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版本更新 · v3arXiv:2501.17563

Search Trees on Trees via LP

Yaniv Sadeh, Haim Kaplan, Uri Zwick

原文摘要

We consider the problem of computing optimal search trees on trees (STTs). STTs generalize binary search trees (BSTs) in which we search nodes in a path (linear order) to search trees that facilitate search over general tree topologies. Golinsky proposed a linear programming (LP) relaxation of the problem of computing an optimal static STT over a given tree topology. He used this LP formulation to compute an STT that is a $2$-approximation to an optimal STT, and conjectured that it is, in fact, an extended formulation of the convex-hull of all depths-vectors of STTs, and thus always gives an optimal solution. In this work we study this LP approach further. We show that the conjecture is false and that Golinsky's LP does not always give an optimal solution. To show this we use what we call the ``normals method''. We use this method to enumerate over vertices of Golinsky's polytope for all tree topologies of no more than 8 nodes. We give a lower bound on the integrality gap of the LP and on the approximation ratio of Golinsky's rounding method. We further enumerate several research directions that can lead to the resolution of the question whether one can compute an optimal STT in polynomial time.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06055

Softmax Attention with Any Temperature in Fixed Dimension: Matching the Best Known Maximum-Inner-Product Bounds

Zixuan Huang, Guozhang Chen

原文摘要

Attention is central to modern AI models for language, vision, and scientific modeling. However, direct evaluation compares every query with every key, requiring quadratic time in the context length, which limits its scalability to long contexts. Consequently, characterizing the complexity bounds of attention is of fundamental importance. For every fixed dimension $d$, we give an algorithm for $m$ queries, $n$ key-value pairs, and additive error $\varepsilon$, with expected time $ \widetilde{\mathrm{O}}\bigl((mn)^{k/(k+1)}+m+n\bigr),\qquad k=\lfloor d/2\rfloor. $ For fixed dimension and a fixed number of value channels, soft-O hides factors of the form $\operatorname{poly}_d\bigl(b,\log(m+n),\log B,\log(1/\varepsilon)\bigr),$ where $b$ bounds input encoding lengths and $B$ bounds coordinate and value magnitudes. In particular, the dependence on $B$ and $1/\varepsilon$ is polylogarithmic, which allows arbitrary positive temperature. When $m=n$ and $d\ge2$, the bound is $\widetilde{\mathrm{O}}(n^{2-2/(\lfloor d/2\rfloor+1)})$: near-linear in dimensions two and three, with exponent $4/3$ in dimensions four and five and $3/2$ in dimensions six and seven. It improves the previous $\widetilde{\mathrm{O}}(n^{2-1/d})$ upper bound and matches the best known batch maximum-inner-product bound in each fixed dimension. Additionally, we reduce integer batch maximum inner product computation to one attention call with the same dimension and input sizes $m,n$.

cs.DS · cs.CC · cs.CG · cs.LG

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2310.14058

Caching Connections in Matchings

Yaniv Sadeh, Haim Kaplan

原文摘要

Motivated by the desire to utilize a limited number of configurable optical switches by recent advances in Software Defined Networks (SDNs), we define an online problem which we call the Caching in Matchings problem. This problem has a natural combinatorial structure and therefore may find additional applications in theory and practice. In the Caching in Matchings problem our cache consists of $k$ matchings of connections between servers that form a bipartite graph. To cache a connection we insert it into one of the $k$ matchings possibly evicting at most two other connections from this matching. This problem resembles the problem known as Connection Caching, where we also cache connections but our only restriction is that they form a graph with bounded degree $k$. Our results show a somewhat surprising qualitative separation between the problems: The competitive ratio of any online algorithm for caching in matchings must depend on the size of the graph. Specifically, we give a deterministic $O(nk)$ competitive and randomized $O(n \log k)$ competitive algorithms for caching in matchings, where $n$ is the number of servers and $k$ is the number of matchings. We also show that the competitive ratio of any deterministic algorithm is $Ω(\max(\frac{n}{k},k))$ and of any randomized algorithm is $Ω(\log \frac{n}{k^2 \log k} \cdot \log k)$. In particular, the lower bound for randomized algorithms is $Ω(\log n)$ regardless of $k$, and can be as high as $Ω(\log^2 n)$ if $k=n^{1/3}$, for example. We also show that if we allow the algorithm to use at least $2k-1$ matchings compared to $k$ used by the optimum then we match the competitive ratios of connection catching which are independent of $n$. Interestingly, we also show that even a single extra matching for the algorithm allows to get substantially better bounds.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v4arXiv:2301.03084

Dynamic Binary Search Trees: Improved Lower Bounds for the Greedy-Future Algorithm

Yaniv Sadeh, Haim Kaplan

原文摘要

Binary search trees (BSTs) are one of the most basic and widely used data structures. The best static tree for serving a sequence of queries (searches) can be computed by dynamic programming. In contrast, when the BSTs are allowed to be dynamic (i.e. change by rotations between searches), we still do not know how to compute the optimal algorithm (OPT) for a given sequence. One of the candidate algorithms whose serving cost is suspected to be optimal up-to a (multiplicative) constant factor is known by the name Greedy Future (GF). In an equivalent geometric way of representing queries on BSTs, GF is in fact equivalent to another algorithm called Geometric Greedy (GG). Most of the results on GF are obtained using the geometric model and the study of GG. Despite this intensive recent fruitful research, the best lower bound we have on the competitive ratio of GF is $\frac{4}{3}$. Furthermore, it has been conjectured that the additive gap between the cost of GF and OPT is only linear in the number of queries. In this paper we prove a lower bound of $2$ on the competitive ratio of GF, and we prove that the additive gap between the cost of GF and OPT can be $Ω(m \cdot \log\log n)$ where $n$ is the number of items in the tree and $m$ is the number of queries.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06024

