On the probabilistic minimum coloring and minimum k-coloring

  • Authors:

  • Cécile Murat;Vangelis Th. Paschos

  • Affiliations:

  • LAMSADE, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, Paris Cedex, France;LAMSADE, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny, Paris Cedex, 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 vi of the input-graph G(V, E) has some presence probability Pi. 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 S1, S2,..., Sk, that minimizes the quantity Σi=1k f(Si), where, for any independent set Si, i = 1,..., k, f(Si) = 1 - Πvj∈si (1 - pj). 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.