Formulations and algorithms for network design problems with connectivity requirements

Recent rapid technological advances (including high-capacity fiber-optic cables) in the telecommunications industry have introduced numerous opportunities for services other than traditional voice transmission. Disruptions in service for these high capacity systems can be (and have been) disastrous. For these reasons, network reliability is critical. This thesis studies the problem of topological design of survivable networks, that is, networks capable of maintaining communication capabilities in response to equipment failure. We study structural properties and develop strong formulations for a general Network Design Problem with Connectivity Requirements (NDC) and for special cases. Using these strong formulations, we devise dual-ascent methods that provide good lower bounds and heuristic solutions with performance guarantees on their degree of suboptimality. In particular: 1. We develop a dual-ascent algorithm for a very general NDC problem. This algorithm optimally solves special cases including the k-edge-disjoint path problem and the knode-disjoint path problem. The algorithm is the most general heuristic procedure of this kind. 2. We investigate the polyhedral structure of a flow formulation for the unitary NDC problem. By projecting out the flow variables, we show that the flow-based formulation implies three new classes of valid inequalities—partition, odd-hole, and combinatorial design—for a well-known cutset formulation of the problem. 3. Using this strong formulation, we devise and empirically test a dual-ascent solution approach for the network design problem with low connectivity requirements. This algorithm requires (on average) 950 seconds of CPU time on a Sun SPARCstation 10 workstation to solve problems with up to 300 nodes and 3000 edges. The algorithm generates solutions known to be within 4 percent of optimality for typical telecommunication applications, and within 1 percent of optimality for classical Steiner tree problems. 4. We develop a new (directed) formulation for the NDC Problem. The results in this theses demonstrate the value of good problem formulations and the effectiveness of dual-ascent solution procedures that exploit a problem’s special structure. Thesis Supervisor: Thomas L. Magnanti Title: George Eastman Professor of Management Science