The two-edge connected hop-constrained network design problem: Valid inequalities and branch-and-cut

This article deals with the Two-edge connected Hop-constrained Network Design Problem (or THNDP for short). Given a weighted graph G = (N,E), an integer L >= 2, and a subset of pairs of nodes D, the problem consists of finding the minimum cost subgraph in G containing at least two edge-disjoint paths of at most L hops between all the pairs in D. First, we show that the THNDP is strongly NP-hard even when the demands in D are rooted at some node s and the costs are unitary. However, if the graph is complete, we prove that the problem in this case can be solved in polynomial time. We give an integer programming formulation of the problem in the space of the design variables when L = 2,3. Then we study the associated polytope. In particular, we consider the case where all the pairs of nodes of D are rooted at a node s. We give several classes of valid inequalities along with necessary and/or sufficient conditions for these inequalities to be facet defining. We also derive separation routines for these inequalities. We finally develop a branch-and-cut algorithm based on these results and discuss some computational results for L = 2,3. (c) 2006 Wiley Periodicals, Inc