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Critical extreme points of the 2-edge connected spanning subgraph polytope

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Abstract

In this paper we study the extreme points of the polytope P(G), the linear relaxation of the 2-edge connected spanning subgraph polytope of a graph G. We introduce a partial ordering on the extreme points of P(G) and give necessary conditions for a non-integer extreme point of P(G) to be minimal with respect to that ordering. We show that, if is a non-integer minimal extreme point of P(G), then G and can be reduced, by means of some reduction operations, to a graph G' and an extreme point of P(G') where G' and satisfy some simple properties. As a consequence we obtain a characterization of the perfectly 2-edge connected graphs, the graphs for which the polytope P(G) is integral.

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References

  1. Baïou, M., Mahjoub, A.R.: Steiner 2-edge connected subgraph polytopes on series parallel graphs. SIAM Journal on Discrete Mathematics 10, 505–514 (1997)

    MATH  MathSciNet  Google Scholar 

  2. Barahona, F., Mahjoub, A.R.: On two-connected subgraphs polytopes. Discrete Mathematics 147, 19–34 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bondy, J.A., Murty, U.S.R.: Graph theory with applications. Universities Press, Belfast, 1976

  4. Boyd, S.C., Hao, T.: An integer polytope related to the design of survivable communication networks. SIAM Journal on Discrete Mathematics 4, 612–630 (1993)

    MATH  MathSciNet  Google Scholar 

  5. Chopra, S.: Polyhedra of the equivalent subgraph problem and some edge connectivity problems. SIAM Journal on Discrete Mathematics 5, 321–337 (1992)

    MATH  MathSciNet  Google Scholar 

  6. Chopra, : The k-edge connected spanning subgraph polyhedron. SIAM Journal on Discrete Mathematics 7, 245–259 (1994)

    MATH  MathSciNet  Google Scholar 

  7. Chopra, S., Stoer, M.: Private Communication

  8. Christofides, N., Whitlock, C.A.: Network synthesis with connectivity constraints: a survey. In: J.P. Brans, (ed) Operational Research 81, North Holland 1981, pp 705–723

  9. Cornuéjols, G., Fonlupt, J., Naddef, D.: The traveling salesman problem on a graph and some related integer polyhedra. Mathematical programming 33, 1–27 (1985)

    MATH  MathSciNet  Google Scholar 

  10. Coullard, R., Rais, A., Rardin, R.L., Wagner, D.K.: Linear time algorithm for the 2-connected Steiner subgraph problem on special classes of gaphs. Networks 23, 195–206 (1993)

    MATH  MathSciNet  Google Scholar 

  11. Coullard, R., Rais, A., Rardin, R.L., Wagner, D.K.: The dominant of the 2-connected Steiner subgraph polytope for W4-free graphs. Discrete Applied Mathematics 66, 33–43 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Didi Biha, M., Mahjoub, A.R.: k-edge connected polyhedra and series-parallel graphs. Operations Research Letters 19, 71–78 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  13. Didi Biha, M., Kerivin, H., Mahjoub, A.R.: On the polytope of the (1,2)-survivable network design problem. Preprint, 2005

  14. Dinits, E.A.: Algorithm for solution of a problem of maximum flow in a network unit with power estimation. Soviet Mathematic's Doklady 11, 1277–1280 (1970)

    MATH  Google Scholar 

  15. Edmonds, J., Karp, R.M.: Theoretical improvement in algorithm efficiency for network flow problems. Journal of Association for Computing Machinery 19, 254–264 (1972)

    Google Scholar 

  16. Erikson, R.E., Monma, C.L., Veinott, A.F. Jr.: Send and split method for minimum-concave network flows. Mathematics of Operations Research 12, 634–664 (1987)

    MathSciNet  Google Scholar 

  17. Fonlupt, J., Mahjoub, A.R.: Critical extreme points of the 2-edge connected spanning subgraph polytope. Technical Report RR-04-17 LIMOS, Université Blaise Pascal, Clermont-Ferrand, France, 2004

