On cut-based inequalities for capacitated network design polyhedra

Abstract

In this article, we study capacitated network design problems. We unify and extend polyhedral results for directed, bidirected, and undirected link capacity models. Valid inequalities based on a network cut are known to be strong in several special cases. We show that regardless of the link model, facets of the polyhedra associated with such a cut translate to facets of the original network design polyhedra if the two subgraphs defined by the network cut are (strongly) connected. Our investigation of the facial structure of the cutset polyhedra allows to complement existing polyhedral results for the three variants by presenting facet-defining flow-cutset inequalities in a unifying way. In addition, we present a new class of facet-defining inequalities, showing as well that flowcutset inequalities alone do not suffice to give a complete description for single-commodity, single-module cutset polyhedra in the bidirected and undirected case -in contrast to a known result for the directed case. The practical importance of the theoretical investigations is highlighted in an extensive computational study on 27 instances from the Survivable Network Design Library (SNDlib).

A Flow Formulation for the Optimum Communication Spanning Tree

“…Constraints (23)- (27) link the flow to the a variables. The complete HL-MCF formulation is given by the objective function (MCF) subject to constraints (1)- (2) and (8)- (27). Lemma 2.1 Formulation HL-MCF is at least as strong as formulation MCF.…”

“…A solutionȳ,x,z,w,ā of the master problem defines a feasible solution for the R-HL-MCF if and only if for each i ∈ F there exist fractional flow values satisfying (21)- (27) with y i =ȳ i and a =ā.…”

“…In [12] a special case of the problem where the dominant extreme points of the demand polyhedron have a disjoint support is studied. For the RND problem with integer capacities and a single traffic matrix (Network Loading, NL) see [4,6,22,27]. For the problem with integer capacities and demand uncertainty (Robust Network Loading, RNL) see [2,3,17,20].…”

Models and algorithms for network design problems

“…In order to reinforce the (weak) LP relaxation of (ND), many researchers have looked at strong cutting planes, including Atamturk (2002); Bienstock and Gunluk (1996); Bienstock et al (1998); Dahl and Stoer (1998); Raack et al (2011). More recently, Achterberg and Raack (2010) have developed a procedure to automatically detect a network design structure within a general MILP, generating strong cutting planes accordingly.…”

“…Finally, they address a more complex formulation, taking node hardware and survivability into account (Baier et al, 2007). They also extend and test di erent sorts of cuts coming from mono-layer models (Raack et al, 2011). Belotti et al (2006) and Capone et al (2007) study multi-layered design with statistical multiplexing, the motivation being that routing di erent commodities on the same capacity results in less variation of the ow on the capacity.…”