out} \in \mathcal{T}$ satisfies $\mathrm{dist}_{T_{in}\cup T_{out}}(u \to v) \le \mathrm{dist}_G(u \to v) + (k-1)\mathrm{dist}_G(u \leftrightarrows v).$ We call this a \emph{$k$-strong (roundtrip) tree cover}. While prior techniques partition distances into geometric scales and build separate structures for each scale, our construction handles all distances simultaneously within a single, unified hierarchy. This yields the first directed tree cover for general $k$ matching the classic Thorup-Zwick stretch-size trade-off. Our cover immediately implies improved constructions of roundtrip spanners, path-reporting distance oracles, and compact routing schemes, as well as their source-wise and prioritized variants. We also provide faster algorithms for these structures: for general $k \ge 4$, all the above objects can be constructed in $\widetilde{O}(\min\{mn, n^{2.5}\})$ expected time, breaking the $mn$ barrier in dense directed graphs.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04730

Finding Integer Roots of Polynomials in Nearly Optimal Time

Yansong Feng, Itamar Nir

原文摘要

Let $f\in\mathbb Z[x]$ be a polynomial of degree $n$ with integer coefficients of absolute value less than $2^b$. Using Pan's approximate factorization theorem, we give a deterministic algorithm that finds all distinct integer roots of $f$ in $\widetilde O(nb)$ bit operations. This matches the bit size of the input up to logarithmic factors and does not require $f$ to be square-free. Since we were unable to verify parts of the running time analysis of Pan's theorem, we also give a self-contained deterministic algorithm with running time $\widetilde O(nb+\min(n,b)^3b^2)$, which is $\widetilde O(nb)$ whenever $b\leq n^{1/4}$. This answers a question of Harvey and Hittmeir (2022), who asked whether integer roots can be found deterministically in quasilinear time, conditionally on Pan's theorem, and unconditionally for $b\leq n^{1/4}$. The fastest previous deterministic algorithm, due to Nir, takes $\widetilde O(n^{3/2}b)$ bit operations and works only for square-free polynomials. Instead of $p$-adic lifting, we combine approximate complex root finding with exact polynomial evaluation. We first approximate the complex roots of the polynomial formed by the top $\min(n,b)+1$ coefficients of $f$. Every integer root $r$ of $f$ with $|r|\geq 2$ lies within distance $2$ of some complex root of this truncated polynomial. Rounding the real parts of the approximate roots therefore gives a list of $O(\min(n,b))$ candidate integers that is guaranteed to contain all integer roots of $f$. Although a single candidate can have up to $b$ bits, a bound via the Mahler measure shows that the bit lengths of all candidates sum to $O(b)$. This allows us to test all candidates by exact multipoint evaluation in $\widetilde O(nb)$ bit operations.

math.NT · cs.DS · cs.SC

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04721

Pure Differentially Private Query Release via Laplace Input Smoothing

Zongrui Zou

原文摘要

We study the offline release of a public family of $k$ statistical queries on a database of $N$ records from a public universe of size $d$, under pure $\varepsilon$-differential privacy. We give an efficient algorithm that, with probability at least $1-β$, answers all queries simultaneously to error $$O\left(\min\left\{1,\sqrt{\frac{\log(d/β)\log k}{N\varepsilon}}\right\}\right).$$ The output is a synthetic histogram on the data universe, and its running time depends polynomially on $N,d$ and $k$. This resolves a conjecture of Nikolov and Ullman (2021) with an efficient, polynomial-time construction. Fitzsimons (2026) recently established the same square-root error scale via an information-theoretic construction that relies on a computationally intractable Lipschitz envelope over the exponential space of possible outputs. In contrast, we introduce an input-side smoothing approach: we run a selection-only multiplicative-weights process after a single sparse Laplace perturbation of the input histogram. Departing from standard first-order stability analyses, we investigate the stability of stability by establishing a uniform lower bound on the second-order log-derivatives of the output transcript across $T$ rounds of multiplicative weight updates. We then develop an analytic smoothing principle showing that this curvature bound allows a sparse, coordinate-wise perturbation to guarantee pure differential privacy globally, attaining optimal utility through a conceptually simple, polynomial-time pure-DP mechanism.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.19505

Counting Linear Extensions Faster than $2^n$

Keigo Oka

原文摘要

We give a deterministic algorithm that counts the linear extensions of any $n$-element poset in $O^*(33^{n/6}) \le O^*(1.791^n)$ time and space, breaking the $O^*(2^n)$ barrier of standard order-ideal dynamic programming without assumptions on height, width, or dimension. Our algorithm fixes an antichain $B$ and tracks order ideals only on the remaining core $V \setminus B$. For each element $v \in B$ that has successors in the core, we guess which of those successors appears first in the linear extension, reducing all upper constraints on $v$ to a single deadline. An element $v \in B$ becomes available once all of its core predecessors have been placed; available elements with the same deadline then face identical remaining constraints, so the dynamic program only needs to record how many available elements have each deadline and weight transitions by the number of choices for the next element. Taking $B$ to be a maximum antichain and balancing this dynamic program against a Dilworth chain partition yields an $O^*(1.89^n)$ algorithm. To obtain our main bound, we use a minimum-cut computation in an auxiliary bipartite network to find an antichain $B$ whose comparability neighborhood $R$ satisfies $|B| \ge 5|R|$ while the remainder $V \setminus (B \cup R)$ partitions into stars with at most five leaves. Since a $j$-leaf star has $2^j+1$ independent sets and $(2^j+1)^{1/(j+1)} \le 33^{1/6}$ for $1 \le j \le 5$, combining the star bound on the remainder with our state bound on $B \cup R$ gives $O^*(33^{n/6})$ time and space.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.11285