Optimal LCA and Sublinear Estimation for Bipartite Matching in $Δ^{O(1/\varepsilon)}$ Queries

Edith Cohen, Mohammad Roghani

原文摘要

A local computation algorithm (LCA) for matching answers ``who is the mate of $v$?'' by inspecting only a small part of the graph, with all answers consistent with one matching. For $n$-vertex graphs of maximum degree $Δ\ge2$, Levi et al. [Algorithmica'17] give a $(1-\varepsilon)$-approximate matching LCA using $(Δ/\varepsilon)^{O(1/\varepsilon^2)} \operatorname{polylog}(n)$ queries per invocation, while Yoshida et al. [SIAM J. Comput.'12] estimate the maximum matching size within additive error $\varepsilon n$ using $Δ^{O(1/\varepsilon^2)}$ queries. Behnezhad, Roghani, and Rubinstein [FOCS'23] proved a $Δ^{Ω(1/\varepsilon)}$ lower bound for both tasks, even on bipartite graphs and even when the LCA's matching guarantee is relaxed to additive $\varepsilon n$ error. This leaves a gap between linear and quadratic dependence on $1/\varepsilon$ in the exponent. We close this gap on bipartite graphs. We give an LCA that answers each mate query with $Δ^{O(1/\varepsilon)}$ queries for a $(1-\varepsilon)$-approximate matching, and an estimator for the maximum matching size within $\varepsilon n$ using $Δ^{O(1/\varepsilon)}$ queries. On general graphs, the same algorithm gives a half-integral fractional matching and estimates the maximum fractional matching size. Our approach runs $O(\varepsilon^{-1}\logΔ)$ rounds of a deterministic local optimization method in which vertices propose to neighbors and update prices according to incoming loads. We design its updates to admit exact recursive sampling with constant expected branching, obtaining the improved randomized bound without exploring full dependency neighborhoods. To our knowledge, this is the first LCA construction based on exact recursive sampling of the iterates of a deterministic continuous optimization method. We expect this approach to have applications beyond matching.

cs.DS

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2610.18828

Settling the Approximability of Submodular Facility Location

Adam N. Elmachtoub, Bento Natura, Meike Neuwohner

原文摘要

In submodular facility location (SFL), introduced by Svitkina and Tardos (SODA 2006), clients $\mathcal{C}$ in a metric space are assigned to facilities $\mathcal{F}$. A facility serving a set $S \subseteq \cal C$ of clients incurs cost $g(S)$, where $g$ is a monotone submodular function common to all facilities. The best known approximation factor for SFL is $O(\log\log|\mathcal{C}|)$, due to Abbasi, Adamczyk, Bosch-Calvo, Byrka, Grandoni, Sornat, and Tinguely (ICALP 2024), improving on the $O(\log|\mathcal{C}|)$-approximation of Svitkina and Tardos. We show that this is best possible up to a constant factor in the value-oracle model by showing that there exists no algorithm with only polynomially many queries to $g$ with approximation ratio $o(\log\log|\mathcal{C}|)$. This bound holds for algorithms with unlimited computation time, and when $g$ is restricted to matroid rank functions. We also show that the natural linear programming relaxation has an integrality gap of $Θ(\log\log|\mathcal{C}|)$. Our lower bound follows by constructing an extension of Vondrák's symmetry-gap approach (FOCS 2009) for matroid rank functions.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.06008

Faster Cluster Decomposition with Stronger Bi-Criteria Guarantee

He Sun, Yong Zhang, Yacong Zhou

原文摘要

Partitioning the vertex set of a graph $G=(V,E)$ into clusters is a classical problem in computer science. As vertices within clusters should be well connected and cuts between clusters sparse, Oveis Gharan and Trevisan (SODA 2014, [OGT14]) proposed to measure the partition quality using inner and outer conductance, i.e., the bi-criteria guarantee. Specifically, when partitioning $V$ into vertex-disjoint subsets $P_1,\ldots, P_k$, every subgraph induced by $P_i$ should have high conductance, while the conductance of $P_i$ in $G$ should be low. They studied the existence of such a partition and designed a polynomial-time algorithm achieving this guarantee. For more than a decade, their algorithm remained the only one achieving this guarantee on its output clusters. In this paper, we study faster cluster decomposition algorithms under the same setup as [OGT14] and prove three results: 1. We make their existential result algorithmic: Our algorithms run in nearly-linear time, and achieve the same bi-criteria guarantee as the one in [OGT14] up to constant factors, including the asymptotically tight dependence on $k$. 2. We prove that their existential result holds algorithmically under a much weaker condition on the initial disjoint subsets. To achieve this, we develop a general algorithm framework that boosts the approximation guarantee of a solution: given an $α$-approximate solution to the $k$-way conductance problem, if $ρ_G(k+1)/ρ_G(k)=Ω\left(α/\varepsilon\right)$, then a $(1+\varepsilon)$-approximate solution can be obtained using only $O(k(1+\logα))$ minimum $s$-$t$ cut computations. 3. We present a nearly-linear time algorithm that finds clusters with the bi-criteria guarantee. Our algorithm has the same guarantee as the one in [OGT14], which, for any unweighted graph of $n$ vertices and $m$ edges, runs in $\widetilde{O}(knm^2)$ time.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05982

