About equivalent interval colorings of weighted graphs

• Authors: Mathieu Bouchard;Mirjana angalović;Alain Hertz
• Venue: Discrete Applied Mathematics
• Year: 2009

Given a graph G=(V,E) with strictly positive integer weights @w"i on the vertices i@?V, a k-interval coloring of G is a function I that assigns an interval I(i)@?{1,...,k} of @w"i 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)0@? 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 @g"i"n"t(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., @w"i=1 for all vertices i@?V). Two k-interval colorings I"1 and I"2 are said equivalent if there is a permutation @p of the integers 1,...,k such that @?@?I"1(i) if and only if @p(@?)@?I"2(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, assuming that they can be identified very quickly. 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.