Max Independent Set Remains NP-hard when Excluding a Planar Induced Minor (and More)

Édouard Bonnet, Yeonsu Chang

原文摘要

We show that Max Independent Set (MIS) remains NP-hard in graphs excluding the $5 \times 5$ grid as an induced minor. This negatively answers a question of Dallard--Milanič--Štorgel that has attracted sustained attention, and refutes a conjecture by Gartland and Lokshtanov. Unless $N$-variable 3-SAT can be solved in time $2^{\mathcal O(N^{1-\varepsilon})}$ for some $\varepsilon > 0$, MIS requires $2^{Ω(n^{1/2-o(1)})}$ time on $n$-vertex graphs in this class. This complements a $2^{\widetilde{O}(n^{2/3})}$-time algorithm on any $H$-induced-minor-free class by Korhonen and Lokshtanov. Exploiting a connection between low twin-width or reduced parameters on one hand and planar-induced-minor-freeness on the other, we further show that 3-Coloring, Feedback Vertex Set and Dominating Set are NP-complete on weakly sparse graphs of reduced component max-leaf (hence twin-width) at most 3, even when a witnessing contraction sequence is given. Classes of bounded reduced component max-leaf (by any constant) and of twin-width at most 3 are known to exclude a fixed planar graph as an induced minor. The 3-Coloring and Feedback Vertex Set results refute more cases of the Gartland--Lokshtanov conjecture, even when the inputs are weakly sparse. The complexity of MIS in weakly sparse graphs excluding a planar induced minor remains open.

cs.CC · cs.DM · cs.DS · math.CO

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04699

Glauber Dynamics for Sampling Colorings: Rapid Mixing above $(1+1/\mathrm{e})Δ$ via Hidden Hard-Core Models

Xiongxin Yang

原文摘要

We show that for every $γ>0$ and every sufficiently large $Δ$, the Glauber dynamics for proper $q$-colorings of any $n$-vertex graph of maximum degree $Δ$ has spectral gap $Ω_γ(1/n)$, and hence mixing time $O_γ(n^2\log q)$, whenever $q\ge(1+1/\mathrm{e}+γ)Δ$. Our proof establishes a Bochner-type inequality by exploiting the spectral gaps of hard-core models hidden in the color classes. The constant $1+1/\mathrm{e}\approx1.368$ comes from the uniqueness threshold $\approx\mathrm{e}/Δ$ of the hard-core model on the infinite $Δ$-regular tree.

cs.DS · cs.DM · math.CO · math.PR

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2611.03107

Finding and Certifying Weighted Minimum Cuts with Cut Queries

Troy Lee

原文摘要

We study the deterministic and nondeterministic cut query complexity of minimum cut in undirected graphs with arbitrary nonnegative edge weights. For the nondeterministic case, we show that every $n$-vertex graph has an $O(n\log^2(2n))$-query certificate of its exact global minimum-cut value, such that any other nonnegatively weighted graph on the same vertex set that agrees on these queries has the same minimum cut value. A key ingredient is a way to certify a maximum $s$--$t$ flow value with only $O(n)$ cut queries. For the deterministic case we give an algorithm that finds an exact global minimum cut using $O(n^{3/2}\log^3(2n)(\log\log(2n))^2)$ cut queries. Previously no $O(n^{2-\varepsilon})$ deterministic bound was known for arbitrary nonnegatively weighted graphs. The algorithm follows Karger's two-respecting tree approach. The key new ingredient is a deterministic construction of a weak skeleton of the input graph $G$: an explicit weighted graph $H$ in which every minimum cut of $G$ is a $5/4$-approximate minimum cut.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2611.00682

Testing MAX-CUT in the Adjacency Matrix Model: Tight Non-Adaptive Bounds and a Cubic Adaptive Lower Bound

