Pub Date : 2024-11-28DOI: 10.1016/j.jcss.2024.103606
Feodor F. Dragan , Guillaume Ducoffe , Heather M. Guarnera
A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. The class of Helly graphs is the discrete analogue of the class of hyperconvex metric spaces. We study diameter, radius and all eccentricity computations within the Helly graphs. Under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, it was recently shown that the radius and the diameter of an n-vertex m-edge Helly graph can be computed with high probability in time (i.e., subquadratic in ). In this paper, we improve that result by presenting a deterministic -time algorithm which computes not only the radius and the diameter but also all vertex eccentricities in a Helly graph. Furthermore, we give a parameterized linear-time algorithm for this problem on Helly graphs, with the parameter being the Gromov hyperbolicity.
{"title":"Fast deterministic algorithms for computing all eccentricities in (hyperbolic) Helly graphs","authors":"Feodor F. Dragan , Guillaume Ducoffe , Heather M. Guarnera","doi":"10.1016/j.jcss.2024.103606","DOIUrl":"10.1016/j.jcss.2024.103606","url":null,"abstract":"<div><div>A graph is Helly if every family of pairwise intersecting balls has a nonempty common intersection. The class of Helly graphs is the discrete analogue of the class of hyperconvex metric spaces. We study diameter, radius and all eccentricity computations within the Helly graphs. Under plausible complexity assumptions, neither the diameter nor the radius can be computed in truly subquadratic time on general graphs. In contrast to these negative results, it was recently shown that the radius and the diameter of an <em>n</em>-vertex <em>m</em>-edge Helly graph can be computed with high probability in <span><math><mover><mrow><mi>O</mi></mrow><mrow><mo>˜</mo></mrow></mover><mo>(</mo><mi>m</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span> time (<em>i.e.</em>, subquadratic in <span><math><mi>n</mi><mo>+</mo><mi>m</mi></math></span>). In this paper, we improve that result by presenting a deterministic <span><math><mi>O</mi><mo>(</mo><mi>m</mi><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>)</mo></math></span>-time algorithm which computes not only the radius and the diameter but also all vertex eccentricities in a Helly graph. Furthermore, we give a parameterized linear-time algorithm for this problem on Helly graphs, with the parameter being the Gromov hyperbolicity.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"149 ","pages":"Article 103606"},"PeriodicalIF":1.1,"publicationDate":"2024-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143166229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-22DOI: 10.1016/j.jcss.2024.103605
George Karakostas , Stavros G. Kolliopoulos
Motivated by time-sharing systems with deadlines, we introduce the study of the following problem. We are given m machines and n jobs, as well as a set of tolerance capacities for every job j and machine i. Can we assign the jobs so that, if job j ends up on machine i, the total size of jobs that are processed on i is at most ? We define two natural optimization versions: (i) Maximize the total weight of jobs that can be assigned without violating the tolerance capacities. (ii) Minimize the amount by which capacities have to be scaled so that all jobs can be assigned. For (i), we provide constant-factor approximations even in the presence of additional side-constraints. For (ii), we provide a strong inapproximability result and integrality gap lower bounds for two key relaxations.
