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

Zdeněk Dvořák,Ken-Ichi Kawarabayashi

doi:10.1016/j.jctb.2020.09.003

Abstract

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomial-time algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G. We show this is not the case: unless P=NP, for every integer k≥1, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using at most χ(G)+k−1 colors. More generally, for every k≥1 and 1≤β≤4/3, there is no polynomial-time algorithm to color a K4k+1-minor-free graph G using less than βχ(G)+(4−3β)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.Furthermore, we give somewhat weaker non-approximability bound for K4k+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.

Highlights

Robin Thomas asked whether for every proper minor-closed class G, there exists a polynomialtime algorithm approximating the chromatic number of graphs from G up to a constant additive error independent on the class G
We present an 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
The problem of determining the chromatic number of a graph, or even of just deciding whether a graph is colorable using a fixed number c ≥ 3 of colors, is NP-complete [7], and it cannot be solved in polynomial time unless P = NP

Summary

Introduction

For some fixed constant α ≥ 0, the following holds: for every proper minor-closed class G, there exists a polynomial-time algorithm taking as an input a graph G ∈ G and returning an integer c such that χ(G) ≤ c ≤ χ(G) + α. E.g., if H is a t-apex graph, there is a polynomial-time algorithm coloring an H-minor-free graph G using at most min(2χ(G), χ(G) + t + 3) ≤ βχ(G) + (2 − β)(t + 3) colors, for any β such that 1 ≤ β ≤ 2; the combined multiplicative-additive non-approximability bounds of Theorems 2 and 5 are of interest in this context.

Tree-like product of graphs

Non-approximability

Approximation algorithms