Songhua He

原文摘要

We study the query complexity of approximating the MAX-CUT value to additive error $\varepsilon n^2$ in the adjacency matrix model. This was one of the first problems studied in graph property testing [GGR'98]. Previously, the best known upper bound for both adaptive and non-adaptive algorithms was $O(1/\varepsilon^6)$ [RV'07; MS'08; GMRS'22]. The only known lower bound was a trivial $Ω(1/\varepsilon^2)$ bound. We close this gap from both sides for non-adaptive algorithms, proving that its query complexity is exactly $Θ(1/\varepsilon^4)$. We also prove an adaptive $Ω(1/\varepsilon^3)$ lower bound. Our upper bound is achieved by a canonical tester that samples $O(1/\varepsilon^2)$ vertices in expectation and outputs the rescaled MAX-CUT value of the induced subgraph. We prove its correctness by applying spectral truncation to the adjacency matrix, using a geometric argument to control the sampling error of the retained low-rank part and a cut-norm restriction estimate [RV'07] for the remainder. As an application, an analogous proof also yields a tight $O(1/\varepsilon^4)$ non-adaptive query upper bound for MAX-2CSP. For our lower bounds, we show indistinguishability between a uniform random graph and a random graph with a planted hidden partition. For the non-adaptive lower bound, we observe that one plus the $χ^2$-divergence between the query transcript distributions is exactly a normalized Ising partition function, which can be bounded using Boolean hypercontractivity. For the adaptive lower bound, we relate each query's distinguishing power to the prediction, based on previous answers, of whether its endpoints lie in the same part of the hidden partition. We bound the combined advantage of these predictions using the completed transcript, which lets us analyze the adaptively chosen pairs of vertices together despite their dependence on earlier answers.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04635

First Near-Additive All-Pairs Approximate Shortest Paths in Decremental Weighted Graphs

Eden Koveshi, Liam Roditty, Ariel Sapir

原文摘要

We present the first four All-Pairs Approximate Shortest Paths (APASP) algorithms for decremental weighted undirected graphs. We extend the hierarchy induced edge sets of Baswana and Kavitha [FOCS'06 and SICOMP'10] to the decremental setting. At any point in time, for any $u,v$, $W_{i}$ denotes the weight of the $i$-th heaviest edge on any shortest path between them in the current graph. For any $\varepsilon>0$, our first algorithm maintains $(1+\varepsilon,2W_{2})$-APASP in $\tilde{O}(n^{\frac{3}{2}+o(1)}m^{\frac{1}{2}})$ total update time, matching a single static computation of Cohen and Zwick [SODA'97 and JALG'01], up to a subpolynomial factor. It also improves the stretch of the $(2+\varepsilon)$-APASP algorithm of Dory, Forster, Nazari and de Vos [ICALP'24]. Our second algorithm maintains $(1+\varepsilon,2W_{1}+2W_{2})$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time, improving upon Chechik's $(3+\varepsilon)$-APASP algorithm [FOCS'18] for $m=Ω(n^{\frac{11}{6}})$. We also obtain two $(1+\varepsilon,2\cdot(k-1)W_{1})$-APASP algorithms with total update times $\tilde{O}(n^{2-\frac{1}{k}+o(1)}m^{\frac{1}{k}})$ and $\tilde{O}(n^{2+\frac{2}{3k-2}+o(1)})$, adapting the static $+2\sum_{i=1}^{k-1}W_{i}$-APASP algorithms of Cohen and Zwick [SODA'97 and JALG'01] and Roditty and Sapir [FSTTCS'25]. For unweighted graphs, we present $(1+\varepsilon,2)$-APASP in $\tilde{O}(n^{\frac{7}{3}})$ total update time and $(1+\varepsilon,2\cdot(k-1))$-APASP in $\tilde{O}(n^{2+\frac{1}{3k-4}+o(1)})$. Combined with the results of Dory, Forster, Nazari and de Vos [ICALP'24], these match the fastest combinatorial static bounds of Dor, Halperin and Zwick [FOCS'96 and SICOMP'00] up to subpolynomial factors.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.29898

A Sharp Dynamic Haussler Lemma

Shihao Shao, Dacheng Tao

原文摘要

Haussler's covering lemma (JCTA 1995) gives the sharp $(C/δ)^d$ metric-entropy bound for binary VC classes under normalized $\ell_1$ distance. We prove a sharp global dynamic analogue for bounded real-valued classes in normalized $\ell_p$, for every $1\le p<\infty$. Writing $d=\operatorname{sfat}_{\varepsilon/32}(\mathcal F)$, we obtain a global dynamic cover of size at most $(C/\varepsilon^p)^d$ for every horizon and $0<\varepsilon<1/2$. Here a single family of deterministic causal experts, fixed independently of the instance sequence, covers every target on every sequence. For $0<\varepsilon<1/4$, a matching worst-case lower bound $(c/\varepsilon)^{pd}$ holds already for binary classes on one fixed sequence, even with real-valued experts. The constants are universal, independent of $p$. For $p=2$, this resolves the horizon-independent sequential $\ell_2$ covering problem of Rakhlin, Sridharan, and Tewari (PTRF 2015) in a stronger global form. For $p=1$, it gives the sharp $(C/δ)^{\operatorname{Ldim}(\mathcal H)}$ global dynamic covering bound for binary classes, resolving questions of Alon, Ben-Eliezer, Dagan, Moran, Naor, and Yogev (STOC 2021) and Malliaris and Moran (Trans. AMS 2024). Our proof combines rank-decreasing correction schedules and accurate sampled prefixes with exact continuations after target identification. Shared-parent routing distributes these continuations among eligible prefixes, eliminating the logarithmic overhead.

cs.LG · cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04569

Abelian embeddings make useful promises: approximation of detrivialised equations

