Integer programming formulations for thek-edge-connected 3-hop-constrained network design problem

Abstract

In this article, we study the k -edge-connected L-hopconstrained network design problem. Given a weighted graph G = (V , E ), a set D of pairs of nodes, two integers L ≥ 2 and k ≥ 2, the problem consists in finding a minimum weight subgraph of G containing at least k edge-disjoint paths of length at most L between every pair {s, t } ∈ D. We consider the problem in the case where L = 2, 3 and |D| ≥ 2. We first discuss integer programming formulations introduced in the literature. Then, we introduce new integer programming formulations for the problem that are based on the transformation of the initial undirected graph into directed layered graphs. We present a theoretical comparison of these formulations in terms of LP-bound. Finally, these formulations are tested using CPLEX and compared in a computational study for k = 3, 4, 5.

k-node-disjoint hop-constrained survivable networks: polyhedral analysis and branch and cut

“…Edge/node-disjoint Huygens et al [25] IPF, valid inequalities and branch-and-cut algorithm for L = 2, 3 k ≥ 1 Edge-disjoint Bendali et al [2] Characterization of the associated polytope for L = 3 and |D| = 1 k ≥ 1 Edge-disjoint Diarrassouba et al [13] Valid inequalities and branch-and-cut and branch-and-cut-and-price algorithms for L = 2, 3 k = 2 Node-disjoint Diarrassouba et al [12] Valid inequalities and branch-and-cut algorithm for L = 3 , valid inequalities, ILP formulation, valid separation, branch-and-cut, polytope inequalities, separation, characterization [1,20,22,23,27,30] branch-and-cut [20,22,23,27,31] k ≥ 3 L = ∞ ILP formulation, valid inequalities, ILP formulation, separation separation, branch-and-cut, polytope valid inequalities, characterization [1,20,23,27,30] branch-and-cut [3,8,20,23] k = 2 L = 2, 3 ILP formulation, valid inequalities, ILP formulation, valid inequalities separation, branch-and-cut [25,26] polyhedral study, branch-and-cut [2,8,20,23] k = 2 L = 4 ILP formulation, valid inequalities, ILP formulation, valid inequalities separation, branch-and-cut [24,25] branch-and-cut [24]…”

“…ILP formulation, valid inequalities, Considered in this paper separation, branch-and-cut, extended formulation [2,4,5,[7][8][9]13] The remaining of the paper is organized as follows. In Section 2, we give an integer programming formulation for the problem.…”

“…Diarrassouba et al [13] introduced the so-called Steiner SPpartition inequalities for the kEDHP. In what follows, we extend these inequalities to the kNDHP.…”

“…Proof Let x ∈ R E be the solution for which we are separating the natural inequalities (1) and (2). To separate them, we consider the following graph transformation from [2] (see also [13]). Let (s, t) ∈ D and let V st = V \{s, t}, V st be a copy of V st and V st = V st ∪ V st ∪ {s, t}.…”

Two node-disjoint hop-constrained survivable network design and polyhedra

“…Bendali et al give a complete and minimal description of the polytope associated with the problem when L = 3 and . Diarrassouba et al (see also ) consider the problem when and . They present several integer programming formulations for the problem and devise a Branch‐and‐Cut algorithm for the problem in this latter case.…”