Skip to main content
SpringerLink
Log in

Survivable networks, linear programming relaxations and the parsimonious property

  • Published:
Mathematical Programming Submit manuscript

Abstract

We consider the survivable network design problem — the problem of designing, at minimum cost, a network with edge-connectivity requirements. As special cases, this problem encompasses the Steiner tree problem, the traveling salesman problem and thek-edge-connected network design problem. We establish a property, referred to as the parsimonious property, of the linear programming (LP) relaxation of a classical formulation for the problem. The parsimonious property has numerous consequences. For example, we derive various structural properties of these LP relaxations, we present some algorithmic improvements and we perform tight worst-case analyses of two heuristics for the survivable network design problem.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Agrawal, P. Klein and R. Ravi, “When trees collide: An approximation algorithm for the generalized Steiner problem in networks,” in:Proceedings of the 23rd ACM Symposium on Theory of Computing (1991) pp. 134–144.

  2. Y.P. Aneja, “An integer linear programming approach to the Steiner problem in graphs,”Networks 10 (1980) 167–178.

    Google Scholar 

  3. A. Balakrishnan, T.L. Magnanti and R.T. Wong, “A dual-ascent procedure for large-scale uncapacitated network design,”Operations Research 37 (1989) 716–740.

    Google Scholar 

  4. E. Balas and P. Toth, “Branch and Bound methods,” in: E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds.,The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (Wiley, New York, 1985) pp. 361–401.

    Google Scholar 

  5. D. Bienstock, M.X. Goemans, D. Simchi-Levi and D.P. Williamson, “A note on the prize collecting traveling salesman problem,”Mathematical Programming 59 (1993) 413–420.

    Google Scholar 

  6. N. Christofides, “Worst-case analysis of a new heuristic for the traveling salesman problem,” Technical Report, GSIA, Carnegie-Mellon University (Pittsburgh, PA, 1976).

    Google Scholar 

  7. G. Cornuejols, M.L. Fisher and G.L. Nemhauser, “Location of bank accounts to optimize float,”Management Science 23 (1977) 789–810.

    Google Scholar 

  8. J. Edmonds, “Maximum matching and a polyhedron with 0–1 vertices,”Journal of Research of the National Bureau of Standards 69B (1965) 125–130.

    Google Scholar 

  9. J. Edmonds, “Submodular functions, matroids and certain polyhedra,” in: R. Guy et al., eds.,Combinatorial Structures and their Applications (Gordon and Breach, London, 1970), pp. 69–87.

    Google Scholar 

  10. C. El-Arbi, “Une heuristique pour le problème de l'arbre de Steiner,”RAIRO Operations Research 12 (1978) 207–212.

    Google Scholar 

  11. H. Frank and I.T. Frisch,Communication, Transmission and Transportation Networks (Addison-Wesley, Reading, MA, 1971).

    Google Scholar 

  12. M.X. Goemans and D. Bertsimas, “Probabilistic analysis of the Held and Karp lower bound for the Euclidean traveling salesman problem,”Mathematics of Operations Research 16 (1991) 72–89.

    Google Scholar 

  13. M.X. Goemans and D.P. Williamson, “A general approximation technique for constrained forest problems,” in:Proceedings of the third ACM-SIAM Symposium on Discrete Algorithms (1992) pp. 307–315.

  14. R.E. Gomory and T.C. Hu, “Synthesis of a communication network,”SIAM Journal on Applied Mathematics 9 (1961) 551–570.

    Google Scholar 

  15. M. Held and R.M. Karp, “The traveling-salesman problem and minimum spanning trees,”Operations Research 18 (1970) 1138–1162.

    Google Scholar 

  16. M. Held and R.M. Karp, “The traveling salesman problem and minimum spanning trees: Part II,”Mathematical Programming 1 (1971) 6–25.

    Google Scholar 

  17. D. Johnson, talk presented at theMathematical Programming Symposium, Tokyo (1988).

  18. D. Johnson, personal communication (1989).

  19. R.M. Karp, “Reducibility among combinatorial problems”, in: R.E. Miller and J.W. Thatcher, eds.,Complexity of Computer Computations (Plenum, New York, 1972) pp. 85–103.