  18. Fonlupt, J., Naddef, D.: The traveling salesman problem in graphs with some excluded minors. Mathematical Programming 53, 147–172 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  19. Fortz, B., Labbé, M., Maffioli, F.: Solving the two-connected network with bounded meshes Problem. Operations Research 48, 866–877 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fortz, B.: Design of Survivable Networks with Bounded Rings. Kluwer Academic Publishers, Dordrecht 2000

  21. Fortz, B., Labbé, M.: Polyhedral results for two-connected networks with bounded rings. Mathematical Programming 93, 27–54 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Fortz, B., Mahjoub, A.R., McCormick, S.T., Pesneau, P.: The 2-edge connected subgraph problem with bounded rings. Mathematical Programming, to appear

  23. Grötschel, M., Lovász, L., Schrijver, A.: The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 70–89 (1981)

    Google Scholar 

  24. Grötschel, M., Monma, C.L.: Integer polyhedra arising from certain network design problem with connectivity constraints. SIAM Journal on Discrete Mathematics 3, 502–523 (1990)

    MATH  MathSciNet  Google Scholar 

  25. Grötschel, M., Monma, C.L., Stoer, M.: Facets for polyhedra arising in the design of commumication networks with low-connectivity constraints. SIAM Journal on Optimization 2, 474–504 (1992)

    MATH  MathSciNet  Google Scholar 

  26. Grötschel, M., Monma, C.L., Stoer, M.: Design of survivable networks. In: M.O. Ball, et al. (eds) Handbook in Operations Research and Management Science 7, North-Holland, 1995, pp 617-671

  27. Grötschel, M., Monma, C.L., Stoer, M.: Computational results with a cutting plane algorithm for designing communication networks with low-connectivity constraints. Operations Research 40/2, 309–330 (1992)

    Google Scholar 

  28. Grötschel, M., Monma, C.L., Stoer, M.,: Polyhedral and computational investigation for designing communication networks with high survivability requirements. Operations Research 43/6, 1012–1024 (1995)

    Google Scholar 

  29. Huygens, D., Mahjoub, A.R., Pesneau, P.: Two edge-disjoint hop-constrained paths and polyhedra. SIAM Journal on Discrete Mathematics 18, 287–312 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  30. Kerivin, H., Mahjoub, A.R.: Design of survivable networks: A survey. Networks, 46 (1), 1–26 (2005)

    MathSciNet  Google Scholar 

  31. Mahjoub, A.R.: Two-edge connected spanning subgraphs and polyhedra. Mathematical Programming 64, 199–208 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Mahjoub, A.R.: On perfectly 2-edge connected graphs. Discrete Mathematics 170, 153–172 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  33. Pulleyblank, W.R.: Polyhedral combinatorics. In: G.L. Nemhauser, et al. (eds) Optimization Handbooks in OR-MS 1, pp 371–446, 1989

  34. Stoer, M.: Design of survivable networks. Lecture Notes in Mathematics 1531, Springer, Berlin, 1992

  35. Winter, P.: Steiner problem in Halin networks. Discrete Applied Mathematics, 281–294 (1987)

  36. Winter, P: Generalized Steiner problem in series-parallel networks. Journal of Algorithms 7, 549–566 (1986)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Équipe Combinatoire, UFR 921, Université Pierre et Marie Curie, 4 place Jussieu, 75252, Paris Cedex 05, France

    Jean Fonlupt

  2. LIMOS, CNRS UMR 6158, Université Blaise Pascal Clermont II, Complexe Scientifique des Cézeaux, 63177, Aubière Cedex, France

    A. Ridha Mahjoub

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  1. Jean Fonlupt
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  2. A. Ridha Mahjoub
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Correspondence to A. Ridha Mahjoub.

Additional information

Received: May, 2004

Part of this work has been done while the first author was visiting the Research Institute for Discrete Mathematics, University of Bonn, Germany.

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Fonlupt, J., Mahjoub, A. Critical extreme points of the 2-edge connected spanning subgraph polytope. Math. Program. 105, 289–310 (2006). https://doi.org/10.1007/s10107-005-0654-8

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