Probabilistic Coloring of Bipartite and Split Graphs

We revisit in this paper the probabilistic coloring problem ( ) and focus ourselves on bipartite and split graphs. We first give some general properties dealing with the optimal solution. We then show that the unique 2-coloring achieves approximation ratio 2 in bipartite graphs under any system of vertex-probabilities and propose a polynomial algorithm achieving tight approximation ratio 8/7 under identical vertex-probabilities. Then we deal with restricted cases of bipartite graphs. Main results for these cases are the following. Under non-identical vertex-probabilities   is polynomial for stars, for trees with bounded degree and a fixed number of distinct vertex-probabilities, and, consequently, also for paths with a fixed number of distinct vertex-probabilities. Under identical vertex-probabilities,   is polynomial for paths, for even and odd cycles and for trees whose leaves are either at even or at odd levels. Next, we deal with split graphs and show that   is NP-hard, even under identical vertex-probabilities. Finally, we study approximation in split graphs and provide a 2-approximation algorithm for the case of distinct probabilities and a polynomial time approximation schema under identical vertex-probabilities. 1 Preliminaries In minimum 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. Since adjacent vertices are forbidden to be colored with the same color, a feasible coloring can be seen as a partition of V into independent sets. So, the optimal solution of minimum coloring is a minimum-cardinality partition into independent sets. The decision version of this problem was shown to be NP-complete in Karp’s seminal paper ([13]). The chromatic number of a graph is the smallest number of colors that can feasibly color its vertices. In the probabilistic version of minimum coloring, denoted by  , we are given: • a graph G(V,E) of order n, and an n-vector Pr = (p1, . . . , pn) of vertex-probabilities; in other words, an instance of   is a pair (G,Pr); • a modification strategy M, i.e., an algorithm that when receiving a coloring C = (S1, . . . , Sk) for V , called a priori solution, and a subgraph G′ = G[V ′] of G induced by a sub-set V ′ ⊆ V as inputs, it modifies C in order to produce a coloring C ′ for G′. The objective is to determine a coloring C∗ (called optimal a priori solution) of G minimizing the quantity (commonly called functional) E(G,C, M) = ∑ V ⊆V Pr[V ′]|C(V ′, M)| where C(V ′, M) is the solution computed by M(C, V ′) (i.e., by M when executed with inputs the a priori solution C and the subgraph of G induced by V ′) and Pr[V ′] = ∏ i∈V ′ pi ∏ i∈V \V ′(1− pi) (there exist 2 n distinct sets V ′; therefore, explicit computation of E(G,C, M) is, a priori, not polynomial). The complexity of   is the complexity of computing C∗.