Survivable Networks, Linear Programming Relaxations and the Parsimonious Property

We consider the survivable network design problem - the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and the k-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worstcase analyses of two heuristics for the survivable network design problem

Intuitive solution-double techniques for worst-case analysis of some survivable network design problems

"February 2000." Title from cover.Includes bibliographical references (p. 10).by T. Magnanti, A. Balakrishnan, P. Mirchandani

Improved Algorithm for Degree Bounded Survivable Network Design Problem

We consider the Degree-Bounded Survivable Network Design Problem: the objective is to find a minimum cost subgraph satisfying the given connectivity requirements as well as the degree bounds on the vertices. If we denote the upper bound on the degree of a vertex v by b(v), then we present an algorithm that finds a solution whose cost is at most twice the cost of the optimal solution while the degree of a degree constrained vertex v is at most 2b(v) + 2. This improves upon the results of Lau and Singh and that of Lau, Naor, Salavatipour and Singh

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

The parsimonious property of cut covering problems and its applications

Includes bibliographical references (p. 26-27).Supported by a Presidential Young Investigator Award. DDM-91568118 Supported by Draper Laboratory.Dimitris Bertsimas and Chungpiaw Teo

Primal-dual approaches to the Steiner problem

We study several old and new algerithms for computing lower and upper bounds for the Steiner problem in networks using dual-ascent and primal-dual strategies. These strategies have been proven to be very useful. for the algorithmic treatment of the Steiner problem. We show that none of the known algorithms can both generate tight lower bounds empirically and guarantee their quality theoretically; and we present a new algorithm which combines both features. The new algorithm has running time O(re log n) and guarantees a ratio of at most two between the generated upper and lower bounds, whereas the fastest previous algorithm with comparably tight empiricalbounds has running time O(e²) without a constant approximation ratio. We show that the approximation ratio two between the bounds can even be achieved in time O(e + n log n), improving the.previous time bound of O(n² log n). The presented insights can also behelpful for the development of further relaxation based approximation algorithms for the Steiner problem