The k edge-disjoint 3-hop-constrained paths polytope

Given a graph G with two distinguished nodes s and t, a cost on each edge of G and two fixed integers k≥2, L≥2, the k edge-disjoint L-hop-constrained paths problem is to find a minimum cost subgraph of G such that between s and t there are at least k edge-disjoint paths of length at most L. In this paper we consider this problem from a polyhedral point of view. We give an integer programming formulation for the problem and discuss the associated polytope. In particular, we show that when L=3 and k≥2, the linear relaxation of the associated polytope, given by the trivial, the st-cut and the so-called L-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L=2,3 and k≥1. This generalizes the results of Huygens et al. (2004) [1] for k=2 and L=2,3 and those of Dahl et al. (2006) [2] for L=2 and k≥2. This also proves a conjecture in [1].ou