Single-Exponential FPT Algorithms for Minimum Bisection and Matching Cut via a New Unbreakable Decomposition

Soh Kumabe

原文摘要

Minimum Bisection asks whether the vertices of a graph can be partitioned into two parts whose sizes differ by at most one and with at most k edges between the two parts. Cygan, Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Saurabh (SICOMP 2019) gave the first fixed-parameter algorithm, which runs in O*(2^(O(k^3))) time, by introducing an unbreakable decomposition. Cygan, Komosa, Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, Saurabh, and Wahlström (TALG 2021) strengthened the decomposition and improved the running time to O*(2^(O(k log k))). However, whether the running time could be improved to single-exponential remained open. In this paper, we give a deterministic O*(2^(O(k)))-time algorithm for Minimum Bisection. Our algorithm also counts all feasible bisections. To obtain an O*(2^(O(k)))-time algorithm, we introduce two new techniques. First, we give a new algorithm that computes a (2k,k)-edge-unbreakable decomposition with adhesion at most 2k in O*(2^(O(k))) time by relaxing the unbreakability and adhesion guarantees of the decomposition in (TALG 2021). Second, we improve the running time of the dynamic program over the decomposition from O*(2^(O(k log k))) to O*(2^(O(k))) using important cuts, inclusion-exclusion, and formal logarithms and exponentials. Together with color coding, the same techniques give a deterministic O*(2^(O(k)))-time algorithm for Matching Cut, which asks for a cut whose crossing edges form a matching. This improves the O*(2^(O(k log k)))-time algorithm of Aravind and Saxena (IWOCA 2021).

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05981

Single-Variable Connectivity in Random $k$-SAT

Chunyang Wang, Yitong Yin, Yixiao Yu, Yiyao Zhang

原文摘要

We resolve longstanding questions about connectivity, reconstruction,and looseness in random $k$-SAT, for sufficiently large fixed $k$ as the number $n$ of variables tends to infinity. $\textbf{Single-variable connectivity up to reconstruction.}$ At clause densities up to \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*-o_k(1)\bigr),\] a polynomial-time algorithm joins a typical pair of satisfying assignments with high probability, using $O_k(n)$ single-variable flips that preserve satisfiability. Here $γ_*\approx0.871$ arises from a refined tree-reconstruction analysis. $\textbf{Settling the reconstruction threshold.}$ Building on the tools developed for connectivity, we determine the clustering threshold defined by the reconstruction criterion to be \[\frac{2^k}{k}\bigl(\ln k+\ln\ln k+γ_*+o_k(1)\bigr),\] below the classical cavity prediction with constant term $1+O(\ln\ln k/\ln k)$. $\textbf{Shattering before rigidity.}$ We locate the looseness--rigidity transition at a higher density, matching the classical cavity prediction for clustering: \[\frac{2^k}{k} \left(\ln k+\ln\ln k+1+Θ\left(\frac{\ln\ln k}{\ln k}\right)\right).\] Below the transition, we establish a stronger form of looseness: any variable of a typical solution can be reversed by logarithmically many single-variable flips that preserve satisfiability. Above it, reversing most variables of a typical solution requires changing linearly many variables (rigidity). Yet throughout an interval between reconstruction and rigidity, the solution space already splits into groups of exponentially small relative size separated by linear Hamming distance and linear energy barriers (shattering), while logarithmic looseness persists.

cs.DS · cs.DM

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arXiv ↗PDF ↗
版本更新 · v3arXiv:2502.14379

Achieving adaptivity and optimality for multi-armed bandits using Exponential-Kullback Leibler Maillard Sampling

Hao Qin, Kwang-Sung Jun, Chicheng Zhang

原文摘要

We study the problem of $K$-armed bandits with reward distributions belonging to a one-parameter exponential distribution family. In the literature, several criteria have been proposed to evaluate the performance of such algorithms, including Asymptotic Optimality, Minimax Optimality, Sub-UCB, and variance-adaptive regret bound. Thompson Sampling-based and Upper Confidence Bound-based algorithms have been employed to achieve some of these criteria. However, none of these algorithms simultaneously satisfy all the aforementioned criteria. In this paper, we design an algorithm, Exponential Kullback-LeiblerMaillard Sampling (abbrev. Exp-KL-MS), that achieves multiple optimality criteria simultaneously, including Asymptotic Optimality, minimax ratio with a $\sqrt{\ln (K)}$ factor, Sub-UCB, and a variance-adaptive regret bound. Our algorithm design follows the Minimum Empirical Divergence framework~\citep{honda2011asymptotically,maillard2011apprentissage}, with the exploration probability of arm $a$ proportional to $\text{exp}\left(-L(N_{t-1, a}) \text{KL}(\hatμ_{t-1, a}, \max_{a'} \hatμ_{t-1, a'})\right)$, where $L(\cdot)$ is an inverse temperature function, $N_{t-1, a}$ is the number of times arm $a$ has been pulled before time $t$, $\hatμ_{t-1, a}$ is the empirical mean of arm $a$ before time $t$, and $\text{KL}(\cdot, \cdot)$ is the Kullback-Leibler divergence between two distributions in the one-parameter exponential distribution family. Our analysis allows different choices of inverse temperature function $L(k)$. We also provide numerical simulations demonstrating the effectiveness of our algorithms.

