Marcin Pilipczuk, Michal Pilipczuk, Paweł Rzaͅżewski
{"title":"通过压缩诱导路径空间求无pt图中独立集的拟多项式时间算法","authors":"Marcin Pilipczuk, Michal Pilipczuk, Paweł Rzaͅżewski","doi":"10.1137/1.9781611976496.23","DOIUrl":null,"url":null,"abstract":"In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] showed a quasi-polynomial-time algorithm for Maximum Weight Independent Set in $P_t$-free graphs, that is, graphs excluding a fixed path as an induced subgraph. Their algorithm runs in time $n^{\\mathcal{O}(\\log^3 n)}$, where $t$ is assumed to be a constant. Inspired by their ideas, we present an arguably simpler algorithm with an improved running time bound of $n^{\\mathcal{O}(\\log^2 n)}$. Our main insight is that a connected $P_t$-free graph always contains a vertex $w$ whose neighborhood intersects, for a constant fraction of pairs $\\{u,v\\} \\in \\binom{V(G)}{2}$, a constant fraction of induced $u-v$ paths. Since a $P_t$-free graph contains $\\mathcal{O}(n^{t-1})$ induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound. We also show that the same approach can be used to obtain quasi-polynomial-time algorithms for related problems, including Maximum Weight Induced Matching and 3-Coloring.","PeriodicalId":93491,"journal":{"name":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","volume":"76 1","pages":"204-209"},"PeriodicalIF":0.0000,"publicationDate":"2020-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"20","resultStr":"{\"title\":\"Quasi-polynomial-time algorithm for Independent Set in Pt-free graphs via shrinking the space of induced paths\",\"authors\":\"Marcin Pilipczuk, Michal Pilipczuk, Paweł Rzaͅżewski\",\"doi\":\"10.1137/1.9781611976496.23\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] showed a quasi-polynomial-time algorithm for Maximum Weight Independent Set in $P_t$-free graphs, that is, graphs excluding a fixed path as an induced subgraph. Their algorithm runs in time $n^{\\\\mathcal{O}(\\\\log^3 n)}$, where $t$ is assumed to be a constant. Inspired by their ideas, we present an arguably simpler algorithm with an improved running time bound of $n^{\\\\mathcal{O}(\\\\log^2 n)}$. Our main insight is that a connected $P_t$-free graph always contains a vertex $w$ whose neighborhood intersects, for a constant fraction of pairs $\\\\{u,v\\\\} \\\\in \\\\binom{V(G)}{2}$, a constant fraction of induced $u-v$ paths. Since a $P_t$-free graph contains $\\\\mathcal{O}(n^{t-1})$ induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound. We also show that the same approach can be used to obtain quasi-polynomial-time algorithms for related problems, including Maximum Weight Induced Matching and 3-Coloring.\",\"PeriodicalId\":93491,\"journal\":{\"name\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"volume\":\"76 1\",\"pages\":\"204-209\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2020-09-28\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"20\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1137/1.9781611976496.23\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Proceedings of the SIAM Symposium on Simplicity in Algorithms (SOSA)","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1137/1.9781611976496.23","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
Quasi-polynomial-time algorithm for Independent Set in Pt-free graphs via shrinking the space of induced paths
In a recent breakthrough work, Gartland and Lokshtanov [FOCS 2020] showed a quasi-polynomial-time algorithm for Maximum Weight Independent Set in $P_t$-free graphs, that is, graphs excluding a fixed path as an induced subgraph. Their algorithm runs in time $n^{\mathcal{O}(\log^3 n)}$, where $t$ is assumed to be a constant. Inspired by their ideas, we present an arguably simpler algorithm with an improved running time bound of $n^{\mathcal{O}(\log^2 n)}$. Our main insight is that a connected $P_t$-free graph always contains a vertex $w$ whose neighborhood intersects, for a constant fraction of pairs $\{u,v\} \in \binom{V(G)}{2}$, a constant fraction of induced $u-v$ paths. Since a $P_t$-free graph contains $\mathcal{O}(n^{t-1})$ induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound. We also show that the same approach can be used to obtain quasi-polynomial-time algorithms for related problems, including Maximum Weight Induced Matching and 3-Coloring.