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Mathematical Programming

, Volume 105, Issue 2–3, pp 181–200 | Cite as

Network reinforcement

  • Francisco Barahona
Article

Abstract

We give an algorithm for the following problem: given a graph G=(V,E) with edge-weights and a nonnegative integer k, find a minimum cost set of edges that contains k disjoint spanning trees. This also solves the following reinforcement problem: given a network, a number k and a set of candidate edges, each of them with an associated cost, find a minimum cost set of candidate edges to be added to the network so it contains k disjoint spanning trees. The number k is seen as a measure of the invulnerability of a network. Our algorithm has the same asymptotic complexity as |V| applications of the minimum cut algorithm of Goldberg & Tarjan.

Keywords

Network reinforcement Spanning trees 

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References

  1. 1.
    Ahuja, R.K., Orlin, J.B., Stein, C., Tarjan, R.E.: Improved algorithms for bipartite network flow. SIAM J. Comput. 23 (5), 906–933 (1994)MathSciNetGoogle Scholar
  2. 2.
    Anglès d'Auriac, J.C., Iglói, F., Preissmann, M., Sebő, A.: Optimal cooperation and submodularity for computing Potts' partition functions with a large number of states. J. Phys. A 35 (33), 6973–6983 (2002)Google Scholar
  3. 3.
    Baïou, M., Barahona, F., Mahjoub, A.R.: Separation of partition inequalities. Math. Oper. Res. 25 (2), 243–254 (2000)MathSciNetGoogle Scholar
  4. 4.
    Barahona, F.: Separating from the dominant of the spanning tree polytope. Op. Res. Lett. 12, 201–203 (1992)MATHMathSciNetGoogle Scholar
  5. 5.
    Cheng, E., Cunningham, W.H.: A faster algorithm for computing the strength of a network. Inf. Process. Lett. 49, 209–212 (1994)MATHGoogle Scholar
  6. 6.
    Cunningham, W.H.: Minimum cuts, modular functions, and matroid polyhedra. Networks 15, 205–215 (1985)MATHMathSciNetGoogle Scholar
  7. 7.
    Cunningham, W.H.: Optimal attack and reinforcement of a network. J. ACM 32, 549–561 (1985)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: R.K. Guy, E. Milner, N. Sauer (eds) Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 69–87Google Scholar
  9. 9.
    Gabow, H.N.: Algorithms for graphic polymatroids and parametric Open image in new window-sets. J. Algorithms 26 (1), 48–86 (1998)MathSciNetGoogle Scholar
  10. 10.
    Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18 (1), 30–55 (1989)MathSciNetGoogle Scholar
  11. 11.
    Goldberg, A.V., Tarjan, R.E.: A new approach to the maximum-flow problem. J. Assoc. Comput. Mach. 35(4), 921–940 (1988)Google Scholar
  12. 12.
    Grötschel, M., Lovász, L., Schrijver, A.: Geometric algorithms and combinatorial optimization, Algorithms and Combinatorics, vol. 2, second edn. Springer, Berlin, 1993Google Scholar
  13. 13.
    Gusfield, D.: Connectivity and edge-disjoint spanning trees. Inform. Process. Lett. 16 (2), 87–89 (1983)MathSciNetGoogle Scholar
  14. 14.
    Gusfield, D.: Computing the strength of a graph. SIAM J. Comput. 20 (4), 639–654 (1991)MathSciNetGoogle Scholar
  15. 15.
    Hao, J., Orlin, J.B.: A faster algorithm for finding the minimum cut in a directed graph. J. Algorithms 17 (3), 424–446 (1994); Third Annual ACM-SIAM Symposium on Discrete Algorithms (Orlando, FL, 1992)MathSciNetGoogle Scholar
  16. 16.
    Jünger, M., Pulleyblank, W.R.: New primal and dual matching heuristics. Algorithmica 13, 357–380 (1995)MATHMathSciNetGoogle Scholar
  17. 17.
    Nash-Williams, C.S.J.A.: Edge-disjoint spanning trees of finite graphs. J. London Math. Soc. 36, 445–450 (1961)MATHMathSciNetGoogle Scholar
  18. 18.
    Padberg, M.W., Wolsey, A.: Trees and cuts. Ann. Discrete Math. 17, 511–517 (1983)MATHMathSciNetGoogle Scholar
  19. 19.
    Picard, J.C., Queyranne, M.: Selected applications of minimum cuts in networks. INFOR-Canada J. Oper. Res. Inform. Process. 20, 394–422 (1982)MATHGoogle Scholar
  20. 20.
    Roskind, J., Tarjan, R.E.: A note on finding minimum-cost edge-disjoint spanning trees. Math. Oper. Res. 10 (4), 701–708 (1985)MathSciNetGoogle Scholar
  21. 21.
    Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. London Math. Soc. 36, 221–230 (1961)MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  1. 1.IBM T. J. Watson research CenterYorktown HeightsUSA

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