Exact algorithms for a discrete metric labeling problem

We are given a edge-weighted undirected graph G = (V, E) and a set of labels/colors C = {1, 2, . . . , p}. A non-empty subset C_v in C is associated with each vertex v in V. A coloring of the vertices is feasible if each vertex v is colored with a color of C_v. A coloring uniquely defines a subset E_0 in E of edges having different colored endpoints. The problem of finding a feasible coloring which defines a minimum weight E_0 is, in general, NP-hard. In this work we first propose polynomial time algorithms for some special cases, namely when the input graph is a tree, a cactus or with bounded tree-width. Then, an implicit enumeration scheme for finding an optimal coloring in the general case is described and computational results are presented