Additive non-approximability of chromatic number in proper minor-closed classes

TL;DR

This paper proves that, unless P=NP, there are no polynomial-time algorithms to approximate the chromatic number of certain minor-closed graph classes within a fixed additive error, and introduces algorithms with specific additive error bounds.

Contribution

It establishes the first non-trivial non-approximability bounds for chromatic number in proper minor-closed classes and provides approximation algorithms with bounds depending on structural parameters.

Findings

No polynomial-time additive approximation within certain bounds unless P=NP.

First non-trivial non-approximability results for chromatic number in proper minor-closed classes.

Algorithms with additive error depending on the apex number and absence of 4-cycles.

Abstract

Robin Thomas asked whether for every proper minor-closed class C, there exists a polynomial-time algorithm approximating the chromatic number of graphs from C up to a constant additive error independent on the class C. We show this is not the case: unless P=NP, for every integer k>=1, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using at most chi(G)+k-1 colors. More generally, for every k>=1 and 1<=\beta<=4/3, there is no polynomial-time algorithm to color a K_{4k+1}-minor-free graph G using less than beta.chi(G)+(4-3beta)k colors. As far as we know, this is the first non-trivial non-approximability result regarding the chromatic number in proper minor-closed classes. We also give somewhat weaker non-approximability bound for K_{4k+1}-minor-free graphs with no cliques of size 4. On the positive side, we present additive approximation algorithm whose error depends on the apex number of the forbidden minor, and an algorithm with additive error 6 under the additional assumption that the graph has no 4-cycles.