Exact algorithms for a discrete metric labeling problem

We are given a edge-weighted undirected graph G=(V,E)G=(V,E) and a set of labels/colors C={1,2,…,p}C={1,2,…,p}. A non-empty subset Cv⊆CCv⊆C is associated with each vertex v∈Vv∈V. A coloring of the vertices is feasible if each vertex vv is colored with a color of CvCv. A coloring uniquely defines a subset E′⊆EE′⊆E of edges having different colored endpoints. The problem of finding a feasible coloring which defines a minimum weight E′E′ 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.