Tree Independence Number IV. Even-hole-free Graphs

TL;DR

This paper demonstrates that even-hole-free graphs have a polylogarithmic bound on their tree independence number, enabling quasi-polynomial algorithms for NP-hard problems like Maximum Weight Independent Set.

Contribution

It establishes a polylogarithmic bound on the tree independence number for even-hole-free graphs, leading to improved algorithms for certain NP-hard problems.

Findings

Tree independence number is at most polylogarithmic in even-hole-free graphs.

Maximum Weight Independent Set can be solved in quasi-polynomial time on these graphs.

Provides a structural insight enabling efficient algorithms for NP-hard problems.

Abstract

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most c log^10 n. 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 even-hole-free.