{"title":"The Power of Migrations in Dynamic Bin Packing","authors":"Konstantina Mellou, Marco Molinaro, Rudy Zhou","doi":"arxiv-2408.13178","DOIUrl":"https://doi.org/arxiv-2408.13178","url":null,"abstract":"In the Dynamic Bin Packing problem, $n$ items arrive and depart the system in\u0000an online manner, and the goal is to maintain a good packing throughout. We\u0000consider the objective of minimizing the total active time, i.e., the sum of\u0000the number of open bins over all times. An important tool for maintaining an\u0000efficient packing in many applications is the use of migrations; e.g.,\u0000transferring computing jobs across different machines. However, there are large\u0000gaps in our understanding of the approximability of dynamic bin packing with\u0000migrations. Prior work has covered the power of no migrations and $> n$\u0000migrations, but we ask the question: What is the power of limited ($leq n$)\u0000migrations? Our first result is a dichotomy between no migrations and linear migrations:\u0000Using a sublinear number of migrations is asymptotically equivalent to doing\u0000zero migrations, where the competitive ratio grows with $mu$, the ratio of the\u0000largest to smallest item duration. On the other hand, we prove that for every\u0000$alpha in (0,1]$, there is an algorithm that does $approx alpha n$\u0000migrations and achieves competitive ratio $approx 1/alpha$ (in particular,\u0000independent of $mu$); we also show that this tradeoff is essentially best\u0000possible. This fills in the gap between zero migrations and $> n$ migrations in\u0000Dynamic Bin Packing. Finally, in light of the above impossibility results, we introduce a new\u0000model that more directly captures the impact of migrations. Instead of limiting\u0000the number of migrations, each migration adds a delay of $C$ time units to the\u0000item's duration; this commonly appears in settings where a blackout or set-up\u0000time is required before the item can restart its execution in the new bin. In\u0000this new model, we prove a $O(min (sqrt{C}, mu))$-approximation, and an\u0000almost matching lower bound.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Adaptive complexity of log-concave sampling","authors":"Huanjian Zhou, Baoxiang Wang, Masashi Sugiyama","doi":"arxiv-2408.13045","DOIUrl":"https://doi.org/arxiv-2408.13045","url":null,"abstract":"In large-data applications, such as the inference process of diffusion\u0000models, it is desirable to design sampling algorithms with a high degree of\u0000parallelization. In this work, we study the adaptive complexity of sampling,\u0000which is the minimal number of sequential rounds required to achieve sampling\u0000given polynomially many queries executed in parallel at each round. For\u0000unconstrained sampling, we examine distributions that are log-smooth or\u0000log-Lipschitz and log strongly or non-strongly concave. We show that an almost\u0000linear iteration algorithm cannot return a sample with a specific exponentially\u0000small accuracy under total variation distance. For box-constrained sampling, we\u0000show that an almost linear iteration algorithm cannot return a sample with\u0000sup-polynomially small accuracy under total variation distance for log-concave\u0000distributions. Our proof relies upon novel analysis with the characterization\u0000of the output for the hardness potentials based on the chain-like structure\u0000with random partition and classical smoothing techniques.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142202686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Approximating $δ$-Covering","authors":"Tim A. Hartmann, Tom Janßen","doi":"arxiv-2408.04517","DOIUrl":"https://doi.org/arxiv-2408.04517","url":null,"abstract":"$delta$-Covering, for some covering range $delta>0$, is a continuous\u0000facility location problem on undirected graphs where all edges have unit\u0000length. The facilities may be positioned on the vertices as well as on the\u0000interior of the edges. The goal is to position as few facilities as possible\u0000such that every point on every edge has distance at most $delta$ to one of\u0000these facilities. For large $delta$, the problem is similar to dominating set,\u0000which is hard to approximate, while for small $delta$, say close to $1$, the\u0000problem is similar to vertex cover. In fact, as shown by Hartmann et al. [Math.