cs.LG · cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05932

Faster Dynamic Maximal Independent Set against Adaptive Adversaries

Hongyi Chen, Shuyan Fan, Tianyi Zhang

原文摘要

We study the fully dynamic maximal independent set (MIS) problem: maintaining an MIS in an undirected graph on $n$ vertices with at most $m$ edges under edge insertions and deletions. Against an oblivious adversary, which fixes the edge update sequence in advance, the randomized algorithms of Behnezhad et al. (2019) and Chechik and Zhang (2019) achieve polylogarithmic update time. Against an adaptive adversary, which may choose each update after observing the previously reported MISs, the best previously known update bound was $O(m^{2/3})$, given by the deterministic algorithm of Gupta and Khan (2018, 2021). Assuming the combinatorial Boolean matrix multiplication hypothesis, Bernstein et al. (2026) proved that any combinatorial algorithm against an adaptive adversary requires at least $m^{1/2-o(1)}$ update time even under edge insertions alone, leaving a polynomial gap between upper and lower bounds. We present a randomized fully dynamic MIS algorithm with $\widetilde O(m^{5/8})=\widetilde O(m^{2/3-1/24})$ update time against adaptive adversaries. This gives the first polynomial improvement over the $m^{2/3}$ bound for adaptive fully dynamic MIS.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05920

Faster Isomorphism for Groups of Class 2

Xiaorui Sun

原文摘要

The finite-group isomorphism problem asks whether two groups, given by their Cayley tables, are isomorphic. Despite decades of study, obtaining an $N^{o(\log N)}$-time algorithm for general groups of order $N$ remains a major open challenge. Groups of nilpotency class $2$ have long been a central obstacle to achieving this goal. Recent breakthroughs have established $N^{o(\log N)}$-time algorithms for $p$-groups of class $2$ with odd $p$ (Sun, STOC '23; Ivanyos-Mendoza-Qiao-Sun-Zhang, FOCS '24; Grochow-Qiao-Stange-Sun, STOC '25) and for groups of Frattini class $2$ (Grochow-Ivanyos-Qiao-Sun, FOCS '26). However, the problem remained open for general groups of class $2$. In this paper, we present an algorithm for testing isomorphism of groups of class $2$, with running time $N^{O((\log N)^{25/26} \log \log N)}$. Our algorithm combines a tensor-based restriction of the search space, an encoding of group relations by alternating matrix tuples, and an orbit computation to recover the remaining central power information.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05916

Improved approximation algorithms for capacitated cluster editing

Cong Ma, Guangting Chen, Yong Chen, Guohui Lin, Jiaxuan Ma, Lusheng Wang

原文摘要

We study the minimum-disagreement cluster editing problem that, from a complete edge-signed graph, where a positive edge represents two vertices agreeing to each other while a negative represents a disagreement, constructs disjoint cliques of upper bounded size such that the total number of negative edges inside cliques and positive edges outside cliques is minimized. When the sizes of the candidate cliques are unbounded, the problem is NP-hard and APX-hard, and many approximation algorithms with improving ratios have been proposed in the literature and the state-of-the-art approximation ratio of $α= 1.3865 + ε$. The capacitated variant, in which the candidate cliques have sizes no greater than a given $k$, is less studied and the best approximation ratio is $5.37$. In this paper, we study the latter problem. We first present a downsizing procedure that achieves an approximation ratio of $α_k = 2 α+ 1$, based on any $α$-approximation algorithm for the uncapacitated problem. Secondly, we propose a hybrid of cluster-LP sampling and pivoting and show that for every fractional capacitated cluster-LP solution of cost $L$, the hybrid using ordinary pivoting has an expected cost at most $\frac 95 L + \frac 1{10} \OPT_k$, and the hybrid using conditional pivoting has an expected cost at most $\frac {306}{203} L + \frac {50}{203} \OPT_k$. A cardinality-constrained adaptation of the approximate dual-separation framework of García-Soriano and Schohn supplies the required fractional solution and yields a randomized $(1.9 + ε)$-approximation algorithm and a randomized $(1.7537 + ε)$-approximation algorithm, respectively.

cs.DS · cs.LG

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05878

Improved Bounds on Distance Preservers

Yanyu Chen, Yu Chen, Zihan Tan, Mingyang Yang

原文摘要

Given an undirected unweighted graph $G$ on $n$ vertices and a set $P$ of $p$ pairs of its vertices, a distance preserver is a sparse subgraph of $G$ that exactly preserves the shortest-path distances between all pairs in $P$. In this paper, we prove a new upper bound of $O(n^{8/9} p^{4/9})$ for distance preservers when $p\in [n^{1/2}, n]$. Our result improves the previous bound of $O\big(n^{2/3}p^{2/3}+np^{1/3}\big)$ by Bodwin and Vassilevska Williams and $O(\min\{n+n^{1/2}p,\,np^{1/2}\})$ by Coppersmith and Elkin, particularly in the regime $p\in [n^{7/10}, n]$, and narrows the gap toward the conjectured $O(n^{2/3}p^{2/3}+n)$ bound. Using the techniques developed for distance preservers, we are also able to obtain pairwise additive $+2$ spanners with $O(n+n^{9/10}p^{2/5})$ edges, pairwise additive $+4$ spanners with $O(n+n^{24/25}p^{8/25})$ edges, and new bounds for sourcewise spanners, improving upon previous results in certain parameter regimes.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05839

