Designing capacitated survivable networks: polyhedral analysis and algorithms

• Authors: Deepak Rajan;Alper Atamturk
• Venue: Designing capacitated survivable networks: polyhedral analysis and algorithms
• Year: 2004

In this dissertation, we develop new methodologies and efficient algorithms to solve the capacitated survivable network design problem. Given the graph and demands between pairs of nodes, we wish to install integer multiples of capacity on the edges and route the demand while minimizing costs. A network is said to be survivable if all demands can be rerouted under the failure of any one of its edges. Traditionally, one uses some variant of the following two approaches: protection or restoration. Protection schemes can be further classified into dedicated protection and shared protection. We propose a method that uses failure-flow patterns for rerouting of disrupted flow. A hybrid between dedicated and shared protection schemes, our method imposes no restrictions on the network, but explicitly introduces slack on the directed cycles used as failure-flow patterns for rerouting disrupted flow. Using failure-flow patterns results in a much smaller formulation than other approaches in terms of number of constraints; we handle the exponential number of directed cycle variables using column generation. We study the arc-set and cut-set polyhedra associated with the problem to generate strong valid inequalities. We develop various families of inequalities, and describe efficient separation algorithms for these and other classes of inequalities. By pricing out directed cycle variables and separating the valid inequalities in a branch-and-cut framework, we show that it is possible to solve much larger problem instances using directed cycles (than shared protection), without significant loss in capacity efficiency. Furthermore, the cuts added improve overall performance by an order of magnitude. The following two directions of research show significant promise in solving capacitated survivable network design problems more effectively. The first considers directed p-cycles as failure-flow patterns for obtaining higher capacity efficiency. Preliminary results are very encouraging; however, pricing sub-problems and polyhedral structure change significantly. The second, approached from two complementary directions, involves the development of stronger classes of inequalities. In the former direction, we develop problem-specific metric-type inequalities for design of survivable networks using various failure-flow patterns. In the latter, we develop problem-independent strong valid inequalities for the mixed-integer knapsack set, a relaxation of any mixed-integer program.