The Complexity of Minimum Convex Coloring

Abstract

A coloring of the vertices of a graph is called convex if each subgraph induced by all vertices of the same color is connected. We consider three variants of recoloring a colored graph with minimal cost such that the resulting coloring is convex. Two variants of the problem are shown to be ${\mathcal{NP}}$-hard on trees even if in the initial coloring each color is used to color only a bounded number of vertices. For graphs of bounded treewidth, we present a polynomial-time (2 + ε)-approximation algorithm for these two variants and a polynomial-time algorithm for the third variant. Our results also show that, unless ${\mathcal{NP}} \subseteq DTIME(n^{O(\log \log n)})$, there is no polynomial-time approximation algorithm with a ratio of size (1 − o(1))ln ln n for the following problem: Given pairs of vertices in an undirected graph of bounded treewidth, determine the minimal possible number l for which all except l pairs can be connected by disjoint paths.