Polynomial-time Algorithm for Maximum Weight Independent Set on P-6-free Graphs

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.