Shyan S. Akmal, V. V. Williams, Ryan Williams, Zixuan Xu
{"title":"Faster Detours in Undirected Graphs","authors":"Shyan S. Akmal, V. V. Williams, Ryan Williams, Zixuan Xu","doi":"10.48550/arXiv.2307.01781","DOIUrl":"https://doi.org/10.48550/arXiv.2307.01781","url":null,"abstract":"The $k$-Detour problem is a basic path-finding problem: given a graph $G$ on $n$ vertices, with specified nodes $s$ and $t$, and a positive integer $k$, the goal is to determine if $G$ has an $st$-path of length exactly $text{dist}(s, t) + k$, where $text{dist}(s, t)$ is the length of a shortest path from $s$ to $t$. The $k$-Detour problem is NP-hard when $k$ is part of the input, so researchers have sought efficient parameterized algorithms for this task, running in $f(k)text{poly}(n)$ time, for $f$ as slow-growing as possible. We present faster algorithms for $k$-Detour in undirected graphs, running in $1.853^k text{poly}(n)$ randomized and $4.082^k text{poly}(n)$ deterministic time. The previous fastest algorithms for this problem took $2.746^k text{poly}(n)$ randomized and $6.523^k text{poly}(n)$ deterministic time [Bez'akov'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact that detecting a path of a given length in an undirected graph is easier if we are promised that the path belongs to what we call a\"bipartitioned\"subgraph, where the nodes are split into two parts and the path must satisfy constraints on those parts. Previously, this idea was used to obtain the fastest known algorithm for finding paths of length $k$ in undirected graphs [Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017]. Our work has direct implications for the $k$-Longest Detour problem: in this problem, we are given the same input as in $k$-Detour, but are now tasked with determining if $G$ has an $st$-path of length at least $text{dist}(s, t) + k.$ Our results for k-Detour imply that we can solve $k$-Longest Detour in $3.432^k text{poly}(n)$ randomized and $16.661^k text{poly}(n)$ deterministic time. The previous fastest algorithms for this problem took $7.539^k text{poly}(n)$ randomized and $42.549^k text{poly}(n)$ deterministic time [Fomin et al., STACS 2022].","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124957893","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":"Evaluating Restricted First-Order Counting Properties on Nowhere Dense Classes and Beyond","authors":"Jannik Dreier, Daniel Mock, P. Rossmanith","doi":"10.48550/arXiv.2307.01832","DOIUrl":"https://doi.org/10.48550/arXiv.2307.01832","url":null,"abstract":"It is known that first-order logic with some counting extensions can be efficiently evaluated on graph classes with bounded expansion, where depth-$r$ minors have constant density. More precisely, the formulas are $exists x_1 ... x_k #y varphi(x_1,...,x_k, y)>N$, where $varphi$ is an FO-formula. If $varphi$ is quantifier-free, we can extend this result to nowhere dense graph classes with an almost linear FPT run time. Lifting this result further to slightly more general graph classes, namely almost nowhere dense classes, where the size of depth-$r$ clique minors is subpolynomial, is impossible unless FPT=W[1]. On the other hand, in almost nowhere dense classes we can approximate such counting formulas with a small additive error. Note those counting formulas are contained in FOC({<}) but not FOC1(P). In particular, it follows that partial covering problems, such as partial dominating set, have fixed parameter algorithms on nowhere dense graph classes with almost linear running time.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126555542","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}
Benjamin Bergougnoux, V. Chekan, R. Ganian, Mamadou Moustapha Kant'e, Matthias Mnich, Sang-il Oum, Michal Pilipczuk, E. J. V. Leeuwen
