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

Given a graph GG with two distinguished nodes ss and tt, a cost on each edge of GG and two fixed integers k≥2k≥2, L≥2L≥2, the kk edge-disjoint LL-hop-constrained paths problem is to find a minimum cost subgraph of GG such that between ss and tt there are at least kk edge-disjoint paths of length at most LL. 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=3L=3 and k≥2k≥2, the linear relaxation of the associated polytope, given by the trivial, the stst-cut and the so-called LL-path-cut inequalities, is integral. As a consequence, we obtain a polynomial time cutting plane algorithm for the problem when L=2,3L=2,3 and k≥1k≥1. This generalizes the results of Huygens et al. (2004) [1] for k=2k=2 and L=2,3L=2,3 and those of Dahl et al. (2006) [2] for L=2L=2 and k≥2k≥2. This also proves a conjecture in [1].