On the directed hop-constrained shortest path problem

In this paper we discuss valid inequalities for the directed hop-constrained shortest path problem. We give complete linear characterizations of the hop-constrained path polytope when the maximum number of hops is equal to 2 or 3. We also present a lifted version of the "jump" inequalities introduced by Dahl (Oper. Res. Lett. 25 (1999) 97) and show that this class of inequalities subsumes inequalities contained in the complete linear description for the case H = 3 as well as a large class of facet defining inequalities for the case H = 4. We use a minmax result by Robacker (Research Memorandum RM-1660, The Rand Corporation, Santa Monica, 1956) to present a framework for deriving a large class of cut-like inequalities for the hop-constrained path problem. A simple relation between the hop-constrained path polytope and the knapsack polytopes is also presented. (C) 2003 Elsevier B.V. All rights reserved.