{"title":"Optimal dynamic program for r-domination problems over tree decompositions","authors":"G. Borradaile, Hung Le","doi":"10.4230/LIPIcs.IPEC.2016.8","DOIUrl":null,"url":null,"abstract":"There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems are optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to $r$-domination problems. In $r$-dominating set, one wished to find a minimum subset $S$ of vertices such that every vertex of $G$ is within $r$ hops of some vertex in $S$. In connected $r$-dominating set, one additionally requires that the set induces a connected subgraph of $G$. We give a $O((2r+1)^{\\mathrm{tw}} n)$ time algorithm for $r$-dominating set and a $O((2r+2)^{\\mathrm{tw}} n^{O(1)})$ time algorithm for connected $r$-dominating set in $n$-vertex graphs of treewidth $\\mathrm{tw}$. We show that the running time dependence on $r$ and $\\mathrm{tw}$ is the best possible under SETH. This adds to earlier observations that a \"+1\" in the denominator is required for connectivity constraints.","PeriodicalId":137775,"journal":{"name":"International Symposium on Parameterized and Exact Computation","volume":"19 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2015-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"29","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"International Symposium on Parameterized and Exact Computation","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4230/LIPIcs.IPEC.2016.8","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 29
Abstract
There has been recent progress in showing that the exponential dependence on treewidth in dynamic programming algorithms for solving NP-hard problems are optimal under the Strong Exponential Time Hypothesis (SETH). We extend this work to $r$-domination problems. In $r$-dominating set, one wished to find a minimum subset $S$ of vertices such that every vertex of $G$ is within $r$ hops of some vertex in $S$. In connected $r$-dominating set, one additionally requires that the set induces a connected subgraph of $G$. We give a $O((2r+1)^{\mathrm{tw}} n)$ time algorithm for $r$-dominating set and a $O((2r+2)^{\mathrm{tw}} n^{O(1)})$ time algorithm for connected $r$-dominating set in $n$-vertex graphs of treewidth $\mathrm{tw}$. We show that the running time dependence on $r$ and $\mathrm{tw}$ is the best possible under SETH. This adds to earlier observations that a "+1" in the denominator is required for connectivity constraints.