Data Reduction for Graph Coloring Problems

TL;DR

This paper investigates how effectively polynomial-time data reduction can simplify graph coloring problems when parameterized by structural modifications, providing bounds and relations to kernelization complexity.

Contribution

It extends Cai's work by establishing bounds and relationships for kernelization of q-Coloring based on structural parameters and coloring complexity.

Findings

Polynomial kernels depend on bounds for q-List-Coloring on certain graph classes.

Existence of polynomial kernels is linked to functions bounding NO-instances.

Provides upper and lower bounds for kernels in parameterized graph coloring.

Abstract

This paper studies the kernelization complexity of graph coloring problems with respect to certain structural parameterizations of the input instances. We are interested in how well polynomial-time data reduction can provably shrink instances of coloring problems, in terms of the chosen parameter. It is well known that deciding 3-colorability is already NP-complete, hence parameterizing by the requested number of colors is not fruitful. Instead, we pick up on a research thread initiated by Cai (DAM, 2003) who studied coloring problems parameterized by the modification distance of the input graph to a graph class on which coloring is polynomial-time solvable; for example parameterizing by the number k of vertex-deletions needed to make the graph chordal. We obtain various upper and lower bounds for kernels of such parameterizations of q-Coloring, complementing Cai's study of the time complexity with respect to these parameters. Our results show that the existence of polynomial kernels for q-Coloring parameterized by the vertex-deletion distance to a graph class F is strongly related to the existence of a function f(q) which bounds the number of vertices which are needed to preserve the NO-answer to an instance of q-List-Coloring on F.