受有截止日期的分时系统的启发，我们引入了对以下问题的研究。我们给定了 m 台机器和 n 个作业，以及每个作业 j 和机器 i 的一组容差能力 uij≥0。我们能否分配作业，使作业 j 最终在机器 i 上处理时，在机器 i 上处理的作业的总大小最多为 uij？我们定义了两个自然优化版本：(i) 在不违反容差能力的情况下，最大化可分配作业的总重量。(ii) 最小化ρ≥1，ρ≥1 是为使所有工作都能分配而必须缩放的容量。对于 (i)，我们提供了恒因子近似值，即使存在额外的附带约束。对于 (ii)，我们提供了一个强大的不可逼近性结果和两个关键松弛的积分差距下限。
{"title":"Time-sharing scheduling with tolerance capacities","authors":"George Karakostas , Stavros G. Kolliopoulos","doi":"10.1016/j.jcss.2024.103605","DOIUrl":"10.1016/j.jcss.2024.103605","url":null,"abstract":"<div><div>Motivated by time-sharing systems with deadlines, we introduce the study of the following problem. We are given <em>m</em> machines and <em>n</em> jobs, as well as a set of <em>tolerance capacities</em> <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>≥</mo><mn>0</mn></math></span> for every job <em>j</em> and machine <em>i</em>. Can we assign the jobs so that, if job <em>j</em> ends up on machine <em>i</em>, the total size of jobs that are processed on <em>i</em> is at most <span><math><msub><mrow><mi>u</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub></math></span>? We define two natural optimization versions: (i) Maximize the total weight of jobs that can be assigned without violating the tolerance capacities. (ii) Minimize the amount <span><math><mi>ρ</mi><mo>≥</mo><mn>1</mn></math></span> by which capacities have to be scaled so that all jobs can be assigned. For (i), we provide constant-factor approximations even in the presence of additional side-constraints. For (ii), we provide a strong inapproximability result and integrality gap lower bounds for two key relaxations.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103605"},"PeriodicalIF":1.1,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142720552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.jcss.2024.103603
Zhiyi Tang
Though embedding problems have been considered for several regular graphs [1], [2], [3], it is still an open problem for hypercube into torus [4], [2]. In the paper, we prove the conjecture mathematically and obtain the minimum wirelength of embedding for hypercube into Cartesian product of paths and/or cycles. In addition, we explain that Gray code embedding is an optimal strategy in such embedding problems.
{"title":"Embedding hypercubes into torus and Cartesian product of paths and/or cycles for minimizing wirelength","authors":"Zhiyi Tang","doi":"10.1016/j.jcss.2024.103603","DOIUrl":"10.1016/j.jcss.2024.103603","url":null,"abstract":"<div><div>Though embedding problems have been considered for several regular graphs <span><span>[1]</span></span>, <span><span>[2]</span></span>, <span><span>[3]</span></span>, it is still an open problem for hypercube into torus <span><span>[4]</span></span>, <span><span>[2]</span></span>. In the paper, we prove the conjecture mathematically and obtain the minimum wirelength of embedding for hypercube into Cartesian product of paths and/or cycles. In addition, we explain that Gray code embedding is an optimal strategy in such embedding problems.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103603"},"PeriodicalIF":1.1,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.jcss.2024.103604
Andreas Emil Feldmann , Anish Mukherjee , Erik Jan van Leeuwen
In the well-known Survivable Network Design Problem (SNDP), we are given an undirected graph G with edge costs, a set R of terminal vertices, and an integer demand for every terminal pair . The task is to compute a subgraph H of G of minimum cost, such that for every terminal pair there are at least disjoint paths between s and t in H. Depending on the type of disjointness, we obtain several variants of SNDP that have been widely studied in the literature: if the paths are required to be edge-disjoint we obtain EC-SNDP, while if they must be internally vertex-disjoint we obtain VC-SNDP. Another important case is the element-connectivity variant (LC-SNDP), where the paths must be disjoint on edges and non-terminals, i.e., they may only share terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size ℓ, the sum of demands D, the number of terminals k, and the maximum demand .
