Polynomial-time algorithm for Maximum Weight Independent Set on $P_6$-free graphs

TL;DR

This paper presents a polynomial-time algorithm for solving the Maximum Weight Independent Set problem specifically on $P_6$-free graphs, extending previous results for smaller path-free graphs and improving computational complexity.

Contribution

It introduces a novel polynomial-time algorithm for $P_6$-free graphs, advancing the understanding of independent set problems in restricted graph classes.

Findings

Polynomial-time algorithm for $P_6$-free graphs

Enumeration of polynomial-size family of vertex subsets

Improves upon previous algorithms for related graph classes

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 $P_6$-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. (SODA 2014), and the quasipolynomial-time algorithm on $P_6$-free graphs of Lokshtanov et al (SODA 2016). The main technical contribution leading to our main result is enumeration of a polynomial-size family $\mathcal{F}$ of vertex subsets with the following property: for every maximal independent set $I$ in the graph, $\mathcal{F}$ contains all maximal cliques of some minimal chordal completion of $G$ that does not add any edge incident to a vertex of $I$.