The Complexity of Minimum Convex Coloring

  • Frank Kammer
  • Torsten Tholey
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5369)

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.

Keywords

Convex Coloring Maximum Disjoint Paths Problem 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Frank Kammer
    • 1
  • Torsten Tholey
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

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