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

Scheduling Algorithms for Parallel Execution of Computer Programs

Computer Scienc

A Survey of Network Design Problems

This report is a survey of the design of various types of networks that frequently occur in the study of transportation and communication problems. The report contains a general framework which facilitates comparisons between problems. We discuss a large number of different network design problems and give computational experience for the various solution techniques.Supported in part by the U.S. Department of Transportation under contract DOT-TSC-1058, Transportation Advanced Research Program (TARP)

Analysis of Linear Programming Relaxations for a Class of Connectivity Problems

We consider the analysis of linear programming (LP) relaxations for a class of connectivity problems. The central problem in the class is the survivable network design problem - the problem of designing a minimum cost undirected network satisfying prespecified connectivity requirements between every pair of vertices. This class includes a number of classical combinatorial optimization problems as special cases such as the Steiner tree problem, the traveling salesman problem, the k-person traveling salesman problem and the k-edge-connected network problem. We analyze a classical linear programming relaxation for this class of problems under three perspectives: structural, worst-case and probabilistic. Our analysis rests mainly upon a deep structural property, the parsimonious property, of this LP relaxation. Roughly stated, the parsimonious property says that, if the cost function satisfies the triangle inequality, there exists an optimal solution to the LP relaxation for which the degree of each vertex is the smallest it can possibly be. The numerous consequences of the parsimonious property make it particularly important. First, several special cases of the parsimonious property are interesting properties by themselves. For example, we derive the monotonicity of the Held-Karp lower bound for the traveling salesman problem and the fact that this bound is a relaxation on the 2-connected network problem. Another consequence is the fact that vertices with no connectivity requirement, such as Steiner vertices in the undirected Steiner tree problem, are unnecessary for the LP relaxation under consideration. From the parsimonious property, it also follows that the LP relaxation bounds corresponding to the Steiner tree problem, the kedge-connected network problem or even the Steiner k-edge-connected network problem can be computed a la Held and Karp.Secondly, we use the parsimonious property to perform worst-case analyses of the duality gap corresponding to these LP relaxations. For this purpose, we introduce two heuristics for the survivable network design problem and present bounds dependent on the actual connectivity requirements. Among other results, we show that the value of the LP relaxation of the Steiner tree problem is within twice the value of the minimum spanning tree heuristic and that several generalizations of the Steiner tree problem, including the k-edge-connected network problem, can also be approximated within a factor of 2 (in some cases, even smaller than 2). We also introduce a new relaxation a la Held and Karp for the k-person traveling salesman problem and show that a variation of an existing heuristic is within times the value of this relaxation. We show that most of our bounds are tight and we investigate whether the bound of 3 for the Held-Karp lower bound is tight.We also perform a probabilistic analysis of the duality gap of these LP relaxations. The model we consider is the Euclidean model. We generalize Steele's theorem on the asymptotic behavior of Euclidean functionals in a way that is particularly convenient for the analysis of LP relaxations. We show that, under the Euclidean model, the duality gap is almost surely a constant and we provide theoretical and empirical bounds on these constants for different problems. From this analysis, we conclude that the undirected LP relaxation for the Steiner tree problem is fairly loose. Finally, we consider the use of directed relaxations for undirected problems. We establish in which settings a related parsimonious property holds and show that, for the Steiner tree problem, the directed relaxation strictly improves upon the undirected relaxation in the worst-case. This latter result uses an elementary but powerful property of linear programs

Optimal routing of urban traffic.

Massachusetts Institute of Technology. Dept. of City and Regional Planning. Thesis. 1965. Ph.D.Ph.D