Survivable network design using polyhedral approaches

We consider the problem of designing a survivable telecommunication network using facilities of a fixed capacity. Given a graph G = (V, E), the traffic demand among the nodes, and the cost of installing facilities on the edges of G, we wish to design the minimum cost network, so that under any single edge failure, the network permits the flow of all traffic using the remaining capacity. The problem is modeled as a mixed integer program, which can be converted into a pure integer program by applying the well-known Japanese Theorem on multi-commodity flows. Using a key theorem that characterizes the facet inequalities of this integer program, we derive several families of 3- and 4-partition facets, which help to achieve extremely tight lower bounds on the problem. Using these bounds, problems of up to 20 nodes and 40 edges have been solved optimally in a pervious work. Using heuristic approaches based on this framework, we solve problems of up to 40 nodes and 80 edges to obtain solutions that are approximately within 5% of optimal solutions.