    Google Scholar 

  20. R.M. Karp, “Probabilistic analysis of partitioning algorithms for the traveling salesman problem in the plane,”Mathematics of Operations Research 2 (1977) 209–224.

    Google Scholar 

  21. L. Kou, G. Markowsky and L. Berman, “A fast algorithm for Steiner trees,”Acta Informatica 15 (1981) 141–145.

    Google Scholar 

  22. J.B. Kruskal, “On the shortest spanning subtree of a graph and the traveling salesman problem,”Proceedings of the American Mathematical Society 7 (1956) 48–50.

    Google Scholar 

  23. E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan and D.B. Shmoys, eds.,The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization (Wiley, New York, 1985).

    Google Scholar 

  24. L. Lovász, “On some connectivity properties of Eulerian graphs,”Acta Mathematica Academiae Scientiarum Hungaricae 28 (1976) 129–138.

    Google Scholar 

  25. N. Maculan, “The Steiner problem in graphs,”Annals of Discrete Mathematics 31 (1987) 185–212.

    Google Scholar 

  26. K. Mehlhorn, “A faster approximation algorithm for the Steiner problem in graphs,”Information Processing Letters 27 (1988) 125–128.

    Google Scholar 

  27. C.L. Monma and C.-W. Ko, “Methods for designing survivable communication networks,” NATO Advanced Research Workshop (Denmark, 1989).

    Google Scholar 

  28. C.L. Monma, B.S. Munson and W.R. Pulleyblank, “Minimum weight two-connected spanning networks,”Mathematical Programming 46 (1990) 153–171.

    Google Scholar 

  29. C.L. Monma and D.F. Shallcross, “Methods for designing communication networks with certain two-connected survivability constraints,”Operations Research 37 (1989) 531–541.

    Google Scholar 

  30. J. Plesnik, “A bound for the Steiner tree problem in graphs,”Mathematica Slovaca 31 (1981) 155–163.

    Google Scholar 

  31. R.C. Prim, “Shortest connection networks and some generalizations,”The Bell System Technical Journal 36 (1957) 1389–1401.

    Google Scholar 

  32. S. Sahni and T. Gonzalez, “P-complete approximation problems,”Journal of the Association for Computing Machinery 23 (1976) 555–565.

    Google Scholar 

  33. D. Shmoys and D. Williamson, “Analyzing the Held-Karp TSP bound: A monotonicity property with application,”Information Processing Letters 35 (1990) 281–285.

    Google Scholar 

  34. S. Sridhar and R. Chandrasekaran, “Integer solution to synthesis of communication network,” Technical Report, The University of Texas (Dallas, TX, 1989).

    Google Scholar 

  35. K. Steiglitz, P. Weiner and D.J. Kleitman, “The design of minimum-cost survivable networks,”IEEE Transactions on Circuit Theory CT-16(4) (1969) 455–460.

    Google Scholar 

  36. H. Takahashi and A. Matsuyama, “An approximate solution for the Steiner problem in graphs,”Mathematica Japonica 24 (1980) 573–577.

    Google Scholar 

  37. P. Winter, “Steiner problem in networks: a survey,”Networks 17 (1987) 129–167.

    Google Scholar 

  38. L.A. Wolsey, “Heuristic analysis, linear programming and Branch and Bound,”Mathematical Programming Study 13 (1980) 121–134.

    Google Scholar 

  39. R.T. Wong, “A dual ascent approach for Steiner tree problems on a directed graph,”Mathematical Programming 28 (1984) 271–287.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Massachusetts Institute of Technology, 02139, Cambridge, MA, USA

    Michel X. Goemans

  2. Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA, USA

    Dimitris J. Bertsimas

Authors
  1. Michel X. Goemans
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dimitris J. Bertsimas
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

The research of both authors was partially supported by the National Science Foundation under grant ECS-8717970 and the Leaders for manufacturing program at MIT.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Goemans, M.X., Bertsimas, D.J. Survivable networks, linear programming relaxations and the parsimonious property. Mathematical Programming 60, 145–166 (1993). https://doi.org/10.1007/BF01580607

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01580607

Key words

Search

Navigation