{"title":"无小图上连通-k-子图覆盖的逼近算法和 FPT 算法","authors":"Pengcheng Liu, Zhao Zhang, Yingli Ran, Xiaohui Huang","doi":"10.1017/s0960129523000439","DOIUrl":null,"url":null,"abstract":"<p>Given a graph G, the minimum Connected-<span>k</span>-Subgraph Cover problem (MinC<span>k</span>SC) is to find a minimum vertex subset <span>C</span> of <span>G</span> such that every connected subgraph of <span>G</span> on <span>k</span> vertices has at least one vertex in <span>C</span>. If furthermore the subgraph of <span>G</span> induced by <span>C</span> is connected, then the problem is denoted as MinC<span>k</span>SC<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline1.png\"><span data-mathjax-type=\"texmath\"><span>$_{con}$</span></span></img></span></span>. In this paper, we first present a PTAS for MinC<span>k</span>SC on an <span>H</span>-minor-free graph, where <span>H</span> is a graph with a constant number of vertices. Then, we design an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline2.png\"><span data-mathjax-type=\"texmath\"><span>$O((\\omega+1)(2(k-1)(\\omega+2))^{3\\omega+3})|V|$</span></span></img></span></span>-time FPT algorithm for MinC<span>k</span>SC<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline3.png\"><span data-mathjax-type=\"texmath\"><span>$_{con}$</span></span></img></span></span> on a graph with treewidth <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline4.png\"><span data-mathjax-type=\"texmath\"><span>$\\omega$</span></span></img></span></span>, based on which we further design an <span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline5.png\"><span data-mathjax-type=\"texmath\"><span>$O(2^{O(\\sqrt{t}\\log t)}|V|^{O(1)})$</span></span></img></span></span> time subexponential FPT algorithm for MinC<span>k</span>SC<span><span><img data-mimesubtype=\"png\" data-type=\"\" src=\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline6.png\"><span data-mathjax-type=\"texmath\"><span>$_{con}$</span></span></img></span></span> on an <span>H</span>-minor-free graph, where <span>t</span> is an upper bound of solution size.</p>","PeriodicalId":49855,"journal":{"name":"Mathematical Structures in Computer Science","volume":null,"pages":null},"PeriodicalIF":0.4000,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs\",\"authors\":\"Pengcheng Liu, Zhao Zhang, Yingli Ran, Xiaohui Huang\",\"doi\":\"10.1017/s0960129523000439\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given a graph G, the minimum Connected-<span>k</span>-Subgraph Cover problem (MinC<span>k</span>SC) is to find a minimum vertex subset <span>C</span> of <span>G</span> such that every connected subgraph of <span>G</span> on <span>k</span> vertices has at least one vertex in <span>C</span>. If furthermore the subgraph of <span>G</span> induced by <span>C</span> is connected, then the problem is denoted as MinC<span>k</span>SC<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline1.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$_{con}$</span></span></img></span></span>. In this paper, we first present a PTAS for MinC<span>k</span>SC on an <span>H</span>-minor-free graph, where <span>H</span> is a graph with a constant number of vertices. Then, we design an <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline2.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$O((\\\\omega+1)(2(k-1)(\\\\omega+2))^{3\\\\omega+3})|V|$</span></span></img></span></span>-time FPT algorithm for MinC<span>k</span>SC<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline3.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$_{con}$</span></span></img></span></span> on a graph with treewidth <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline4.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$\\\\omega$</span></span></img></span></span>, based on which we further design an <span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline5.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$O(2^{O(\\\\sqrt{t}\\\\log t)}|V|^{O(1)})$</span></span></img></span></span> time subexponential FPT algorithm for MinC<span>k</span>SC<span><span><img data-mimesubtype=\\\"png\\\" data-type=\\\"\\\" src=\\\"https://static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240109155214832-0728:S0960129523000439:S0960129523000439_inline6.png\\\"><span data-mathjax-type=\\\"texmath\\\"><span>$_{con}$</span></span></img></span></span> on an <span>H</span>-minor-free graph, where <span>t</span> is an upper bound of solution size.</p>\",\"PeriodicalId\":49855,\"journal\":{\"name\":\"Mathematical Structures in Computer Science\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.4000,\"publicationDate\":\"2024-01-10\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Mathematical Structures in Computer Science\",\"FirstCategoryId\":\"94\",\"ListUrlMain\":\"https://doi.org/10.1017/s0960129523000439\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Mathematical Structures in Computer Science","FirstCategoryId":"94","ListUrlMain":"https://doi.org/10.1017/s0960129523000439","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
给定一个图 G,最小连通子图覆盖问题(MinCkSC)就是找到 G 的最小顶点子集 C,使得 G 的 k 个顶点上的每个连通子图都至少有一个顶点在 C 中。在本文中,我们首先提出了在 H-minor-free 图上的 MinCkSC 的 PTAS,其中 H 是具有恒定顶点数的图。然后,我们为具有树宽 $\omega$ 的图上的 MinCkSC$_{con}$ 设计了一个 $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$ 时的 FPT 算法、在此基础上,我们进一步为无 H 小数图上的 MinCkSC$_{con}$ 设计了一种 $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ 时间的亚指数 FPT 算法,其中 t 是解大小的上限。
Approximation Algorithm and FPT Algorithm for Connected-k-Subgraph Cover on Minor-Free Graphs
Given a graph G, the minimum Connected-k-Subgraph Cover problem (MinCkSC) is to find a minimum vertex subset C of G such that every connected subgraph of G on k vertices has at least one vertex in C. If furthermore the subgraph of G induced by C is connected, then the problem is denoted as MinCkSC$_{con}$. In this paper, we first present a PTAS for MinCkSC on an H-minor-free graph, where H is a graph with a constant number of vertices. Then, we design an $O((\omega+1)(2(k-1)(\omega+2))^{3\omega+3})|V|$-time FPT algorithm for MinCkSC$_{con}$ on a graph with treewidth $\omega$, based on which we further design an $O(2^{O(\sqrt{t}\log t)}|V|^{O(1)})$ time subexponential FPT algorithm for MinCkSC$_{con}$ on an H-minor-free graph, where t is an upper bound of solution size.
期刊介绍:
Mathematical Structures in Computer Science is a journal of theoretical computer science which focuses on the application of ideas from the structural side of mathematics and mathematical logic to computer science. The journal aims to bridge the gap between theoretical contributions and software design, publishing original papers of a high standard and broad surveys with original perspectives in all areas of computing, provided that ideas or results from logic, algebra, geometry, category theory or other areas of logic and mathematics form a basis for the work. The journal welcomes applications to computing based on the use of specific mathematical structures (e.g. topological and order-theoretic structures) as well as on proof-theoretic notions or results.