在著名的 "可存活网络设计问题"（SNDP）中，我们给定了一个带边成本的无向图 G、一组终端顶点 R 以及每个终端对 s,t∈R 的整数需求 ds,t。我们的任务是计算代价最小的 G 子图 H，使得对于每个终端对 s,t∈R，H 中的 s 和 t 之间至少有 ds,t 互不相交的路径。根据互不相交的类型，我们会得到 SNDP 的几种变体，这些变体已在文献中得到广泛研究：如果路径必须是边互不相交，我们会得到 EC-SNDP；如果路径必须是内部顶点互不相交，我们会得到 VC-SNDP。另一种重要情况是元素连通性变体（LC-SNDP），即路径必须在边和非终端上不相交，也就是说，它们只能共享终端。在这项工作中，我们将阐明上述问题的参数化复杂性。我们考虑了几个自然参数，包括解大小 ℓ、需求总和 D、终端数 k 和最大需求量 dmax。
{"title":"The parameterized complexity of the survivable network design problem","authors":"Andreas Emil Feldmann , Anish Mukherjee , Erik Jan van Leeuwen","doi":"10.1016/j.jcss.2024.103604","DOIUrl":"10.1016/j.jcss.2024.103604","url":null,"abstract":"<div><div>In the well-known <span>Survivable Network Design Problem (SNDP)</span>, we are given an undirected graph <em>G</em> with edge costs, a set <em>R</em> of terminal vertices, and an integer demand <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> for every terminal pair <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>R</mi></math></span>. The task is to compute a subgraph <em>H</em> of <em>G</em> of minimum cost, such that for every terminal pair <span><math><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>R</mi></math></span> there are at least <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> disjoint paths between <em>s</em> and <em>t</em> in <em>H</em>. Depending on the type of disjointness, we obtain several variants of SNDP that have been widely studied in the literature: if the paths are required to be edge-disjoint we obtain <span>EC-SNDP</span>, while if they must be internally vertex-disjoint we obtain <span>VC-SNDP</span>. Another important case is the element-connectivity variant (<span>LC-SNDP</span>), where the paths must be disjoint on edges and non-terminals, i.e., they may only share terminals. In this work we shed light on the parameterized complexity of the above problems. We consider several natural parameters, which include the solution size <em>ℓ</em>, the sum of demands <em>D</em>, the number of terminals <em>k</em>, and the maximum demand <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>max</mi></mrow></msub></math></span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103604"},"PeriodicalIF":1.1,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-13DOI: 10.1016/j.jcss.2024.103602
Wen Li , Ralf Klasing , Yaping Mao , Bo Ning
Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let G be a graph with vertex set and edge set . For any subset M in and an edge e in , let be the set of pairs such that where and . M is called a distance-edge-monitoring set if every edge e of G is monitored by some vertex of M, that is, the set is nonempty. The distance-edge-monitoring number of G, denoted by , is defined as the smallest size of distance-edge-monitoring sets of G. For two graphs of order , respectively, in this paper, we prove that , where □ is the Cartesian product operation. Moreover, we characterize the networks attaining the upper and lower bounds and show their applications on some known networks. We also obtain the distance-edge-monitoring numbers of join, corona, cluster, and some specific networks.
Foucaud 等人最近在网络监控领域提出并开始研究一个新的图论概念。假设 G 是一个具有顶点集 V(G) 和边集 E(G) 的图。对于 V(G) 中的任意子集 M 和 E(G) 中的边 e，设 P(M,e) 是一对 (x,y) 的集合，使得 dG(x,y)≠dG-e(x,y) 其中 x∈M 和 y∈V(G) 。如果 G 的每一条边 e 都受到 M 的某个顶点的监控，即集合 P(M,e) 非空，则 M 称为距离边监控集。对于阶数分别为 m,n 的两个图 G、H，本文证明了 max{mdem(H),ndem(G)}≤dem(G□H)≤mdem(H)+ndem(G)-dem(G)dem(H) ，其中 □ 是笛卡尔乘积运算。此外，我们还描述了达到上界和下界的网络的特征，并展示了它们在一些已知网络中的应用。我们还得到了 join、corona、cluster 和一些特定网络的距离边监控数。