Tamio-Vesa Nakajima, Zephyr Verwimp, Stanislav Živný

原文摘要

We study the following family of search problems: given a system of equations $x+y+z\equiv 0\pmod p$ promised to admit a solution that is nonconstant on every equation, find a $c$-colouring of the variables that makes no equation monochromatic. For $p\in\{2,3\}$, the problem recovers known tractable Promise Constraint Satisfaction Problems (PCSPs). For $p > 3$, we obtain a complete complexity classification: the problem is polynomial-time solvable for $c \geq 4$, yet even its decision version is NP-hard for $c = 3$, and it is not well-defined for $c = 2$. Our main conceptual contribution is a general tractability framework for PCSPs based on Abelian embeddings, a notion introduced by Bhangale, Khot, and Minzer [STOC 2022] in the study of quantitative approximation of satisfiable CSPs. We show that suitable Abelian embeddings yield polynomial-time PCSP reductions realised by singleton algorithms in the sense of Zhuk [FOCS 2026]. On the hardness side, the promise relation is far removed from the reconfigurable regime covered by the conditional hardness framework of Banakh, Barsukov, and Nakajima [LICS 2026]; our proof instead relies on a new additive-combinatorial rigidity theorem classifying special $3$-colourings of $\mathbb{Z}_p$.

cs.CC · cs.DM · cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v3arXiv:2609.38044

A Tale of Two Walks: From Kac to KMP and Back

Qian Chen, Jingcheng Liu, Minglong Qin, Leonard J. Schulman, Fang Song, Penghui Yao, Mingnan Zhao

原文摘要

We reveal an unexpected connection between two interacting particle systems: Kac's walk and the Kipnis-Marchioro-Presutti (KMP) process. We first establish a mixing-time analogue of Aldous's conjecture for the KMP process on general graphs: the mixing time of the $t$-particle process on any connected $d$-vertex graph is at most $O(\log d)$ times the corresponding single-particle mixing time, uniformly in $t$. As an application, we improve the mixing-time bound for coordinate hit-and-run on the $n$-dimensional standard simplex from $\widetilde{O}(n^3)$ [KV26] to $O(n\log n)$, while removing the dependence on the initial distribution. We then show that the twirling channel induced by the parallel Kac's walk on the symmetric subspace can be exactly encoded by a Markov chain on partitions, which naturally lifts to a parallel KMP chain on the complete graph. We prove that $O(\log d+\log(1/\varepsilon))$ repetitions of the parallel Kac's walk suffice to approximate Haar twirling on the symmetric subspace of $(\mathbb{C}^d)^{\otimes t}$ to error $\varepsilon$, uniformly in all $t$. Our main technical contribution is a new structural property of KMP-type dynamics: conditional product structure, which holds for both the standard KMP processes on general graphs and the parallel KMP processes on complete graphs. Conditioned on suitable auxiliary randomness, the evolution of the labeled KMP process factorizes into independent single-particle updates. Combining this structure with an exact coupling, we remove the dependence on the number of particles from the mixing bounds for both unlabeled KMP models. This bypasses the $\log(π_{\min}^{-1})$ dependency from spectral-gap arguments, which is often prohibitively large for many-particle systems. Using this framework, we prove mixing bounds that are sharp up to logarithmic factors for both unlabeled KMP models.

quant-ph · cs.DS · math.PR

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04559

Distributionally Robust Search: The Power of Information and the Limits of Moments

Nicole Megow, Cliff Stein, Jose Verschae

原文摘要

We study binary search for a target drawn from a distribution on $n$ ordered elements. The cost is the expected number of comparison queries. In our distributionally robust framework, the algorithm knows only an uncertainty set containing the distribution and minimizes its worst-case approximation ratio against an optimal strategy that knows the distribution. Collections of uncertainty sets capture different forms of information: quantiles, moments, general linear measurements, or approximate CDFs. This framework quantifies how much and what form of information is needed for near-optimal search. Our main result separates moments from general linear measurements. Although $n-1$ moments determine the distribution exactly, for every constant $δ\in(0,1)$, the first $n^{1-δ}$ exact moments can be as uninformative for search as no information: the worst-case approximation ratio remains $Ω(\log n)$. In contrast, for every constant $\varepsilon\in(0,1)$, $O(n^\varepsilon\log^4 n)$ suitably chosen linear measurements suffice for an $O(1/\varepsilon)$-approximation. The limitation lies in the measurements chosen, rather than linearity itself. Using LP duality and polynomial bounds of Rakhmanov and Bernstein, we construct distributions with identical moments and mass concentrated at different positions. For the upper bound, we combine dyadic sketches and Johnson--Lindenstrauss embeddings with a truncated median-search algorithm that converts uniform CDF approximation into search guarantees. Measuring information in bits, we prove a lower bound of $Ω(1/\varepsilon)$ on the approximation ratio for summaries of any form using at most $\frac12 n^\varepsilon\log n$ bits, for every sufficiently small constant $\varepsilon>0$. Quantiles match this within constant factors. For Kolmogorov balls of radius $n^{-\varepsilon}$ the optimal approximation ratio is $Θ(1/\varepsilon)$.

cs.DS · cs.DM

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2607.10667

Constant-factor approximation of MinCostCSP with a conservative majority polymorphism

