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

Authors:
F. Bendali;I. Diarrassouba;A. R. Mahjoub;J. Mailfert
Affiliations:
Laboratoire LIMOS, CNRS UMR 6158, Université Blaise Pascal-Clermont II, Complexe scientifique des Cézeaux, 63177 Aubière cedex, France;Laboratoire LIMOS, CNRS UMR 6158, Université Blaise Pascal-Clermont II, Complexe scientifique des Cézeaux, 63177 Aubière cedex, France;Laboratoire LAMSADE, CNRS UMR 7024, Université Paris-Dauphine, Place du Maréchal De Lattre de Tassigny 75775, Paris, France;Laboratoire LIMOS, CNRS UMR 6158, Université d'Auvergne, 49 Bd François Mitterand, 63000 Clermont-Ferrand, France
Venue:
Discrete Optimization
Year:
2010

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Abstract

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].