{"title":"Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth","authors":"Benjamin Bergougnoux, V. Chekan, R. Ganian, Mamadou Moustapha Kant'e, Matthias Mnich, Sang-il Oum, Michal Pilipczuk, E. J. V. Leeuwen","doi":"10.48550/arXiv.2307.01285","DOIUrl":"https://doi.org/10.48550/arXiv.2307.01285","url":null,"abstract":"Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on $n$-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth $d$ and using $k$ labels, we can solve - Independent Set in time $2^{O(dk)}cdot n^{O(1)}$ using $O(dk^2log n)$ space; - Max Cut in time $n^{O(dk)}$ using $O(dklog n)$ space; and - Dominating Set in time $2^{O(dk)}cdot n^{O(1)}$ using $n^{O(1)}$ space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with $d$ if one wishes to keep the space complexity polynomial.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121719664","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":"Effective Resistances in Non-Expander Graphs","authors":"Dongrun Cai, Xue-gang Chen, Pan Peng","doi":"10.48550/arXiv.2307.01218","DOIUrl":"https://doi.org/10.48550/arXiv.2307.01218","url":null,"abstract":"Effective resistances are ubiquitous in graph algorithms and network analysis. In this work, we study sublinear time algorithms to approximate the effective resistance of an adjacent pair $s$ and $t$. We consider the classical adjacency list model for local algorithms. While recent works have provided sublinear time algorithms for expander graphs, we prove several lower bounds for general graphs of $n$ vertices and $m$ edges: 1.It needs $Omega(n)$ queries to obtain $1.01$-approximations of the effective resistance of an adjacent pair $s$ and $t$, even for graphs of degree at most 3 except $s$ and $t$. 2.For graphs of degree at most $d$ and any parameter $ell$, it needs $Omega(m/ell)$ queries to obtain $c cdot min{d, ell}$-approximations where $c>0$ is a universal constant. Moreover, we supplement the first lower bound by providing a sublinear time $(1+epsilon)$-approximation algorithm for graphs of degree 2 except the pair $s$ and $t$. One of our technical ingredients is to bound the expansion of a graph in terms of the smallest non-trivial eigenvalue of its Laplacian matrix after removing edges. We discover a new lower bound on the eigenvalues of perturbed graphs (resp. perturbed matrices) by incorporating the effective resistance of the removed edge (resp. the leverage scores of the removed rows), which may be of independent interest.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127935645","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}
Xiangyun Ding, Xiaojun Dong, Yan Gu, Youzhe Liu, Yihan Sun
{"title":"Efficient Parallel Output-Sensitive Edit Distance","authors":"Xiangyun Ding, Xiaojun Dong, Yan Gu, Youzhe Liu, Yihan Sun","doi":"10.4230/LIPIcs.ESA.2023.40","DOIUrl":"https://doi.org/10.4230/LIPIcs.ESA.2023.40","url":null,"abstract":"Given two strings $A[1..n]$ and $B[1..m]$, and a set of operations allowed to edit the strings, the edit distance between $A$ and $B$ is the minimum number of operations required to transform $A$ into $B$. Sequentially, a standard Dynamic Programming (DP) algorithm solves edit distance with $Theta(nm)$ cost. In many real-world applications, the strings to be compared are similar and have small edit distances. To achieve highly practical implementations, we focus on output-sensitive parallel edit-distance algorithms, i.e., to achieve asymptotically better cost bounds than the standard $Theta(nm)$ algorithm when the edit distance is small. We study four algorithms in the paper, including three algorithms based on Breadth-First Search (BFS) and one algorithm based on Divide-and-Conquer (DaC). Our BFS-based solution is based on the Landau-Vishkin algorithm. We implement three different data structures for the longest common prefix (LCP) queries needed in the algorithm: the classic solution using parallel suffix array, and two hash-based solutions proposed in this paper. Our DaC-based solution is inspired by the output-insensitive solution proposed by Apostolico et al., and we propose a non-trivial adaption to make it output-sensitive. All our algorithms have good theoretical guarantees, and they achieve different tradeoffs between work (total number of operations), span (longest dependence chain in the computation), and space. We test and compare our algorithms on both synthetic data and real-world data. Our BFS-based algorithms outperform the existing parallel edit-distance implementation in ParlayLib in all test cases. By comparing our algorithms, we also provide a better understanding of the choice of algorithms for different input patterns. We believe that our paper is the first systematic study in the theory and practice of parallel edit distance.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"33 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121710554","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}