\u0000Program. 22], $delta$-Covering for all unit-fractions $delta$ is polynomial\u0000time solvable, while for all other values of $delta$ the problem is NP-hard. We study the approximability of $delta$-Covering for every covering range\u0000$delta>0$. For $delta geq 3/2$, the problem is log-APX-hard, and allows an\u0000$mathcal O(log n)$ approximation. For every $delta < 3/2$, there is a\u0000constant factor approximation of a minimum $delta$-cover (and the problem is\u0000APX-hard when $delta$ is not a unit-fraction). We further study the dependency\u0000of the approximation ratio on the covering range $delta < 3/2$. By providing\u0000several polynomial time approximation algorithms and lower bounds under the\u0000Unique Games Conjecture, we narrow the possible approximation ratio, especially\u0000for $delta$ close to the polynomial time solvable cases.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"21 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946268","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Paweł Gawrychowski, Egor Gorbachev, Tomasz Kociumaka
{"title":"Core-Sparse Monge Matrix Multiplication: Improved Algorithm and Applications","authors":"Paweł Gawrychowski, Egor Gorbachev, Tomasz Kociumaka","doi":"arxiv-2408.04613","DOIUrl":"https://doi.org/arxiv-2408.04613","url":null,"abstract":"The task of min-plus matrix multiplication often arises in the context of\u0000distances in graphs and is known to be fine-grained equivalent to the All-Pairs\u0000Shortest Path problem. The non-crossing property of shortest paths in planar\u0000graphs gives rise to Monge matrices; the min-plus product of $ntimes n$ Monge\u0000matrices can be computed in $O(n^2)$ time. Grid graphs arising in sequence\u0000alignment problems, such as longest common subsequence or longest increasing\u0000subsequence, are even more structured. Tiskin [SODA'10] modeled their behavior\u0000using simple unit-Monge matrices and showed that the min-plus product of such\u0000matrices can be computed in $O(nlog n)$ time. Russo [SPIRE'11] showed that the\u0000min-plus product of arbitrary Monge matrices can be computed in time\u0000$O((n+delta)log^3 n)$ parameterized by the core size $delta$, which is\u0000$O(n)$ for unit-Monge matrices. In this work, we provide a linear bound on the core size of the product\u0000matrix in terms of the core sizes of the input matrices and show how to solve\u0000the core-sparse Monge matrix multiplication problem in $O((n+delta)log n)$\u0000time, matching the result of Tiskin for simple unit-Monge matrices. Our\u0000algorithm also allows $O(log delta)$-time witness recovery for any given\u0000entry of the output matrix. As an application of this functionality, we show\u0000that an array of size $n$ can be preprocessed in $O(nlog^3 n)$ time so that\u0000the longest increasing subsequence of any sub-array can be reconstructed in\u0000$O(l)$ time, where $l$ is the length of the reported subsequence; in\u0000comparison, Karthik C. S. and Rahul [arXiv'24] recently achieved\u0000$O(l+n^{1/2}log^3 n)$-time reporting after $O(n^{3/2}log^3 n)$-time\u0000preprocessing. Our faster core-sparse Monge matrix multiplication also enabled\u0000reducing two logarithmic factors in the running times of the recent algorithms\u0000for edit distance with integer weights [Gorbachev & Kociumaka, arXiv'24].","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"171 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946267","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Travis Gagie, Giovanni Manzini, Gonzalo Navarro, Marinella Sciortino
{"title":"Movelet Trees","authors":"Travis Gagie, Giovanni Manzini, Gonzalo Navarro, Marinella Sciortino","doi":"arxiv-2408.04537","DOIUrl":"https://doi.org/arxiv-2408.04537","url":null,"abstract":"We combine Nishimoto and Tabei's move structure with a wavelet tree to show\u0000how, if $T [1..n]$ is over a constant-sized alphabet and its Burrows-Wheeler\u0000Transform (BWT) consists of $r$ runs, then we can store $T$ in $O left( r log\u0000frac{n}{r} right)$ bits such that when given a pattern $P [1..m]$, we can\u0000find the BWT interval for $P$ in $O (m)$ time.