Abstract (EN)

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 View the MathML source, where, for any independent set View the MathML source, f(Si)=1-∏vjset membership, variantSi(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 kgreater-or-equal, slanted3. 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.