Makespan Bounds for Synchronous Pebble Motion on Graphs

Lucy McManamon, Jingjin Yu

原文摘要

We study the largest minimum makespan of labeled pebble motion on graphs under a model that permits synchronous pebble movement. While classical pebble motion on graphs allows only one pebble to move at a time, our model permits any number of pebbles to move to an adjacent vertex, provided that no two pebbles collide. At full occupancy every vertex is assigned a pebble and a legal round consists of disjoint cycle rotations. We prove a series of makespan bounds for this problem on graphs with varying structure and pebble density. For cycles with a chord, we extend a known quadratic bound at span two to every fixed span. For random cubic graphs, we convert permutation routing via matchings into legal schedules in two ways, applying cycle-cover batching at full occupancy and disjoint vacancy-delivery paths below full occupancy. For simple 3-connected planar cubic graphs, we prove a tight linear class maximum at every pebble count. We extend the results to increasing degree, common-neighbor conditions, and vertex connectivity. The results describe how graph connectivity, cycle structure, expansion, planarity, and occupancy affect worst-case routing time.

cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05813

Towards the Query Complexity of Sampling under Strong Log-Concavity and Log-Sobolev Inequalities

Huanjian Zhou, Taiji Suzuki

原文摘要

We study the query complexity of sampling from smooth distributions on $\mathbb R^d$ using first-order oracles. For $d$-dimensional strongly log-concave targets with condition number $κ=2$, we prove that any randomized sampler requires $\exp({Ω(\log^{1/3} d)})$ queries even from a constant warm start. This rules out polylogarithmic dimension dependence and strictly separates strongly log-concave sampling from Gaussian sampling. For targets with $L$-smooth potentials satisfying an $α$-log-Sobolev inequality with $κ= L/α$, we prove an $Ω(κ)$ query lower bound for $d\gtrsim\log^2κ$. This shows that log-Sobolev sampling is strictly harder than strongly log-concave sampling. Our proofs construct instance-dependent tests that target samples typically pass but outputs of algorithms making few queries typically fail. For strongly log-concave targets, we embed a sequence of hidden Gaussian into high-degree tensor polynomial with smooth gates, forcing any algorithm to uncover the layers sequentially. The log-Sobolev construction hides a constant fraction of the target mass in angular caps while controlling the log-Sobolev constant via delicate estimates.

cs.CC · cs.DS · math.ST · stat.ML

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05807

Near-Optimal Complexity of Finding Approximate Local Minima of Nonconvex Finite Sums

Qihao Zhou

原文摘要

We study the component-oracle complexity of finding approximate local minima of nonconvex finite sums under mean-squared averaged smoothness with parameter $L$ and a $ρ$-Lipschitz average Hessian. Each query returns the value, gradient, and one Hessian-vector product of a component. We show that the complexity in the high-accuracy regime $\varepsilon\le c_0L^2/(nρ)$ is $\tildeΘ(n^{3/4}L^{1/2}ρ^{1/4}Δ\varepsilon^{-7/4})$ for a sufficiently large initial gap $Δ$. We attain the upper bound by adapting the inverse solver and eigenvector routine of FastCubic to averaged smoothness. Our lower bound assigns the bridge edges of a nested chain to random components; a replicated spectral construction gives the same bound for unrestricted randomized algorithms. Under individual smoothness, a factor of $n^{1/4}$ remains for $\varepsilon\le cL^2/(n^2ρ)$, up to logarithmic factors. In the complementary low-accuracy regime, we propose PathVR, which uses $\tilde O(n+\sqrt{n}LΔ\varepsilon^{-2})$ queries and matches a lower bound from dense weak hiding.

math.OC · cs.DS

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arXiv ↗PDF ↗
新论文 · v1arXiv:2611.05794

Near linear mixing of random-cluster Glauber dynamics on random regular graphs

Zhangsong Li

原文摘要

We study the random-cluster model on locally treelike graphs at low temperatures. This is a model of dependent percolation parametrized by an edge probability $p \in (0,1)$ and a clustering weight $q \in [1,\infty)$, generalizing independent Bernoulli percolation ($q=1$) and closely related to the classical ferromagnetic Ising and Potts spin systems at integer $q$. For $q>2$, at parameter $p$ below the tree uniqueness threshold $p_{\mathsf u}(q,d)$, it is expected that sampling is easy and local Markov chains mix rapidly on all graphs with degree bounded by $d$. On typical graphs, the same is predicted at $p>p_{\mathsf s}(q,d)$, where $p_{\mathsf s}(q,d)$ is a second uniqueness transition point on the $d$-regular wired tree. Our result confirms this prediction for random regular graphs, showing that the random-cluster Glauber dynamics mix in time $O(n^{1+o(1)})$ on a random $d$-regular graph for all fixed $q>2$ and $p>p_{\mathsf s}$. For comparison, the best prior sampling results for these models that held for all $p>p_{\mathsf s}$ required $q\gtrsim \log d$ \cite{BGPZ26}. The key difficulty is that simply utilizing the contraction of cavity messages on $d$-regular trees does not suffice to cover the entire regime $q>2$. To address this issue, motivated by \cite{LS26}, we develop a careful bootstrap argument to show the shrinkage of disagreement under grand coupling. The core ideas of the proof were generated using GPT-6 and the Pharos system; the proof was then refined and simplified by the author.

