Tree independence number II. Three-path-configurations

TL;DR

This paper proves that 3PC-free graphs have a poly-logarithmic bound on their tree-independence number, enabling quasi-polynomial algorithms for certain NP-hard problems on these graphs.

Contribution

It introduces the class of 3PC-free graphs and establishes their poly-logarithmic tree-independence number, facilitating efficient algorithms for NP-hard problems.

Findings

3PC-free graphs have poly-logarithmic tree-independence number.

Maximum Weight Independent Set can be solved in quasi-polynomial time on 3PC-free graphs.

Other NP-hard problems are also efficiently solvable on these graphs.

Abstract

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant $c$ such that every $n$-vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most $c (\log n)^2$. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.