Abstract
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. (SODA 2014), and the quasipolynomial-time algorithm on P6-free graphs of Lokshtanov et al (SODA 2016). 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.