{"title":"论多路近分隔器的参数化复杂性","authors":"B. Jansen, S. K. Roy","doi":"10.48550/arXiv.2310.04332","DOIUrl":null,"url":null,"abstract":"We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \\subseteq V(G)$, it asks whether there is a vertex set $S \\subseteq V(G) \\setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k \\log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T \\subseteq V(G)$ along with a single vertex $x \\in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"263 1","pages":"28:1-28:18"},"PeriodicalIF":0.0000,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the Parameterized Complexity of Multiway Near-Separator\",\"authors\":\"B. Jansen, S. K. Roy\",\"doi\":\"10.48550/arXiv.2310.04332\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \\\\subseteq V(G)$, it asks whether there is a vertex set $S \\\\subseteq V(G) \\\\setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k \\\\log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T \\\\subseteq V(G)$ along with a single vertex $x \\\\in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.\",\"PeriodicalId\":137775,\"journal\":{\"name\":\"International Symposium on Parameterized and Exact Computation\",\"volume\":\"263 1\",\"pages\":\"28:1-28:18\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2023-10-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"International Symposium on Parameterized and Exact Computation\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.48550/arXiv.2310.04332\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Parameterized and Exact Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2310.04332","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On the Parameterized Complexity of Multiway Near-Separator
We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \subseteq V(G)$, it asks whether there is a vertex set $S \subseteq V(G) \setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k \log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T \subseteq V(G)$ along with a single vertex $x \in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.