A. Hevia, Benjamin Kallus, Summer McClintic, Samantha Reisner, Darren Strash, Johnathan Wilson
{"title":"Solving Edge Clique Cover Exactly via Synergistic Data Reduction","authors":"A. Hevia, Benjamin Kallus, Summer McClintic, Samantha Reisner, Darren Strash, Johnathan Wilson","doi":"10.48550/arXiv.2306.17804","DOIUrl":"https://doi.org/10.48550/arXiv.2306.17804","url":null,"abstract":"The edge clique cover (ECC) problem -- where the goal is to find a minimum cardinality set of cliques that cover all the edges of a graph -- is a classic NP-hard problem that has received much attention from both the theoretical and experimental algorithms communities. While small sparse graphs can be solved exactly via the branch-and-reduce algorithm of Gramm et al. [JEA 2009], larger instances can currently only be solved inexactly using heuristics with unknown overall solution quality. We revisit computing minimum ECCs exactly in practice by combining data reduction for both the ECC emph{and} vertex clique cover (VCC) problems. We do so by modifying the polynomial-time reduction of Kou et al. [Commun. ACM 1978] to transform a reduced ECC instance to a VCC instance; alternatively, we show it is possible to ``lift'' some VCC reductions to the ECC problem. Our experiments show that combining data reduction for both problems (which we call emph{synergistic data reduction}) enables finding exact minimum ECCs orders of magnitude faster than the technique of Gramm et al., and allows solving large sparse graphs on up to millions of vertices and edges that have never before been solved. With these new exact solutions, we evaluate the quality of recent heuristic algorithms on large instances for the first time. The most recent of these, textsf{EO-ECC} by Abdullah et al. [ICCS 2022], solves 8 of the 27 instances for which we have exact solutions. It is our hope that our strategy rallies researchers to seek improved algorithms for the ECC problem.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130834801","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":"Maximal k-Edge-Connected Subgraphs in Almost-Linear Time for Small k","authors":"Thatchaphol Saranurak, Wuwei Yuan","doi":"10.48550/arXiv.2307.00147","DOIUrl":"https://doi.org/10.48550/arXiv.2307.00147","url":null,"abstract":"We give the first almost-linear time algorithm for computing the emph{maximal $k$-edge-connected subgraphs} of an undirected unweighted graph for any constant $k$. More specifically, given an $n$-vertex $m$-edge graph $G=(V,E)$ and a number $k = log^{o(1)}n$, we can deterministically compute in $O(m+n^{1+o(1)})$ time the unique vertex partition ${V_{1},dots,V_{z}}$ such that, for every $i$, $V_{i}$ induces a $k$-edge-connected subgraph while every superset $V'_{i}supset V_{i}$ does not. Previous algorithms with linear time work only when $kle2$ {[}Tarjan SICOMP'72{]}, otherwise they all require $Omega(m+nsqrt{n})$ time even when $k=3$ {[}Chechik~et~al.~SODA'17; Forster~et~al.~SODA'20{]}. Our algorithm also extends to the decremental graph setting; we can deterministically maintain the maximal $k$-edge-connected subgraphs of a graph undergoing edge deletions in $m^{1+o(1)}$ total update time. Our key idea is a reduction to the dynamic algorithm supporting pairwise $k$-edge-connectivity queries {[}Jin and Sun FOCS'20{]}.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129159423","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":"Algorithms for Computing Maximum Cliques in Hyperbolic Random Graphs","authors":"Eunjin Oh, Seunghyeok Oh","doi":"10.48550/arXiv.2306.16775","DOIUrl":"https://doi.org/10.48550/arXiv.2306.16775","url":null,"abstract":"In this paper, we study the maximum clique problem on hyperbolic random graphs. A hyperbolic random graph is a mathematical model for analyzing scale-free networks since it effectively explains the power-law degree distribution of scale-free networks. We propose a simple algorithm for finding a maximum clique in hyperbolic random graph. We first analyze the running time of our algorithm theoretically. We can compute a maximum clique on a hyperbolic random graph $G$ in $O(m + n^{4.5(1-alpha)})$ expected time if a geometric representation is given or in $O(m + n^{6(1-alpha)})$ expected time if a geometric representation is not given, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively, and $alpha$ denotes a parameter controlling the power-law exponent of the degree distribution of $G$. Also, we implemented and evaluated our algorithm empirically. Our algorithm outperforms the previous algorithm [BFK18] practically and theoretically. Beyond the hyperbolic random graphs, we have experiment on real-world networks. For most of instances, we get large cliques close to the optimum solutions efficiently.