math.PR · cs.DS · math-ph

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arXiv ↗PDF ↗
版本更新 · v2arXiv:2606.13133

Learning-Augmented Approximation for Unrelated-Machines Makespan Scheduling

Kaito Baba, Evripidis Bampis, Giorgos Mitropoulos

原文摘要

Recently, Antoniadis et al. (ICLR 2025) proposed a framework for incorporating predictions to approximate NP-hard selection problems. Despite its simplicity, this approach tightly matches theoretical lower bounds, making its generalization highly compelling. We address an open question raised in the work of Antoniadis et al., concerning the extension of this approach to other important problems outside the class of selection problems, such as scheduling. We develop a learning-augmented algorithm for the makespan minimization problem on unrelated machines, denoted by $R\|C_{\max}$. By using predictions of heavy job assignments, we achieve a polynomial-time $(1+\varepsilon)$-approximation for accurate predictions that smoothly degrades to a worst-case 2-approximation as the error increases. We conclude our work with an empirical analysis of our method.

cs.DS · cs.LG

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新论文 · v1arXiv:2611.05777

Sublinear Metric Distortion for Matching

Moses Charikar, Alireza Haqi, Jabari Hastings, Yichen Huang, Prasanna Ramakrishnan

原文摘要

We study metric distortion in one-sided matching. Given $n$ agents and $n$ items in an unknown metric space, our goal is to match agents to items so that the total distance between matched pairs is small. However, instead of direct access to the distances in the metric, we are only given each agent's ranking of the items by distance. What is the smallest worst-case approximation factor (called the distortion) that a mechanism can guarantee in expectation? Despite significant interest over the last decade, the best-known guarantee has been the $O(n)$ distortion of Random Serial Dictatorship since the beginning. We give a matching mechanism with distortion $O(n^{\log_4 3})$, the first sublinear upper bound. The key idea is to use a partial matching derived from a truncated version of either Random Serial Dictatorship or the Probabilistic Serial rule. We then complete this matching using a new, general-purpose cluster completion algorithm that ensures unmatched agents receive nearby items. Along the way, we pin down the optimal distortion of Random Serial Dictatorship and Probabilistic Serial up to logarithmic and constant factors, respectively, and give improved lower bounds for truthful mechanisms.

cs.GT · cs.DM · cs.DS

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新论文 · v1arXiv:2611.05769

Noncrossing is All You Need: Beyond Identical Delay Functions in Online Matching with Delays on General Metrics

Yossi Azar, Junhao Gan, Shahar Lewkowicz, Xiao Sun, Seeun William Umboh

原文摘要

Online optimization with delays has emerged as a powerful paradigm with numerous applica- tions, including matching, caching, server management, network design, and set cover. In most of these problems, competitive algorithms are known for arbitrary monotone delay functions, even when different requests are associated with different delay functions. The notable exception is online matching with delays, where all previously known algorithms achieving constant or polylogarithmic competitive ratios require the delay functions to be identical, either linear or concave, and known in advance. We study the online minimum-cost perfect matching with delays problem on general metrics. We introduce the broad class of noncrossing delay functions, which strictly generalizes the classic setting of identical concave delay functions while naturally modeling scenarios in which different requests may experience different delay costs. We present the first deterministic $O(1)$-competitive algorithm for the single-location online matching problem with noncrossing delay functions in the non-clairvoyant setting as well as a randomized $O(lognlogΔ+log^2Δ)$-competitive algorithm of an arbitrary metric space where $n$ is the number of points of the metric space and $Δ$ is its aspect ratio. Our result allows each request to have an arbitrary noncrossing delay function, thereby simultaneously removing the assumptions of identical delay functions, concavity, and clairvoyance that were required by all previous constant or polylogarithmic-competitive algorithms.

cs.DS

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新论文 · v1arXiv:2611.05756

Flow-Optimal Separators for Minor-Free and Induced-Minor-Free Graphs in Linear Time via Follow the Regularized Leader

Hung Le, Jack Spalding-Jamieson

原文摘要

We give a new algorithm for finding balanced separators in minor-free and induced-minor-free graphs. A key technical idea is a new connection to online learning: we formulate a continuous, non-convex optimization problem and solve it using the Follow the Regularized Leader (FTRL) framework. Our implementation of each FTRL step yields a purely combinatorial algorithm that assigns and adjusts vertex weights over multiple rounds. For graphs with $n$ vertices and $m$ edges, we obtain the following results. (1) For $H$-induced-minor-free graphs, we find a balanced separator of size $O(\log(|V(H)|)\sqrt{|V(H)|+|E(H)|}\sqrt{m})$ in expected linear time for fixed $H$. As a corollary, we find balanced separators of size $O(\sqrt{m})$ in string graphs in expected linear time, without requiring string representations. (2) For $K_h$-minor-free graphs, we find a balanced separator of size $O(h\log h\sqrt{n})$ in expected linear time for fixed $h$. Our algorithm unifies two long lines of work on separators in minor-free graphs, dating back to Alon, Seymour, and Thomas (1990): one optimizing separator size and the other optimizing running time. It also gives a natural interpretation of the vertex weighting scheme in a recent linear-time algorithm by Bonnet, Korhonen, Le, Li, and Masarik (2026). The separator bounds for both graph classes are flow-optimal: techniques based on flow-cut duality, including ours, cannot yield balanced separators of asymptotically smaller size.

cs.DS

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新论文 · v1arXiv:2611.05726

Do Language Models Know Which Constraints Belong Together?

