Probabilistic graph-coloring in bipartite and split graphs

We revisit in this paper the stochastic model for minimum graph-coloring introduced in (C. Murat and V. Th. Paschos, On the probabilistic minimum coloring and minimum k-coloring, Discrete Applied Mathematics 154, 2006), and study the underlying combinatorial optimization problem (called probabilistic coloring) in bipartite and split graphs. We show that the obvious 2-coloring of any connected bipartite graph achieves standardapproximation ratio 2, that when vertex-probabilities are constant probabilistic coloring is polynomial and, finally, we propose a polynomial algorithm achieving standardapproximation ratio 8/7. We also handle the case of split graphs. We show that probabilistic coloring is NP-hard, even under identical vertex-probabilities, that it is approximable by a polynomial time standard-approximation schema but existence of a fully a polynomial time standard-approximation schema is impossible, even for identical vertex-probabilities, unless P = NP. We finally study differential-approximation of probabilistic coloring in both bipartite and split graphs. 1 Preliminaries In minimum graph-coloring problem, the objective is to color the vertex-set V of a graph G(V,E) with as few colors as possible so that no two adjacent vertices receive the same color. The decision version of this problem, called graph k-colorability in [13] and defined as: “given a graph G(V,E) and a positive integer k 6 |V |, is G, k-colorable?” was shown to be NP-complete in Karp’s seminal paper ([21]) and remains NP-complete even restricted to graphs of constant (independent on n) chromatic number at least 3 ([13]). Since adjacent vertices are forbidden to be colored with the same color, a feasible coloring is a partition of V into vertex-sets such that, for any such set, no two of its vertices are mutually adjacent. Such sets are usually called independent sets. So, the optimal solution of minimum coloring is a minimum-cardinality partition into independent sets. The chromatic number of a graph is the smallest number of colors that can feasibly color its vertices. In this paper, we use the following stochastic model for combinatorial optimization problems. Consider a generic instance I of a combinatorial optimization problem Π. Assume that Π is not ∗Part of this research has been performed while the first author was in visit at the LAMSADE on a research position funded by the CNRS