About equivalent interval colorings of weighted graphs

Given a graph G = (V,E) with strictly positive integer weights ωi on the vertices i ∈ V , a k-interval coloring of G is a function I that assigns an interval I(i) ⊆ {1, · · · , k} of ωi consecutive integers (called colors) to each vertex i ∈ V . If two adjacent vertices x and y have common colors, i.e. I(i)∩ I(j) 6= ∅ for an edge [i, j] in G, then the edge [i, j] is said conflicting. A k-interval coloring without conflicting edges is said legal. The interval coloring problem (ICP) is to determine the smallest integer k, called interval chromatic number of G and denoted χint(G), such that there exists a legal k-interval coloring of G. For a fixed integer k, the k-interval graph coloring problem (k-ICP) is to determine a k-interval coloring of G with a minimum number of conflicting edges. The ICP and k-ICP generalize classical vertex coloring problems where a single color has to be assigned to each vertex (i.e., ωi = 1 for all vertices i ∈ V ). Two k-interval colorings I1 and I2 are said equivalent if there is a permutation π of the integers 1, · · · , k such that l ∈ I1(i) if and only if π(l) ∈ I2(i) for all vertices i ∈ V . As for classical vertex coloring, the efficiency of algorithms that solve the ICP or the k-ICP can be increased by avoiding considering equivalent k-interval colorings. To this purpose, we define and prove a necessary and sufficient condition for the equivalence of two k-interval colorings. We then show how a simple tabu search algorithm for the k-ICP can possibly be improved by forbidding the visit of equivalent solutions.