Valid inequalities for the k-Color Shortest Path Problem

Given a digraph D=(V,A) where each arc (i,j)∈A has a cost dij∈R+ and a color c(i,j), a positive integer k, and vertices s,t∈V, the k-Color Shortest Path Problem consists in finding a path from s to t of minimum cost while using at most k distinct arc colors. We propose valid inequalities for the problem that proved to strengthen the linear relaxation of an existing Integer Linear Programming formulation for the problem. One exponential set of valid inequalities defines a new formulation for the problem that is solved by using a branch-and-cut algorithm. We introduce more challenging instances for the problem and present numerical experiments for both the benchmark and the new instances. Finally, we evaluate the individual and the collective use of the valid inequalities. Computational results for the proposed ideas and for existing solution approaches for the problem showed the effectiveness of the new inequalities in handling the new instances, both in terms of execution times and improvement of the linear relaxed solutions.