{"title":"Monitoring the edges of product networks using distances","authors":"Wen Li , Ralf Klasing , Yaping Mao , Bo Ning","doi":"10.1016/j.jcss.2024.103602","DOIUrl":"10.1016/j.jcss.2024.103602","url":null,"abstract":"<div><div>Foucaud et al. recently introduced and initiated the study of a new graph-theoretic concept in the area of network monitoring. Let <em>G</em> be a graph with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and edge set <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. For any subset <em>M</em> in <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and an edge <em>e</em> in <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, let <span><math><mi>P</mi><mo>(</mo><mi>M</mi><mo>,</mo><mi>e</mi><mo>)</mo></math></span> be the set of pairs <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> such that <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>≠</mo><msub><mrow><mi>d</mi></mrow><mrow><mi>G</mi><mo>−</mo><mi>e</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> where <span><math><mi>x</mi><mo>∈</mo><mi>M</mi></math></span> and <span><math><mi>y</mi><mo>∈</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. <em>M</em> is called a <em>distance-edge-monitoring set</em> if every edge <em>e</em> of <em>G</em> is monitored by some vertex of <em>M</em>, that is, the set <span><math><mi>P</mi><mo>(</mo><mi>M</mi><mo>,</mo><mi>e</mi><mo>)</mo></math></span> is nonempty. The <em>distance-edge-monitoring number</em> of <em>G</em>, denoted by <span><math><mi>dem</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, is defined as the smallest size of distance-edge-monitoring sets of <em>G</em>. For two graphs <span><math><mi>G</mi><mo>,</mo><mi>H</mi></math></span> of order <span><math><mi>m</mi><mo>,</mo><mi>n</mi></math></span>, respectively, in this paper, we prove that <span><math><mi>max</mi><mo></mo><mo>{</mo><mi>m</mi><mi>dem</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>,</mo><mi>n</mi><mi>dem</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>}</mo><mo>≤</mo><mi>dem</mi><mo>(</mo><mi>G</mi><mspace></mspace><mo>□</mo><mspace></mspace><mi>H</mi><mo>)</mo><mo>≤</mo><mi>m</mi><mi>dem</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>+</mo><mi>n</mi><mi>dem</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>−</mo><mi>dem</mi><mo>(</mo><mi>G</mi><mo>)</mo><mi>dem</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span>, where □ is the Cartesian product operation. Moreover, we characterize the networks attaining the upper and lower bounds and show their applications on some known networks. We also obtain the distance-edge-monitoring numbers of join, corona, cluster, and some specific networks.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103602"},"PeriodicalIF":1.1,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703524","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-05DOI: 10.1016/j.jcss.2024.103600
Jesper Nederlof, Krisztina Szilágyi
In this paper we investigate the parameterized complexity of counting and detecting small patterns in unit disk graphs: Given an n-vertex unit disk graph G with an embedding of ply p (i.e. G is an intersection graph of closed unit disks, and each point is contained in at most p disks) and a k-vertex unit disk graph P, count the number of (induced) copies of P in G. For general patterns P, we give an time algorithm for counting pattern occurrences. We show this is tight, even for ply : any time algorithm violates the Exponential Time Hypothesis (ETH). Our approach combines tools developed for planar subgraph isomorphism such as ‘efficient inclusion-exclusion’ from Nederlof (2020) [15], and ‘isomorphisms checks’ from Bodlaender et al. (2016) [5] with a different separator hierarchy and a new bound on the number of non-isomorphic separations tailored for unit disk graphs.
