Polynomial-time Algorithm for Maximum Weight Independent Set on <i>P</i> <sub>6</sub> -free Graphs
Andrzej Grzesik, Tereza Klimošová, Marcin Pilipczuk, Michał Pilipczuk
Abstract
In the classic Maximum Weight Independent Set problem, we are given a graph G with a nonnegative weight function on its 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 P 6 -free graph, that is, a graph that has no path on 6 vertices as an induced subgraph. This improves the polynomial-time algorithm on P 5 -free graphs of Lokshtanov et al. [ 15 ] and the quasipolynomial-time algorithm on P 6 -free graphs of Lokshtanov et al. [ 14 ]. 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 .