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"86 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Regularized Unconstrained Weakly Submodular Maximization","authors":"Yanhui Zhu, Samik Basu, A. Pavan","doi":"arxiv-2408.04620","DOIUrl":"https://doi.org/arxiv-2408.04620","url":null,"abstract":"Submodular optimization finds applications in machine learning and data\u0000mining. In this paper, we study the problem of maximizing functions of the form\u0000$h = f-c$, where $f$ is a monotone, non-negative, weakly submodular set\u0000function and $c$ is a modular function. We design a deterministic approximation\u0000algorithm that runs with ${{O}}(frac{n}{epsilon}log frac{n}{gamma\u0000epsilon})$ oracle calls to function $h$, and outputs a set ${S}$ such that\u0000$h({S}) geq\u0000gamma(1-epsilon)f(OPT)-c(OPT)-frac{c(OPT)}{gamma(1-epsilon)}logfrac{f(OPT)}{c(OPT)}$,\u0000where $gamma$ is the submodularity ratio of $f$. Existing algorithms for this\u0000problem either admit a worse approximation ratio or have quadratic runtime. We\u0000also present an approximation ratio of our algorithm for this problem with an\u0000approximate oracle of $f$. We validate our theoretical results through\u0000extensive empirical evaluations on real-world applications, including vertex\u0000cover and influence diffusion problems for submodular utility function $f$, and\u0000Bayesian A-Optimal design for weakly submodular $f$. Our experimental results\u0000demonstrate that our algorithms efficiently achieve high-quality solutions.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946266","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Simple Linear-time Repetition Factorization","authors":"Yuki Yonemoto, Shunsuke Inenaga","doi":"arxiv-2408.04253","DOIUrl":"https://doi.org/arxiv-2408.04253","url":null,"abstract":"A factorization $f_1, ldots, f_m$ of a string $w$ of length $n$ is called a\u0000repetition factorization of $w$ if $f_i$ is a repetition, i.e., $f_i$ is a form\u0000of $x^kx'$, where $x$ is a non-empty string, $x'$ is a (possibly-empty) proper\u0000prefix of $x$, and $k geq 2$. Dumitran et al. [SPIRE 2015] presented an\u0000$O(n)$-time and space algorithm for computing an arbitrary repetition\u0000factorization of a given string of length $n$. Their algorithm heavily relies\u0000on the Union-Find data structure on trees proposed by Gabow and Tarjan [JCSS\u00001985] that works in linear time on the word RAM model, and an interval stabbing\u0000data structure of Schmidt [ISAAC 2009]. In this paper, we explore more\u0000combinatorial insights into the problem, and present a simple algorithm to\u0000compute an arbitrary repetition factorization of a given string of length $n$\u0000in $O(n)$ time, without relying on data structures for Union-Find and interval\u0000stabbing. Our algorithm follows the approach by Inoue et al. [ToCS 2022] that\u0000computes the smallest/largest repetition factorization in $O(n log n)$ time.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"11 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946269","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Lower Bounds for Approximate (& Exact) k-Disjoint-Shortest-Paths","authors":"Rajesh Chitnis, Samuel Thomas, Anthony Wirth","doi":"arxiv-2408.03933","DOIUrl":"https://doi.org/arxiv-2408.03933","url":null,"abstract":"Given a graph $G=(V,E)$ and a set $T={ (s_i, t_i) : 1leq ileq k\u0000}subseteq Vtimes V$ of $k$ pairs, the $k$-vertex-disjoint-paths (resp.\u0000$k$-edge-disjoint-paths) problem asks to determine whether there exist~$k$\u0000pairwise vertex-disjoint (resp. edge-disjoint) paths $P_1, P_2, ..., P_k$ in\u0000$G$ such that, for each $1leq ileq k$, $P_i$ connects $s_i$ to $t_i$. Both\u0000the edge-disjoint and vertex-disjoint versions in undirected graphs are\u0000famously known to be FPT (parameterized by $k$) due to the Graph Minor Theory\u0000of Robertson and Seymour. Eilam-Tzoreff [DAM `98] introduced a variant, known\u0000as the $k$-disjoint-shortest-paths problem, where each individual path is\u0000further required to be a shortest path connecting its pair. They showed that\u0000the $k$-disjoint-shortest-paths problem is NP-complete on both directed and\u0000undirected graphs; this holds even if the graphs are planar and have unit edge\u0000lengths. We focus on four versions of the problem, corresponding to considering\u0000edge/vertex disjointness, and to considering directed/undirected graphs.