Lower bounds and a tabu search algorithm for the minimum deficiency problem

An edge coloring of a graph G=(V,E) is a function c:E -> a"center dot that assigns a color c(e) to each edge eaE such that c(e)not equal c(e') whenever e and e' have a common endpoint. Denoting S (v) (G,c) the set of colors assigned to the edges incident to a vertex vaV, and D (v) (G,c) the minimum number of integers which must be added to S (v) (G,c) to form an interval, the deficiency D(G,c) of an edge coloring c is defined as the sum a (vaV) D (v) (G,c), and the span of c is the number of colors used in c. The problem of finding, for a given graph, an edge coloring with a minimum deficiency is NP-hard. We give new lower bounds on the minimum deficiency of an edge coloring and on the span of edge colorings with minimum deficiency. We also propose a tabu search algorithm to solve the minimum deficiency problem and report experiments on various graph instances, some of them having a known optimal deficiency.