Robust capacitated trees and networks with uniform demands

We are interested in the design of robust (or resilient) capacitated rooted Steiner networks in case of terminals with uniform demands. Formally, we are given a graph, capacity and cost functions on the edges, a root, a subset of nodes called terminals, and a bound k on the number of edge failures. We first study the problem where k = 1 and the network that we want to design must be a tree covering the root and the terminals: we give complexity results and propose models to optimize both the cost of the tree and the number of terminals disconnected from the root in the worst case of an edge failure, while respecting the capacity constraints on the edges. Second, we consider the problem of computing a minimum-cost survivable network, i.e., a network that covers the root and terminals even after the removal of any k edges, while still respecting the capacity constraints on the edges. We also consider the possibility of protecting a given number of edges. We propose three different formulations: a cut-set based formulation, a flow based one, and a bilevel one (with an attacker and a defender). We propose algorithms to solve each formulation and compare their efficiency

Hypergraphic LP Relaxations for Steiner Trees

We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Koenemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of sqrt(3) ~ 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60 ~ 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well.Comment: Revised full version; a shorter version will appear at IPCO 2010

An Exact Algorithm for the Steiner Forest Problem

The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. The problem has famous linear programming based 2-approximations [Agrawal et al., 1995; Goemans and Williamson, 1995; Jain, 2001] whose bottleneck is the fact that the most natural formulation of the problem as an integer linear program (ILP) has an integrality gap of 2. We propose new cut-based ILP formulations for the problem along with exact branch-and-bound based algorithms. While our new formulations cannot improve the integrality gap, we can prove that one of them yields stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation [Balakrishnan et al., 1989; Chopra and Rao, 1994] and the advanced flow-based formulation by Magnanti and Raghavan [Magnanti and Raghavan, 2005]. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that our new branch-and-bound algorithms outperform branch-and-bound algorithms based on the previous formulations. Our formulations can be seen as a cut-based analogon to [Magnanti and Raghavan, 2005], whose existence was an open problem

A Comparison of Steiner Tree Relaxations

There are many (mixed) integer programming formulations of the Steiner problem in networks. The corresponding linear programming relaxations are of great interest particularly, but not exclusively, for computing lower bounds; but not much has been known ab out the relative quality of these relaxations. We compare all classical and some new relaxations from a theoretical point of view with respect to their optimal values. Among other things, we prove that the optimal value of a flowclass relaxation (e.g. the multicommodity flow or the dicut relaxation) cannot be worse than the optimal value of a tree-class relaxation (e.g. degree-constrained spanning tree relaxation) and that the ratio of the corresponding optimal values can be arbitrarily large. Furthermore, we present a new flow based relaxation, which is to the authors' knowledge the strongest linear relaxation of polynomial size for the Steiner problem in networks

An Exact Algorithm for the Steiner Forest Problem

The Steiner forest problem asks for a minimum weight forest that spans a given number of terminal sets. The problem has famous linear programming based 2-approximations [Agrawal et al., 1995; Goemans and Williamson, 1995; Jain, 2001] whose bottleneck is the fact that the most natural formulation of the problem as an integer linear program (ILP) has an integrality gap of 2. We propose new cut-based ILP formulations for the problem along with exact branch-and-bound based algorithms. While our new formulations cannot improve the integrality gap, we can prove that one of them yields stronger linear programming bounds than the two previous strongest formulations: The directed cut formulation [Balakrishnan et al., 1989; Chopra and Rao, 1994] and the advanced flow-based formulation by Magnanti and Raghavan [Magnanti and Raghavan, 2005]. In an experimental evaluation, we show that the linear programming bounds of the new formulations are indeed strong on practical instances and that our new branch-and-bound algorithms outperform branch-and-bound algorithms based on the previous formulations. Our formulations can be seen as a cut-based analogon to [Magnanti and Raghavan, 2005], whose existence was an open problem

Connectivity-splitting models for survivable network design

"January 2000." Title from cover.Includes bibliographical references (p. 24-25).by T.L. Magnanti, A. Balakrishnan, P. Mirchandani