On the probabilistic minimum coloring and minimum k-coloring

  • Authors:

  • Cécile Murat;Vangelis Th. Paschos

  • Affiliations:

  • LAMSADE, CNRS UMR 7024 and Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France;LAMSADE, CNRS UMR 7024 and Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, 75775 Paris Cedex 16, France

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2006

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Abstract

We study a robustness model for the minimum coloring problem, where any vertex v"i of the input-graph G(V,E) has some presence probability p"i. We show that, under this model, the original coloring problem gives rise to a new coloring version (called Probabilistic Min Coloring) where the objective becomes to determine a partition of V into independent sets S"1,S"2,...,S"k, that minimizes the quantity @?"i"="1^kf(S"i), where, for any independent set S"i,i=1,...,k, f(S"i)=1-@?"v"""j"@?"S"""i(1-p"j). We show that Probabilistic Min Coloring is NP-hard and design a polynomial time approximation algorithm achieving non-trivial approximation ratio. We then focus ourselves on probabilistic coloring of bipartite graphs and show that the problem of determining the best k-coloring (called Probabilistic Min k-Coloring) is NP-hard, for any k=3. We finally study Probabilistic Min Coloring and Probabilistic Min k-Coloring in a particular family of bipartite graphs that plays a crucial role in the proof of the NP-hardness result just mentioned, and in complements of bipartite graphs.