1. 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

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

3. 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.

4. 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.

5. Pierre Pesneau5 Cut inequalities Let. Pose Let

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

7. Pierre Pesneau7 Formulation

8. 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

9. 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.

10. 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).

11. 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.

12. 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

13. 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.

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

15. 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.

16. 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.

17. 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 :

18. 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

19. 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.

20. 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.

21. Pierre Pesneau21 Results : random instances NodeKCutCycleMetricSubsetCyclom.Cycle PGap roGap finCPU (s) 20313.027.419.811.09.07.00.150.000.28 20440.21000.4496.8118.256.828.83.460.0054.28 20552.02487.8709.61068.681.828.04.360.00276.30 20661.61838.8419.41656.062.820.64.470.00173.64 201038.8484.2124.41953.626.21.42.500.0042.86 30328.298.259.820.633.414.80.830.008.50 30497.611029.84768.2646.4457.8125.05.991.977200.00 30594.812905.22462.02893.2380.251.66.412.007200.00 306104.412738.01704.46560.8207.418.07.863.857200.00 3010121.27326.4647.823070.0103.40.84.330.965504.88 40353.4311.0175.840.0192.453.61.310.00362.92 404108.87140.62824.2453.0658.4109.05.983.127200.00 405120.88141.81369.21540.6395.241.08.215.547200.00 406118.68776.6932.63227.4254.634.09.266.737200.00 4010152.08886.8607.815942.2127.00.54.322.047200.00 50378.8701.4414.892.6631.0179.82.010.263522.34 504112.24475.41592.6253.6902.683.67.215.547200.00 505111.24982.6787.0928.4497.445.09.347.557200.00 506112.05420.2527.41726.4277.217.410.899.277200.00 5010139.47298.4259.410923.2102.61.66.234.587200.00

22. Pierre Pesneau22 Results : real instances NodeKCutCycleMetricSubsetCyclom.Cycle PGap roGap finCPU (s) 1232462110.00 0.01 1241022 12900.520.000.13 1251428 43401.770.000.29 126881738500.720.000.15 121040200600.830.000.09 173134341151160.510.000.52 17418159111401631.730.002.81 1752897421031622.210.001.47 176971126620.00 0.08 17102000200.00 0.03 303 91501849170.580.009.96 3041101137357688164431024.591.127200.00 3051611316128354185405764.720.987200.00 3061441287424157987264245.501.927200.00 30106961716416682701.410.0058.40 52387584358738371081.310.002092.77 52410736561398175867606.154.607200.00 5251303867728745607409.117.887200.00 5261284506477140030888.096.727200.00 5210110355211872739017.606.427200.00

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

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

25. 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.

26. 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

27. 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

28. 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).