在本文中，我们研究了计算和检测单位盘图中小图案的参数化复杂性：给定一个具有 ply p 嵌入的 n 个顶点单位盘图 G（即 G 是封闭单位盘的交集图，且每个点最多包含在 p 个盘中）和一个 k 个顶点单位盘图 P，计算 P 在 G 中的（诱导）副本数。对于一般图案 P，我们给出了一个 2O(pk/logk)nO(1)时间的算法来计算图案出现次数。我们证明了这一算法的严密性，即使对于 ply p=2 也是如此：任何 2o(n/logn)nO(1)时间算法都违反了指数时间假说 (ETH)。我们的方法结合了为平面子图同构开发的工具，如 Nederlof (2020) [15] 的 "高效包容-排除 "和 Bodlaender 等人 (2016) [5] 的 "同构检查"，以及不同的分离器层次结构和为单位盘图量身定制的非同构分离数量新约束。
{"title":"Algorithms and Turing kernels for detecting and counting small patterns in unit disk graphs","authors":"Jesper Nederlof, Krisztina Szilágyi","doi":"10.1016/j.jcss.2024.103600","DOIUrl":"10.1016/j.jcss.2024.103600","url":null,"abstract":"<div><div>In this paper we investigate the parameterized complexity of counting and detecting small patterns in unit disk graphs: Given an <em>n</em>-vertex unit disk graph <em>G</em> with an embedding of ply <em>p</em> (i.e. <em>G</em> is an intersection graph of closed unit disks, and each point is contained in at most <em>p</em> disks) and a <em>k</em>-vertex unit disk graph <em>P</em>, count the number of (induced) copies of <em>P</em> in <em>G</em>. For general patterns <em>P</em>, we give an <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>p</mi><mi>k</mi><mo>/</mo><mi>log</mi><mo></mo><mi>k</mi><mo>)</mo></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> time algorithm for counting pattern occurrences. We show this is tight, even for ply <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>: any <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>o</mi><mo>(</mo><mi>n</mi><mo>/</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow></msup><msup><mrow><mi>n</mi></mrow><mrow><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></math></span> time algorithm violates the Exponential Time Hypothesis (ETH). Our approach combines tools developed for planar subgraph isomorphism such as ‘efficient inclusion-exclusion’ from Nederlof (2020) <span><span>[15]</span></span>, and ‘isomorphisms checks’ from Bodlaender et al. (2016) <span><span>[5]</span></span> with a different separator hierarchy and a new bound on the number of non-isomorphic separations tailored for unit disk graphs.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103600"},"PeriodicalIF":1.1,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654144","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-31DOI: 10.1016/j.jcss.2024.103601
Chris Köcher , Dietrich Kuske
A cooperating multi-pushdown system consists of a tuple of pushdown systems that can delegate the execution of recursive procedures to sub-tuples; control returns to the calling tuple once all sub-tuples finished their task. This allows the concurrent execution since disjoint sub-tuples can perform their task independently. Because of the concrete form of recursive descent into sub-tuples, the content of the multi-pushdown does not form an arbitrary tuple of words, but can be understood as a Mazurkiewicz trace. For such systems, we prove that the backwards reachability relation efficiently preserves recognizability, generalizing a result and proof technique by Bouajjani et al. for single-pushdown systems. It follows that the reachability relation is decidable for cooperating multi-pushdown systems in polynomial time and the same holds, e.g., for safety and liveness properties given by recognizable sets of configurations.
{"title":"Backwards-reachability for cooperating multi-pushdown systems","authors":"Chris Köcher , Dietrich Kuske","doi":"10.1016/j.jcss.2024.103601","DOIUrl":"10.1016/j.jcss.2024.103601","url":null,"abstract":"<div><div>A cooperating multi-pushdown system consists of a tuple of pushdown systems that can delegate the execution of recursive procedures to sub-tuples; control returns to the calling tuple once all sub-tuples finished their task. This allows the concurrent execution since disjoint sub-tuples can perform their task independently. Because of the concrete form of recursive descent into sub-tuples, the content of the multi-pushdown does not form an arbitrary tuple of words, but can be understood as a Mazurkiewicz trace. For such systems, we prove that the backwards reachability relation efficiently preserves recognizability, generalizing a result and proof technique by Bouajjani et al. for single-pushdown systems. It follows that the reachability relation is decidable for cooperating multi-pushdown systems in polynomial time and the same holds, e.g., for safety and liveness properties given by recognizable sets of configurations.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103601"},"PeriodicalIF":1.1,"publicationDate":"2024-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142654143","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-22DOI: 10.1016/j.jcss.2024.103596
Daniela Bubboloni , Costanza Catalano , Andrea Marino , Ana Silva
We extend the concept of out/in-branchings spanning the vertices of a digraph to temporal graphs, which are digraphs where arcs are available only at prescribed times. While the literature has focused on minimum weight/earliest arrival time Temporal Out-Branchings (tob), we solve the problem for other optimization criteria (travel duration, departure time, number of transfers, total waiting time, traveling time). For some criteria we provide a log linear algorithm for computing such branchings, while for others we prove that deciding the existence of a spanning tob is NP-complete. The same results hold for optimal temporal in-branchings. We also investigate the related problem of computing a spanning temporal subgraph with the minimum number of arcs and optimizing a chosen criterion; this problem turns out to be always NP-hard. The hardness results are quite surprising, as computing optimal paths between nodes is always polynomial-time.