Marcin Kozik, Stanislav Živný

原文摘要

For a relational structure A, the Minimum Cost Constraint Satisfaction Problem is the following problem denoted by MinCostCSP(A): Given an instance of CSP(A) with rational costs on variable-value pairs, find a solution to the instance minimizing the sum of the chosen costs. For the exact minimization, a classification of MinCostCSP(A) in terms of A was established by Takhanov [STACS'10]. We focus on constant-factor approximations of MinCostCSP(A). DeHaan, Huang, and Lee recently showed that if A fails to admit a conservative near-unanimity polymorphism then MinCostCSP(A) is not constant-factor approximable [APPROX'25]. We provide a significant step towards a full classification, by proving a dichotomy for structures A admitting a conservative majority (also known as 3-near-unanimity) polymorphism. Our dichotomy criterion is not in terms of an algebraic condition on A but we show that this is unavoidable. We include a simple argument proving that no such condition exists.

cs.CC · cs.DM · cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2611.01796

Exact Tie-Breaking and Tight Degree Bounds for Minimum-Degree Greedy on Interval Graphs

Ruize Zhao

原文摘要

We study tie-breaking in the minimum-degree greedy algorithm (MDG) for maximum independent set on interval graphs. Given an interval representation of an $n$-vertex graph, we compute the largest and smallest output sizes and construct executions attaining them in $O(n^3)$ time and $O(n^2)$ space. For maximum degree at most four, favorable tie-breaking gives a tight guarantee of $3/4$. For every maximum-degree bound of at least five, the guarantee drops to $2/3$. Both bounds are tight even on connected unit interval graphs whose output size is independent of tie-breaking. Together with the known optimality of favorable tie-breaking for maximum degree at most three, these results give a complete classification by degree bound.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04506

Dynamic Steiner Tree under Edge Updates

Jeff Giliberti, MohammadTaghi Hajiaghayi, Mohammad Mahdavi, Xinchen Yang

原文摘要

The Steiner tree problem is the canonical generalization of minimum spanning tree (MST). When connectivity is required only among a fixed subset of vertices (terminals), the goal is to maintain a minimum-cost tree connecting these terminals as the graph undergoes edge updates. While dynamic MST has been studied since the 1980s~[Frederickson, STOC'83, SICOMP'85], and the first sublinear-in-$n$ dynamic MST algorithm dates back almost thirty-five years to Eppstein, Galil, Italiano, and Nissenzweig~[FOCS'92, JACM'97], achieving sublinear-in-$n$ update time for dynamic Steiner tree has long remained open, as a single edge update can invalidate many routes in the underlying terminal metric closure. We present the first fully dynamic algorithm for Steiner tree that achieves a constant-factor approximation with sublinear-in-$n$ update time. For any fixed $0<ζ<1/2$, our deterministic algorithm maintains an $O_ζ(1)$-approximate Steiner tree for undirected graphs with polynomially bounded edge weights in $\widetilde O(n^{1/2+ζ})$ amortized update time, answering cost queries in $O(1)$ time, and reporting an explicit tree in $\widetilde O(n^{1+ζ})$ time. Our approach combines periodic terminal-metric rebuilds on a dynamic spanner with a top-tree dynamic program that localizes edge deletions and reconnects surviving components through a dynamic distance sparsifier. Along the way, we give a simpler randomized insertion-only algorithm with a better, explicit approximation ratio.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v3arXiv:2609.13693

Polynomial-time Tensor Decomposition under the Kruskal Condition and the Lovitz--Petrov Generalization

Shiri Sivan

原文摘要

An identifiability criterion is a condition on a tensor decomposition that certifies uniqueness when the condition holds. Kruskal's classical condition is one of the best-known deterministic criteria for identifiability. However, no polynomial-time decomposition algorithm is known under the Kruskal condition; verifying the condition itself is NP-hard. Lovitz and Petrov introduced a strictly more general identifiability condition which, in contrast, is polynomial-time verifiable, but no polynomial-time decomposition algorithm was known under this condition either. In this paper we give a randomized polynomial-time algorithm, in the algebraic model over $\mathbb R$, for tensor decomposition under the Lovitz--Petrov condition, and therefore also under the stricter Kruskal condition. Moreover, combining our algorithm with polynomial-time verification of the Lovitz--Petrov condition yields an efficient end-to-end certification procedure: after computing a decomposition, one can deterministically certify in polynomial time that it is unique and therefore of minimum rank.

cs.DS · math.CO

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2604.00954

Fully Scalable MPC Algorithms for Euclidean Clustering: Constant Approximation in Constant Rounds

Artur Czumaj, Shaofeng H. -C. Jiang, Yaonan Jin, Jianing Lou, Weicheng Wang

原文摘要

We study Euclidean $k$-Means and $k$-Median under the Massively Parallel Computation (MPC) model, focusing on the \emph{fully scalable} setting. Previously, fully scalable algorithms for $k$-Means and/or $k$-Median either required $O(\log\log n\cdot\log\log\log n)$ rounds [Cohen-Addad, Kuhn, and Parsaeian; SODA'26] or guaranteed only an $O(\log n)$-approximation [Goranci, Jiang, Kiss, Kong, Qian, and Szilagyi; SODA'26]. In this paper, we close this gap by giving the first fully scalable algorithms for both problems, which work even when $k$ is polynomially larger than the local memory, and \emph{simultaneously} achieve an $O(1)$-approximation and $O(1)$ round complexity. Technically, we devise a new variant of the MP algorithm [Mettu and Plaxton, SICOMP'03] that computes a \emph{fractional solution satisfying the Lagrangian Multiplier Preserving (LMP) property}. Importantly, this LMP guarantee holds even under arbitrary distance distortions, a feature that is crucial for efficient MPC implementations. We then give a novel MPC rounding algorithm that first rounds to a bicriteria solution with $k+O(n^{1-ε})$ centers and then uses an \emph{iterative kernelization} idea to convert it into one with exactly $k$ centers.

cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04457

Entropy-Weighted Soundness for Multi-Answer Birthday Graphs

Alexandru Popa

原文摘要

The vertices of a Boolean birthday graph are partial assignments on $\ell$-subsets of $n$ variables. Although a satisfying assignment induces a $K$-clique for $K=\binom n\ell$, an arbitrary selection of $K$ vertices may repeat some views and omit others. This makes soundness substantially harder than under uniform view sampling. For bounded-occurrence, constant-gap Boolean ternary CSPs whose constraints each involve three distinct variables (allowing arbitrary Boolean predicates), we prove that the graph's maximum induced $K$-vertex density satisfies \[ ρ^*(K)\le 2\exp\!\left(-c\, \frac{\ell^3}{n^2\log(en/\ell)}\right) \] for $2\le\ell\le x_0n$ and $\ell\log(en/\ell)\ge C_0\log(n+1)$, with positive constants depending only on the source parameters. The argument refines the biclique counting method by charging the entropy of an adversarially chosen flattening to mutual information and a combinatorial obstruction energy. In the overlapping parameter range it strengthens the explicit soundness exponent previously established for this graph. We also establish an unconditional stability theorem for arbitrary sets $H$ of exactly $K$ distinct local assignments, with missing views and multiple answers: $f_*(H)\le34(n/\ell)q_{\mathrm{con}}(H)$ for every $1\le\ell\le n$. It gives a complementary additive near-clique soundness bound. We do not derive a new fine-grained hardness theorem.

cs.CC · cs.DS

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04444

Binary Huffman Coding with Unequal Letter Costs Is NP-Complete

L. Sunil Chandran, Rishikesh Gajjala

原文摘要

Binary prefix coding assigns prefix-free binary words to symbols to minimize their frequency-weighted cost. Huffman solved the equal-letter-cost case in polynomial time in 1952 [Proc. IRE'52]. For unequal letter costs, Karp gave an integer-programming formulation in 1961 [IRE Trans. Inf. Theory'61]. Exact polynomial-time algorithms for fixed integer costs were given by Golin and Rote [IEEE Trans. Inf. Theory'98] and Bradford, Golin, Larmore, and Rytter [J. Algorithms'02]. For input-dependent costs, Golin, Mathieu, and Young gave a polynomial-time approximation scheme [SICOMP'12]. The exact complexity when letter costs are part of the input had remained unresolved. We resolve this $65$-year-old problem by proving that deciding whether there exists a binary prefix-free code of frequency-weighted cost at most a given threshold is NP-complete, even when the letter costs are consecutive positive integers $(M,M+1)$ supplied as part of the input. All frequencies, letter costs, and thresholds are encoded in binary. Consequently, finding an optimal code is NP-hard. Our proof uses two polynomial-time reductions, starting from Planar Independent Set. The intermediate problem prescribes the number of leaves at each depth of a full ordered binary tree and minimizes a signed linear score on their sorted right-edge counts. The first reduction represents an independent-set verification program by a tree growing level by level. A fixed-profile identity converts narrow coordinate restrictions into linear penalties that enforce the computation. The second reduction adds large dyadic terms to the symbol frequencies to force prescribed depths. These terms contribute a constant to the weighted number of ones, leaving smaller signed perturbations to reproduce the tree score.

cs.DS · cs.CC · cs.CG · cs.IT

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2609.23711

A Fixed-Parameter Algorithm for 4-Block Integer Programming

Klaus Jansen, Felix Ohnesorge, Corinna Wambsganz

原文摘要

We give a fixed-parameter tractable (FPT) algorithm with running time $f(k,\barΔ)\cdot {|I|}^{O(1)}$ for integer linear programs with 4-block structure, parameterized by the maximum block dimension $k$ and the largest absolute entry $\barΔ$ in the local-column blocks. This result resolves a long-standing open question in parameterized complexity, and answers a conjecture by Eisenbrand and Rothvoss (2026) in the positive. This result has several implications for other block-structured integer programming models, including mixed fracture number and special cases of 4-block programs with large entries outside of the diagonal. Important tools for this algorithm are structural properties of generalized $n$-fold integer programs shown by Ligthart~(2026) and an algorithm by Veselov, Gribanov, Zolotykh and Chirkov~(2020) for optimizing discrete convic functions. Independently and shortly after the first version of our paper appeared, Koutecký, Lassota, and Ligthart (2026) obtained an FPT algorithm for 4-block integer programming.

cs.CC · cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2603.26129

Improved Algorithms for Unrelated Crowd Worker Scheduling in Mobile Social Networks