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"133 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133091725","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":"Improved Algorithms for Online Rent Minimization Problem Under Unit-Size Jobs","authors":"Enze Sun, Zonghan Yang, Yuhao Zhang","doi":"10.48550/arXiv.2306.17241","DOIUrl":"https://doi.org/10.48550/arXiv.2306.17241","url":null,"abstract":"We consider the Online Rent Minimization problem, where online jobs with release times, deadlines, and processing times must be scheduled on machines that can be rented for a fixed length period of $T$. The objective is to minimize the number of machine rents. This problem generalizes the Online Machine Minimization problem where machines can be rented for an infinite period, and both problems have an asymptotically optimal competitive ratio of $O(log(p_{max}/p_{min}))$ for general processing times, where $p_{max}$ and $p_{min}$ are the maximum and minimum processing times respectively. However, for small values of $p_{max}/p_{min}$, a better competitive ratio can be achieved by assuming unit-size jobs. Under this assumption, Devanur et al. (2014) gave an optimal $e$-competitive algorithm for Online Machine Minimization, and Chen and Zhang (2022) gave a $(3e+7)approx 15.16$-competitive algorithm for Online Rent Minimization. In this paper, we significantly improve the competitive ratio of the Online Rent Minimization problem under unit size to $6$, by using a clean oracle-based online algorithm framework.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"87 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130007722","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}
Victor A. Campos, J. Costa, Raul Lopes, Ignasi Sau
{"title":"New Menger-like dualities in digraphs and applications to half-integral linkages","authors":"Victor A. Campos, J. Costa, Raul Lopes, Ignasi Sau","doi":"10.48550/arXiv.2306.16134","DOIUrl":"https://doi.org/10.48550/arXiv.2306.16134","url":null,"abstract":"We present new min-max relations in digraphs between the number of paths satisfying certain conditions and the order of the corresponding cuts. We define these objects in order to capture, in the context of solving the half-integral linkage problem, the essential properties needed for reaching a large bramble of congestion two (or any other constant) from the terminal set. This strategy has been used ad-hoc in several articles, usually with lengthy technical proofs, and our objective is to abstract it to make it applicable in a simpler and unified way. We provide two proofs of the min-max relations, one consisting in applying Menger's Theorem on appropriately defined auxiliary digraphs, and an alternative simpler one using matroids, however with worse polynomial running time. As an application, we manage to simplify and improve several results of Edwards et al. [ESA 2017] and of Giannopoulou et al. [SODA 2022] about finding half-integral linkages in digraphs. Concerning the former, besides being simpler, our proof provides an almost optimal bound on the strong connectivity of a digraph for it to be half-integrally feasible under the presence of a large bramble of congestion two (or equivalently, if the directed tree-width is large, which is the hard case). Concerning the latter, our proof uses brambles as rerouting objects instead of cylindrical grids, hence yielding much better bounds and being somehow independent of a particular topology. We hope that our min-max relations will find further applications as, in our opinion, they are simple, robust, and versatile to be easily applicable to different types of routing problems in digraphs.","PeriodicalId":201778,"journal":{"name":"Embedded Systems and Applications","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131973706","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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