Junbo Jacob Lian, Kaixuan Chen, Hanzhang Qin, Huiling Chen, Chaoyu Zhang, Chung-Piaw Teo

原文摘要

Dantzig-Wolfe reformulation strengthens the linear relaxation of a mixed-integer program by convexifying a chosen set of constraint families, and every candidate costs a column generation run. The choice is hard when families strengthen the bound only together, because a search that adds one family at a time then stops at the linear relaxation. The classical inequalities of lot sizing, network design and facility location rest on such zero-marginal groups. We prove that when each evaluation convexifies at most $s$ of $k$ families, any method that observes only bounds needs a covering number of evaluations, of order $k^w$, to find a hidden group of $w$ families, also on actual Dantzig-Wolfe relaxations. A prediction that misses $e$ members reduces the cost to $Θ(k^e)$, through one staged search that does not need to know $e$, so the evaluations a prediction saves measure what the predictor knows. We study two sources of predictions. A one-pass lookahead at an optimal LP dual is provably blind to groups whose families share no variable, and a language model reads the names and text of the model. JointDW verifies the prediction of the language model with the staged search and interleaves it with the lookahead at a bounded loss. Groups that strengthen the bound only jointly appear in a quarter of the MIPLIB-NL models whose bound can be improved. There the strongest language models identify them about as often as the lookahead, their predictions rest on the names for textbook models, and reasoning about doubles the accuracy of the same model. On larger models with distracting families, the lookahead finds the group in one evaluation of three families. Language models given only the text seldom name it, while one reasoning model given every coefficient names it in all but one answer and then needs fewer evaluations than the lookahead for pairs of families.

cs.LG · cs.AI · cs.DS · math.OC

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新论文 · v1arXiv:2611.05725

Faster Approximate Text-to-Pattern Hamming Distances via Near-Optimal Sumset Estimation

Nick Fischer, Yael Kirkpatrick, Vasileios Nakos

原文摘要

\emph{Text-to-Pattern Hamming Distances} is one of the most studied problems in string algorithms: given a text $T$ of length~$n$ and a pattern~$P$, the task is to compute the Hamming distance between~$P$ and every length-$|P|$ substring of~$T$. As exact algorithms have been stuck at time~$\tilde O(n^{3/2})$ for decades, a long line of work has turned to the $(1+ε)$-approximate version of the problem. Milestones include Karloff's $\tilde O(ε^{-2} n)$-time algorithm [IPL '93], followed by randomized improvements to $\tilde O(ε^{-1} n)$ by Kopelowitz and Porat [FOCS '15] and recently to $\tilde O(ε^{-0.915} n)$ by Chan, Jin, Vassilevska W., and Xu [FOCS '23]. In this work, we significantly improve this bound to $\tilde O(ε^{-2/3} n)$, using a considerably simpler algorithm. Our algorithm follows the framework of Chan, Jin, Vassilevska W., and Xu, which reduces the problem to a variant of 3SUM that we call \emph{Sumset Estimation}. We solve Sumset Estimation optimally (up to logarithmic factors) by replacing their matrix multiplication techniques with ideas from sparse recovery. Specifically, we adapt the classic Count Sketch by replacing its random signs with random complex roots of unity---a simple idea that may find further applications for other additive problems. Finally, we give evidence that our running time might in fact be optimal: we show that the $ε^{-2/3}$ dependence is \emph{best possible} for a certain natural class of algorithms, including ours and all previous algorithms.

cs.DS

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版本更新 · v2arXiv:2605.14994

Sharp Bounds on the Eigenvalues of Kikuchi Graphs and Applications to Quantum Max Cut

Ainesh Bakshi, Arpon Basu, Pravesh Kothari, Anqi Li

原文摘要

We prove that the maximum eigenvalue of the (both signed and unsigned) Laplacian of level $k$ Kikuchi graph of any graph $G$ with $m$ edges is at most $m+k$. This confirms four recent conjectures of Apte, Parekh, and Sud. As applications, we obtain that tensor products of one and two qubit product states achieve an approximation ratio of $5/8$ for Quantum Max Cut and $5/7$ for the XY Hamiltonian. Moreover, combining our bounds with the algorithms analyzed by Apte, Parekh, and Sud, yields efficient algorithms achieving an approximation ratio of $0.614$ for Quantum Max Cut and $0.674$ for the XY Hamiltonian. Finally, we also make modest progress on Brouwer's conjecture and improve Lew's bound on the sum of the top-$k$ eigenvalues of a Graph Laplacian.

quant-ph · cs.DS · math.CO

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新论文 · v1arXiv:2611.06957

An Integrality-Gap Lower Bound of 1.2325 for Minimum Multiway Cut

Dimitar Chakarov, Yury Makarychev

原文摘要

We prove that the integrality gap of the Călinescu-Karloff-Rabani relaxation for Minimum Multiway Cut is at least $1.2325$ when the number of terminals is large. By a result of Manokaran, Naor, Raghavendra, and Schwartz (2008), it is therefore NP-hard, under the Unique Games Conjecture, to approximate Minimum Multiway Cut within a factor of $1.2325$. This improves the previously best known lower bound of $1.20016$ due to Bérczi, Chandrasekaran, Király, and Madan (2020).

