The Complexity of Minimum Convex Coloring

  • Authors:

  • Frank Kammer;Torsten Tholey

  • Affiliations:

  • Institut für Informatik, Universität Augsburg, Augsburg, Germany D-86135;Institut für Informatik, Universität Augsburg, Augsburg, Germany D-86135

  • Venue:
  • ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
  • Year:
  • 2008

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Abstract

A coloring of the vertices of a graph is called convex if eachsubgraph induced by all vertices of the same color is connected. Weconsider three variants of recoloring a colored graph with minimalcost such that the resulting coloring is convex. Two variants ofthe problem are shown to be ${\mathcal{NP}}$-hard on trees even ifin the initial coloring each color is used to color only a boundednumber of vertices. For graphs of bounded treewidth, we present apolynomial-time (2 + ε)-approximation algorithmfor these two variants and a polynomial-time algorithm for thethird variant. Our results also show that, unless ${\mathcal{NP}}\subseteq DTIME(n^{O(\log \log n)})$, there is no polynomial-timeapproximation algorithm with a ratio of size (1 - o(1))lnln n for the following problem: Given pairs of vertices inan undirected graph of bounded treewidth, determine the minimalpossible number l for which all except l pairscan be connected by disjoint paths.