2-edge connected subgraphs with bounded rings B. Fortz Institut de Gestion et d’Administration, Louvain la Neuve, Belgique A. R. Mahjoub LIMOS, CNRS, Université Blaise Pascal, Clermont-Ferrand, France S. T. McCormick Faculty of Commerce, Vancouver, Canada P. Pesneau LIMOS, CNRS, Université Blaise Pascal, Clermont-Ferrand, France

Pierre Pesneau2 Outline Presentation of the problem Polyhedral study Branch&Cut algorithm Computational results

Pierre Pesneau3 Network design Designing the network topology Network survivability 2-edge connectivity Network performance in case of failure bounded ring constraints Problem : Find a 2-edge connected subgraph at minimum cost such that each edge belong to a cycle of length bounded by an integer K.

Pierre Pesneau4 2-node connectivity (Fortz, Labbé, Maffioli 2000) Formulation in terms of edges and cycles Valid inequalities and necessary and sufficient conditions to be facet defining Separation algorithms Branch&Cut algorithm Heuristics.

Pierre Pesneau5 Cut inequalities Let. Pose Let

Pierre Pesneau6 Cycle inequalities T Let be a partition such that, let, let Every solution must verify : Cycle configuration

Pierre Pesneau7 Formulation

Pierre Pesneau8 Cycle inequalities : facets Let G=(V,E) be a complete graph. Let be a partition of V and The associated cycle inequality is facet defining if and only if : –, – and – for all

Pierre Pesneau9 Cycle inequalities : separation Let x be a solution. The separation problem of cycle inequalities for an edge e=st The bounded (s,t)-path cut problem with B=K-1.

Pierre Pesneau10 Bounded path cut problem Let G=(V,E) be a graph, s and t two nodes and B an integer. Bounded (s,t)-path cut : set of edges that cut all (s,t)-path of length Problem : Find a minimum cost bounded (s,t)-path cut (BPCP).

Pierre Pesneau11 BPCP If B=2 the problem is trivial. If B=3 : –The problem is polynomial. –It can be reduced to find a minimum cut in a particular directed graph.

Pierre Pesneau12 BPCP If : Heuristic based on the Primal-Dual method : While C is not a bounded (s,t)-path cut do Find an (s,t)-path P of bounded length Increase until an edge verifies Improve C by removing useless edges of C

Pierre Pesneau13 Cycle inequalities : separation Let G=(V,E) be a graph and e=st be an edge. Calculate a bounded (s,t)-path cut C with B=K-1. If x(C) < x(e), the we get a violated cycle inequality and the associated partition is obtained by a breadth-first search from s in the graph G\C. We strengthen the partition by reducing to a single node.

Pierre Pesneau14 Cyclomatic inequalities Introduced by Fortz, Labbé (1999). Let be a partition of V. Every solution must verify :

Pierre Pesneau15 Cyclomatic inequalities : separation 1st heuristic : Based on the separation of partition inequalities (Cunningham) : Consider : We apply Barahona’s algorithm for the separation of the partition inequalities.

Pierre Pesneau16 Cyclomatic inequalities : separation 2nd heuristic : Let G=(V,E) be a graph and x a solution While and |V|>2 do Find an edge e with the greatest value in x Contract edge e in the graph G If |V|>2 then we have a violated cyclomatic inequality and each element of the associated partition is given by the expansion of the nodes of the graph.

Pierre Pesneau17 Cycle partition inequalities Let be a partition of V such that Let T be the chords of the partition and C be the other edges of the partition. Every solution must verify :

Pierre Pesneau18 Cycle partition inequalities : facets Let G=(V,E) be a complete graph. Let be a partition of V. The cycle partition inequality is facet defining if and only if : – –there is at most one such that – for

Pierre Pesneau19 Cycle partition inequalities : separation Same idea than the separation of cyclomatic inequalities : Let G=(V,E) be a graph and x be a solution While |V|>K+1 do Contract edge e with the greatest value in x Search the order of the nodes of the final graph such that is minimum. If this value is <2K then, the expansion of the nodes of the final graph give a paretition inducing violated cycle partition inequality.

Pierre Pesneau20 Computational results Branch&Cut algorithm. Tree manager : BCP (IBM). Linear solver : CPLEX 7.1. PC PIV 1,7 GHz, 1 Go RAM. Random and real data. Complete graphs. Time limit : 2 hours.

Pierre Pesneau21 Results : random instances NodeKCutCycleMetricSubsetCyclom.Cycle PGap roGap finCPU (s)

Pierre Pesneau22 Results : real instances NodeKCutCycleMetricSubsetCyclom.Cycle PGap roGap finCPU (s)

Pierre Pesneau23 Example of a solution 52 nodes 35 minutes 2047 constraints

Pierre Pesneau24 Perspectives Solve bigger instances (particularly for K=3,4 and 5). Improve separation routines. Find new classes of valid inequalities.

Pierre Pesneau25 Subset inequalities Let T be an edge set such that G\T does not contain a feasible solution. We have : Separation : when we separate cycle inequalities, if two consecutive elements of the partition are reduced to a single node, then T induced a violated subset inequality.

Pierre Pesneau26 Cut inequalities : facets Let G=(V,E) be a complete graph. Let The cut inequality is facet defining if and only if : –either and –or and

Pierre Pesneau27 Cyclomatic inequalities : facets Let G=(V,E) be a complete graph. Let be a partition of V. The cyclomatic inequality is facet defining if and only if and : –either and or and for –or for

Pierre Pesneau28 Metric inequalities Introduced by Fortz, Labbé, Maffioli (2000). Let G=(V,E) be a graph and Let be a set of node potential satisfying : Every solution must verify : where Separation : heuristic of Fortz et al. (2000).