cs.DS

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新论文 · v1arXiv:2611.05713

Incremental Minimum Spanning Trees on a Napkin

Lucian Bicsi, Adrian Budău

原文摘要

In the incremental minimum spanning forest problem, weighted edges arrive one at a time. After each insertion we must report the edge that leaves the minimum spanning forest, and queries ask for the bottleneck distance of two vertices: the smallest possible maximum edge cost on a path between them. Link-cut trees solve both in $O(\log n)$ amortized time. In 2021 we wrote the folklore BottleneckTree, a short competitive-programming structure that stores one parent pointer and one cost per vertex, and conjectured that it also runs in $O(\log n)$ amortized time. We show that this conjecture is false: on a family of inputs, its amortized cost per operation is $Ω(\sqrt{n})$. We then present a structure with the same state, two arrays and a single loop, and prove that it runs in $O(\log n)$ amortized time per insertion and per query. Unlike the AM-tree of Ding, Gu and Sun, which attains the same bound, it stores no subtree sizes and maintains no balance invariant. For a connected graph with distinct costs, the structure can be viewed as an implicit link-cut tree on the Kruskal reconstruction tree: each rotation corresponds to exactly one preferred-child change, and each skip acts on a stale path-parent pointer. In experiments it is 1.3-2.1 times faster than the folklore BottleneckTree on insertions, up to 480 times faster on its adversarial input, and 6.3-16 times faster than an explicit link-cut tree.

cs.DS

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新论文 · v1arXiv:2611.05712

Reachability Diameter: Beyond the Square-Root Barrier, with Polynomial Hardness

Hector Tierno, Shuang Yang

原文摘要

The reachability diameter of a directed graph is the maximum finite shortest-path distance across all pairs of vertices. It has received recent attention under this name, although the underlying quantity has long played a central role in shortcut set and parallel reachability. Notably, Bals et al. establish hardness for every polynomial-factor approximation in nearly-linear time for weighted digraphs. Yet they give $n^{1/2+o(1)}$-approximation in the unweighted regime with nearly-linear running time, and their conditional lower bounds rule out certain constants only, leaving open whether polynomial loss is necessary. We give a randomized algorithm that estimates the reachability diameter of an unweighted digraph within a factor $n^{1/3+o(1)}$, with high probability, in $\widetilde{O}(m+n)$ time. We complement our algorithm with a lower bound conditional on the 3-uniform Hyperclique hypothesis, ruling out an $n^{2/15-\varepsilon}$-approximation in $\widetilde{O}(m+n)$ time for every fixed $\varepsilon\in(0,2/15)$, establishing the necessity of polynomial loss for nearly-linear time algorithms.

cs.DS

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版本更新 · v3arXiv:2604.05152

Polynomial and Pseudopolynomial Algorithms for Two Classes of Bin Packing Instances

Renan Fernando Franco da Silva, Vinícius Loti de Lima, Rafael C. S. Schouery, Jean-François Côté, Manuel Iori

原文摘要

The Bin Packing Problem (BPP) and Cutting Stock Problem (CSP) are classical combinatorial optimization problems extensively studied since the 1960s. State-of-the-art exact algorithms are based on set-cover and arc-flow models whose linear relaxation, rounded up, matches the integer optimum for most benchmark instances, a condition known as the Integer Round-up Property (IRUP). In 2016, Delorme et al. showed that most existing instances could be solved within ten minutes by approaches exploiting this property. This motivated them to introduce two new classes of instances, Augmented IRUP (AI) and Augmented Non-IRUP (ANI), designed to challenge the exact algorithms available at the time. Although these classes have motivated significant advances over the past decade, 13 out of 500 AI and ANI instances remain unsolved within standard time limits in the literature. In this paper, we show that while AI and ANI are particularly hard for mixed-integer programming methods, the BPP restricted to these classes is not strongly NP-hard. We present polynomial-time algorithms for the AI class and pseudopolynomial-time algorithms for the ANI class, which solve all such instances orders of magnitude faster than previous approaches. They are also straightforward to adapt to the Skiving Stock Problem, the dual counterpart of the CSP. In addition, they can be used as preprocessing routines in exact methods, as their runtime is independent of the instance class, although they are guaranteed to return an optimality status only for instances belonging to the class for which they were designed.

cs.DS

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新论文 · v1arXiv:2611.05669

The Harmonic Online $k$-Server Algorithm Is Poly(k)-Competitive

Yichen Huang

原文摘要

The online $k$-server problem is one of the most fundamental problems in online algorithms. In this problem, $k$ servers are located in a metric space and serve a sequence of requests at points in the metric space by moving a server there. The Harmonic algorithm is a very natural algorithm: it chooses a server with probability inversely proportional to its distance from the request. It is also highly efficient, being memoryless and answering each request in $O(k)$ time. However, its competitive ratio has remained unsettled between the $O(2^k \log k)$ upper bound (Bartal and Grove, JACM 2000) and the $k(k+1)/2$ lower bound (Raghavan and Snir, 1994, IBM J. Res. Dev.). We prove that Harmonic is $2k^4$-competitive on every metric space against adaptive online adversaries. Our analysis uses a potential combining corrected electrical energy with the distances of online servers to their assigned adversary locations.

cs.DS

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