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oai:drops-oai.dagstuhl.de:6939 2024-03-06T10:38:47Z ddc:004 open_access

H-Free Graphs, Independent Sets, and Subexponential-Time Algorithms Bacsó, Gábor Marx, Dániel Tuza, Zsolt independent set scattered set subexponential algorithms H-free graphs It is an outstanding open question in algorithmic graph theory to determine the complexity of the MAXIMUM INDEPENDENT SET problem on P_t-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t at most 5 [Lokshtanov et al., SODA 2014, 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t=5 [Discrete App. Math., 158 (2010), 1041-1044], we show that for any t at least 5, there is an algorithm for MAXIMUM INDEPENDENT SET on P_t-free graphs whose running time is subexponential in the number of vertices. SCATTERED SET is the generalization of MAXIMUM INDEPENDENT SET where the vertices of the solution are required to be at distance at least $d$ from each other. We give a complete characterization of those graphs H for which SCATTERED SET on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges): * If every component of H is a path, then d-SCATTERED SET on H-free graphs with n vertices and m edges can be solved in time 2^{(n+m)^{1-O(1/|V(H)|)}}, even if d is part of the input. * Otherwise, assuming ETH, there is no 2^{o(n+m)} time algorithm for d-SCATTERED SET for any fixed d at least 3 on H-free graphs with n vertices and m edges. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Gábor Bacsó and Dániel Marx and Zsolt Tuza 2017 Is Part Of LIPIcs, Volume 63, 11th International Symposium on Parameterized and Exact Computation (IPEC 2016) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.IPEC.2016.3 urn:nbn:de:0030-drops-69397 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2016.3 eng https://creativecommons.org/licenses/by/3.0/legalcode