\u0000Building on the reduction of Chitnis [SIDMA `23] for $k$-edge-disjoint-paths on\u0000planar DAGs, we obtain the following inapproximability lower bound for each of\u0000the four versions of $k$-disjoint-shortest-paths on $n$-vertex graphs: - Under\u0000Gap-ETH, there exists a constant $delta>0$ such that for any constant\u0000$0<epsilonleq frac{1}{2}$ and any computable function $f$, there is no\u0000$(frac{1}{2}+epsilon)$-approx in $f(k)cdot n^{deltacdot k}$ time. We\u0000further strengthen our results as follows: Directed: Inapprox lower bound for\u0000edge-disjoint (resp. vertex-disjoint) paths holds even if the input graph is a\u0000planar (resp. 1-planar) DAG with max in-degree and max out-degree at most $2$.\u0000Undirected: Inapprox lower bound for edge-disjoint (resp. vertex-disjoint)\u0000paths hold even if the input graph is planar (resp. 1-planar) and has max\u0000degree $4$.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946273","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joakim Blikstad, Ola Svensson, Radu Vintan, David Wajc
{"title":"Deterministic Online Bipartite Edge Coloring","authors":"Joakim Blikstad, Ola Svensson, Radu Vintan, David Wajc","doi":"arxiv-2408.03661","DOIUrl":"https://doi.org/arxiv-2408.03661","url":null,"abstract":"We study online bipartite edge coloring, with nodes on one side of the graph\u0000revealed sequentially. The trivial greedy algorithm is $(2-o(1))$-competitive,\u0000which is optimal for graphs of low maximum degree, $Delta=O(log n)$ [BNMN\u0000IPL'92]. Numerous online edge-coloring algorithms outperforming the greedy\u0000algorithm in various settings were designed over the years (e.g., AGKM FOCS'03,\u0000BMM SODA'10, CPW FOCS'19, BGW SODA'21, KLSST STOC'22, BSVW STOC'24), all\u0000crucially relying on randomization. A commonly-held belief, first stated by\u0000[BNMN IPL'92], is that randomization is necessary to outperform greedy. Surprisingly, we refute this belief, by presenting a deterministic algorithm\u0000that beats greedy for sufficiently large $Delta=Omega(log n)$, and in\u0000particular has competitive ratio $frac{e}{e-1}+o(1)$ for all\u0000$Delta=omega(log n)$. We obtain our result via a new and surprisingly simple\u0000randomized algorithm that works against adaptive adversaries (as opposed to\u0000oblivious adversaries assumed by prior work), which implies the existence of a\u0000similarly-competitive deterministic algorithm [BDBKTW STOC'90].","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946272","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Finding longer cycles via shortest colourful cycle","authors":"Andreas Björklund, Thore Husfeldt","doi":"arxiv-2408.03699","DOIUrl":"https://doi.org/arxiv-2408.03699","url":null,"abstract":"We consider the parameterised $k,e$-Long Cycle problem, in which you are\u0000given an $n$-vertex undirected graph $G$, a specified edge $e$ in $G$, and a\u0000positive integer $k$, and are asked to decide if the graph $G$ has a simple\u0000cycle through $e$ of length at least $k$. We show that the problem can be\u0000solved in $1.731^koperatorname{poly}(n)$ time, improving over the previously\u0000best known $2^koperatorname{poly}(n)$ time algorithm and solving an open\u0000problem [Fomin et al., TALG 2024]. When the graph is bipartite, we can solve\u0000the problem in $2^{k/2}operatorname{poly}(n)$ time, matching the fastest known\u0000algorithm for finding a cycle of length exactly $k$ in an undirected bipartite\u0000graph [Bj\"orklund et al., JCSS 2017]. Our results follow the approach taken by [Fomin et al., TALG 2024], which\u0000describes an efficient algorithm for finding cycles using many colours in a\u0000vertex-coloured undirected graph. Our contribution is twofold. First, we\u0000describe a new algorithm and analysis for the central colourful cycle problem,\u0000with the aim of providing a comparatively short and self-contained proof of\u0000correctness. Second, we give tighter reductions from $k,e$-Long Cycle to the\u0000colourful cycle problem, which lead to our improved running times.","PeriodicalId":501525,"journal":{"name":"arXiv - CS - Data Structures and Algorithms","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141946270","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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