Chi-Yeh Chen

原文摘要

This paper studies task scheduling with unrelated crowd workers in mobile social networks, where the service time of a task depends on the crowd worker that performs it, and the objective is to minimize the total weighted completion time. For identical crowd workers, we tighten the approximation ratio of the Largest-Ratio-First (LRF) algorithm and update the competitive ratio of its online version. For unrelated crowd workers, we show that the time-indexed rounding framework of Im and Li remains valid when worker-dependent contact delays are added, which yields a randomized algorithm with an expected approximation ratio of 1.45; this improves upon the expected ratio of $1.5+ε$ obtained in our previous work for identical crowd workers. We also derandomize this algorithm. To improve computational efficiency, we further propose an algorithm that exploits the optimality of the Smith-ratio order on a single crowd worker, and we prove that it is a deterministic 1.5-approximation algorithm and that its polynomial-time version achieves an approximation ratio arbitrarily close to 1.5. Experiments on synthetic instances and on instances derived from real contact traces show that this algorithm outperforms three variants of LRF in all tested settings, with the largest advantage when the crowd workers are highly heterogeneous, performs on par with a local-search heuristic for unrelated machines, and is within 0.4\% of the optimum on average on small instances that can be solved exactly.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.01320

Dense Language Generation Made Simple: Deterministic, Randomized, and Multi-Order Algorithms

Ziyi Cai, Shuangping Li, Yiheng Shen, Kangning Wang, Peng Zhang

原文摘要

Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.

cs.DS · cs.AI · cs.CL · cs.DM · cs.LG

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04365

Space-Optimal Algorithms for Spanning Forests in Dynamic Streams at Every Confidence Level: With and Without Deletion Constraints

Zemin Chao, Jianhe Cen, Hengzhao Ma, Hongzhi Wang

原文摘要

We characterize the optimal space for one-pass spanning-forest computation in dynamic streams of $v$-vertex simple graphs with failure probability at most $δ$. Nelson and Yu proved an $Ω(v\log^2v\log(v/δ))$-bit lower bound, establishing the AGM algorithm's optimality for each fixed $0<ε<1$ and $2^{-v^{1-ε}}<δ\le1/4$. We resolve the remaining high-confidence regime: for every $0<δ\le1/4$, matching lower bounds and an algorithm give \[ Θ\!\left(\min\left\{v^2,\;v\log(v/δ)\log^2\!\left(2+\frac{v}{\log(v/δ)}\right)\right\}\right) \] bits, improving space by up to a factor of $Θ((\log v/\log\log v)^2)$ in this regime. We also study streams with at most $Z$ previously inserted edges deleted and not yet reinserted at report time, where the integer $Z\ge0$ is given in advance. Total deletions, restorations, and intermediate absence counts are unrestricted. For a single report and every $0<δ\le1/4$, matching lower bounds and a one-pass randomized algorithm yield optimal space $Θ(\mathcal L(v,Z,\log(1/δ)))$ bits, where $x_Z(t)=\min\{Z+1,v,\sqrt{vt}\}$ and \[ \mathcal L(v,Z,t)=v\log v\min\{\log^2v,\log^2(Z+2)\} +vt\log\!\left(1+\frac{x_Z(t)}t\right)\log\!\left(1+\frac v{x_Z(t)}\right). \] At zero error, a lower bound for all zero-error algorithms and a matching deterministic algorithm give optimal space \[ Θ\!\left(\min\{v^2,v(Z+1)\log(1+v/(Z+1))\}\right) \] bits. Together, these results characterize the memory required across all confidence levels and report-time edge-loss bounds.

cs.DS · cs.CC

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.06724

An FPRAS for Counting Common Bases of Two Matroids

Xiaoyu Chen, Kuikui Liu

原文摘要

We design the first fully polynomial-time randomized approximation scheme ($\mathsf{FPRAS}$) for approximately counting and uniformly sampling common bases of two matroids given by their independence oracles. Moreover, our algorithms generalize far beyond this to Hadamard products of two probability measures on the Boolean cube satisfying a simple nonnegative curvature condition. These algorithmic primitives have myriad applications in statistical physics, polyhedral combinatorics, the theory of quantum many-body systems, and beyond. The central difficulty is that the hard constraints imposed by the two input matroids (or the two input measures, more generally) obstruct traditional approaches based on local Markov chains. Relaxing these constraints can restore rapid mixing, but configurations that satisfy the original hard constraints may become exponentially rare if this is done naïvely. Our main contribution is a novel approach for approximate counting that marries Markov chain Monte Carlo (MCMC) with convex programming to overcome these obstacles: - We relax the intersection by imposing an overlap penalty on the product measure formed by the two input measures. We prove, via an integrated Bochner-type method, that this "soft intersection" satisfies a Poincaré Inequality uniformly over all external fields. - We use samples from the relaxed distribution to solve a dual maximum entropy convex program inspired by the Gurvits capacity. This program reweights the soft intersection measure using external fields to boost the aggregate probability of configurations satisfying the hard intersection constraint, without changing their relative weights. We bound the success probability directly using the uniform Poincaré Inequality and smallness of the gradient norm. AI Disclosure: GPT-5.6 Sol Ultra and GPT-6 Astra Ultra were heavily used to develop the ideas in this paper.

