The separation problem of rounded capacity inequalities: Some polynomial cases

In this paper we are interested in the separation problem of the so-called rounded capacity inequalities which are involved in the two-index formulation of the CVRP (Capacitated Vehicle Routing Problem) polytope. In a recent work (Diarrassouba, 2015, [1]), we have investigated the theoretical complexity of that problem in the general case. Several authors have devised heuristic separation algorithms for rounded capacity inequalities for solving the CVRP. In this paper, we investigate the conditions under which this separation problem can be solved in polynomial time, and this, in the context of the CVRP or for solving other combinatorial optimization problems in which rounded capacity inequalities are involved. We present four cases in which they can be separated in polynomial time and reduce the problem to O(n2) maximum flow computations.