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Pub Date : 2024-10-21DOI: 10.1016/j.jcss.2024.103597
Bart M.P. Jansen , Jari J.H. de Kroon , Michał Włodarczyk
The celebrated notion of important separators bounds the number of small -separators in a graph which are ‘farthest from S’ in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of k-secluded vertex sets: sets with an open neighborhood of size at most k. In this terminology, the bound on important separators says that there are at most maximal k-secluded connected vertex sets C containing S but disjoint from T. We generalize this statement significantly: even when we demand that avoids a finite set of forbidden induced subgraphs, the number of such maximal subgraphs is and they can be enumerated efficiently. This enumeration algorithm allows us to give improved parameterized algorithms for Connectedk-Secluded-Free Subgraph and for deleting into scattered graph classes.
著名的重要分隔符概念限定了图中在技术意义上 "离 S 最远 "的小 (S,T) 分隔符的数量。在本文中，我们针对无向图引入了这一强大算法基本原理的广义化，用 k 个排除顶点集来表述：具有大小至多为 k 的开放邻域的集合。在这个术语中，重要分隔符的约束是指最多有 4k 个最大的 k-secluded连通顶点集 C，其中包含 S 但与 T 不相交。我们对这一声明进行了显著的概括：即使我们要求 G[C] 避免有限的禁止诱导子图集 F，这种最大子图的数量也是 2O(k)，而且可以高效地枚举出来。有了这种枚举算法，我们就能给出改进的参数化算法，用于连接 k-Secluded F-Free Subgraph 和删除成分散图类。
{"title":"Single-exponential FPT algorithms for enumerating secluded F-free subgraphs and deleting to scattered graph classes","authors":"Bart M.P. Jansen , Jari J.H. de Kroon , Michał Włodarczyk","doi":"10.1016/j.jcss.2024.103597","DOIUrl":"10.1016/j.jcss.2024.103597","url":null,"abstract":"<div><div>The celebrated notion of important separators bounds the number of small <span><math><mo>(</mo><mi>S</mi><mo>,</mo><mi>T</mi><mo>)</mo></math></span>-separators in a graph which are ‘farthest from <em>S</em>’ in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of <em>k-secluded</em> vertex sets: sets with an open neighborhood of size at most <em>k</em>. In this terminology, the bound on important separators says that there are at most <span><math><msup><mrow><mn>4</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span> maximal <em>k</em>-secluded connected vertex sets <em>C</em> containing <em>S</em> but disjoint from <em>T</em>. We generalize this statement significantly: even when we demand that <span><math><mi>G</mi><mo>[</mo><mi>C</mi><mo>]</mo></math></span> avoids a finite set <span><math><mi>F</mi></math></span> of forbidden induced subgraphs, the number of such maximal subgraphs is <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msup></math></span> and they can be enumerated efficiently. This enumeration algorithm allows us to give improved parameterized algorithms for <span>Connected</span> <em>k</em><span>-Secluded</span> <span><math><mi>F</mi></math></span><span>-Free Subgraph</span> and for deleting into scattered graph classes.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103597"},"PeriodicalIF":1.1,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142554741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-21DOI: 10.1016/j.jcss.2024.103599
Juhi Chaudhary , Meirav Zehavi
A matching M in a graph G is an acyclic matching if the subgraph of G induced by the endpoints of the edges of M is a forest. Given a graph G and , Acyclic Matching asks whether G has an acyclic matching of size at least ℓ. In this paper, we prove that assuming , there does not exist any -approximation algorithm for Acyclic Matching that approximates it within a constant factor when parameterized by ℓ. Our reduction also asserts -inapproximability for Induced Matching and Uniquely Restricted Matching. We also consider three below-guarantee parameters for Acyclic Matching, viz. , , and , where , is the matching number, and is the independence number of G. Also, we show that Acyclic Matching does not exhibit a polynomial kernel with respect to vertex cover number (or vertex deletion distance to clique) plus the size of the matching unless .