cs.DS · cs.CC · cs.DM · math.PR

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04353

XOR-Functions and Beyond in the Random Query Model

Sumegha Garg, Janani Sundaresan, Santhoshini Velusamy

原文摘要

We study time--space tradeoffs for computing Boolean functions $f:\{0,1\}^n \to \{0,1\}$ in the random query model. On input $x$, a branching program receives independent samples $(i,x(i))$, where $i$ is uniform in $[n]$. We want to evaluate $f(x)$ while minimizing the width $2^S$ and the length $T$ of the branching program. There is a simple upper bound of $T = \tilde{O}(n)$, $S = O(n)$: coupon-collector shows that $\tilde{O}(n)$ samples suffice, and they can be stored in $O(n)$ space. Raz and Zhan (ITCS 2020) introduced this model and studied the zero-error setting. In the bounded-error setting, Dinur (SODA 2024) proved $TS = \tildeΩ(n \cdot inf[f])$, where $inf[f]$ is the total influence of $f$. For many functions with low influence such as Equality and Tribes, this lower bound can leave a substantial gap from the best known upper bounds. Our main result is a stronger lower bound of $T(S+\log T) = Ω(n \cdot sens[g])$ for any XOR-function $g(x,y)= f(x \oplus y)$, where $sens[g]$ is the sensitivity of $g$. Through simple corollaries, we get tight nearly-quadratic time--space lower bounds for Tribes and Equality, despite their low influence. Our proof introduces a new notion of information complexity tailored to the random query model along with a corresponding direct-sum argument which may be of independent interest. Dinur also proved a tight bound of $TS = Ω(n^2)$ for Majority, whose influence is only $Θ(\sqrt{n})$, and asked whether an alternative characterization of Boolean functions could give a unified proof of tight results for a broader class of functions, including Majority. We answer this open question by proving $TS = \tildeΩ(n \cdot dgn[f])$, where $dgn[f]$ is the degeneracy of the sensitivity graph of $f$ (for Majority, this is $Θ(n)$). We also prove that this bound is tight for all symmetric functions up to logarithmic factors.

cs.DS

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.05953

Fast mixing of the SYK model at high temperatures

Yiyi Cai, Yongtao Zhan, Alexander Zlokapa

原文摘要

The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions then applies two key new ideas: a Bochner inequality for quantum dynamics, and an especially simple SYK analysis based on a comparison to classical dynamics.

quant-ph · cond-mat.dis-nn · cs.DS · math-ph

查看论文
arXiv ↗PDF ↗
新论文 · v1arXiv:2611.04336

Gaussian Difference Dynamics: Breaking the $2^{n/2}$ Barrier for the Shortest Vector Problem and Discrete Gaussian Sampling

Yiming Gao, Xuyuan Han, Honggang Hu

原文摘要

We give classical randomized algorithms in a word RAM model that break the $2^{n/2}$ barrier for both discrete Gaussian sampling (DGS) and the Shortest Vector Problem (SVP). For every fixed $α>1$, shifted DGS on arbitrary lattices at width $σ\geαη_{1/2}(L)$ takes $$ (3/2)^{n/2+o(n)}\text{ time} \qquad\text{and}\qquad (4/3)^{n/2+o(n)}\text{ space}. $$ These bounds match the Becker--Ducas--Gama--Laarhoven (BDGL) heuristic sieving exponents. At widths at least $3η_{1/2}(L)/2$, the same total cost suffices for an entire batch of $\lfloor(4/3)^{n/2}\rfloor$ samples jointly close to independent target Gaussians. For SVP on arbitrary lattices, our algorithm returns a shortest nonzero vector with high probability in $$ 2^{0.4130n+o(n)}\text{ time} \qquad\text{and}\qquad 2^{0.3138n+o(n)}\text{ space}. $$ As a corollary, we obtain centered DGS at every width with the same exponents, answering a question of Stephens-Davidowitz (SODA 2016). We introduce Gaussian difference dynamics (GDD), a framework for tracking expected counts and pairwise clustering of vectors through repeated differencing. A closed recursion for the first two counting moments controls these quantities relative to a Gaussian reference, without requiring independence within intermediate lists. Accurate moment bounds let us extract a Gaussian sample by weighted selection, while the weaker bounds needed to establish a shortest vector representation allow us to retain more differences. We perform these transitions on thin spherical shells, where the Gaussian rejection weights can be renormalized at subexponential cost. For batch sampling, we combine this geometry with the double rejection technique of Doulgerakis and Laarhoven to control the joint distribution of outputs while reusing vectors.

cs.DS · cs.CR

查看论文
arXiv ↗PDF ↗
版本更新 · v2arXiv:2608.25491

A Spectral Local-to-Global Principle for Spin Systems on Graphs with Girth At Least Five

Xiaoyu Chen, Kuikui Liu

原文摘要

It is proved that, for every $δ\in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+δ)Δ$ and the underlying graph has girth at least $5$ and maximum degree $Δ= Ω_δ(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+δ)(1-β)Δ$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

cs.DS · cs.DM · math.PR

查看论文
arXiv ↗PDF ↗