In the classic Maximum Weight Independent Set problem we are given a graph G with a nonnegative weight function on vertices, and the goal is to find an independent set in G of maximum possible weight. While the problem is NP-hard in general, we give a polynomial-time algorithm working on any P6-free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P5-free graphs of Lokshtanov et al. [11], and the quasipolynomial-time algorithm on P6-free graphs of Lokshtanov et al. [12]. The main technical contribution leading to our main result is enumeration of a polynomial-size family ℱ of vertex subsets with the following property: for every maximal independent set I in the graph, ℱ contains all maximal cliques of some minimal chordal completion of G that does not add any edge incident to a vertex of I.

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cover image Proceedings
Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms
Pages: 1257 - 1271
Editor: Timothy M. Chan, University of Illinois at Urbana-Champaign, USA
ISBN (Online): 978-1-61197-548-2


Published online: 2 January 2019




The majority of this work was done while Andrzej Grzesik held a post-doc position of Warsaw Centre of Mathematics and Computer Science (WCMCS). WCMCS also supported a visit of Tereza Klimošová in Warsaw, during which this research was initiated. The work of Ma. Pilipczuk is supported by the Polish National Science Centre grant UMO-2013/09/B/ST6/03136. Mi. Pilipczuk was supported by the Foundation for Polish Science (FNP) via the START stipend programme. Klimošová was supported by ANR project Stint under reference ANR-13-BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d'Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR), by Center of Excellence – ITI, project P202/12/G061 of GA CR and by Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).

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