如果 M 的边的端点所诱导的 G 子图是一个森林，那么图 G 中的匹配 M 就是非循环匹配。给定一个图 G 和 ℓ∈N，非循环匹配问 G 是否有大小至少为 ℓ 的非循环匹配。在本文中，我们证明了假设 W[1]⊈FPT 时，不存在任何 FPT 近似算法，可以在以ℓ 为参数时，以常数因子内逼近 Acyclic Matching。我们的还原也证明了诱导匹配和唯一限制匹配的 FPT 近似性。我们还考虑了 Acyclic Matching 的三个低于保证的参数，即 n2-ℓ、MM(G)-ℓ 和 IS(G)-ℓ，其中 n=V(G), MM(G) 是匹配数，IS(G) 是 G 的独立数。此外，我们还证明，除非 NP⊆coNP/poly，否则无循环匹配并不表现出关于顶点覆盖数（或顶点到小块的删除距离）加上匹配大小的多项式内核。
{"title":"Parameterized results on acyclic matchings with implications for related problems","authors":"Juhi Chaudhary , Meirav Zehavi","doi":"10.1016/j.jcss.2024.103599","DOIUrl":"10.1016/j.jcss.2024.103599","url":null,"abstract":"<div><div>A matching <em>M</em> in a graph <em>G</em> is an <em>acyclic matching</em> if the subgraph of <em>G</em> induced by the endpoints of the edges of <em>M</em> is a forest. Given a graph <em>G</em> and <span><math><mi>ℓ</mi><mo>∈</mo><mi>N</mi></math></span>, <span>Acyclic Matching</span> asks whether <em>G</em> has an acyclic matching of <em>size</em> at least <em>ℓ</em>. In this paper, we prove that assuming <span><math><mi>W</mi><mo>[</mo><mn>1</mn><mo>]</mo><mo>⊈</mo><mi>FPT</mi></math></span>, there does not exist any <span><math><mi>FPT</mi></math></span>-approximation algorithm for <span>Acyclic Matching</span> that approximates it within a constant factor when parameterized by <em>ℓ</em>. Our reduction also asserts <span><math><mi>FPT</mi></math></span>-inapproximability for <span>Induced Matching</span> and <span>Uniquely Restricted Matching</span>. We also consider three below-guarantee parameters for <span>Acyclic Matching</span>, viz. <span><math><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>ℓ</mi></math></span>, <span><math><mrow><mi>MM</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>ℓ</mi></math></span>, and <span><math><mrow><mi>IS</mi><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mi>ℓ</mi></math></span>, where <span><math><mi>n</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, <span><math><mi>MM</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the <em>matching number</em>, and <span><math><mi>IS</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the <em>independence number</em> of <em>G</em>. Also, we show that <span>Acyclic Matching</span> does not exhibit a polynomial kernel with respect to vertex cover number (or vertex deletion distance to clique) plus the size of the matching unless <span><math><mrow><mi>NP</mi></mrow><mo>⊆</mo><mrow><mi>coNP</mi></mrow><mo>/</mo><mrow><mi>poly</mi></mrow></math></span>.</div></div>","PeriodicalId":50224,"journal":{"name":"Journal of Computer and System Sciences","volume":"148 ","pages":"Article 103599"},"PeriodicalIF":1.1,"publicationDate":"2024-10-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142530067","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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