The labeled maximum matching problem

Given a graph G where a label is associated with each edge, we address the problem of looking for a maximum matching of G using the minimum number of different labels, namely the labeled maximum matching problem. It is a relatively new problem whose application is related to the timetabling problem. We prove it is NP-complete and present four different mathematical formulations. Moreover, we propose an exact algorithm based on a branch-and-bound approach to solve it. We evaluate the performance of our algorithm on a wide set of instances and compare our computational times with the ones required by CPLEX to solve the proposed mathematical formulations. Test results show the effectiveness of our procedure, that hugely outperforms the solver.