Skip to main content

On Two-Stage Stochastic Minimum Spanning Trees

  • Conference paper
Integer Programming and Combinatorial Optimization (IPCO 2005)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 3509))

  • 1253 Accesses

Abstract

We consider the undirected minimum spanning tree problem in a stochastic optimization setting. For the two-stage stochastic optimization formulation with finite scenarios, a simple iterative randomized rounding method on a natural LP formulation of the problem yields a nearly best-possible approximation algorithm.

We then consider the Stochastic minimum spanning tree problem in a more general black-box model and show that even under the assumptions of bounded inflation the problem remains log n-hard to approximate unless P = NP; where n is the size of graph. We also give approximation algorithm matching the lower bound up to a constant factor.

Finally, we consider a slightly different cost model where the second stage costs are independent random variables uniformly distributed between [0,1]. We show that a simple thresholding heuristic has cost bounded by the optimal cost plus \(\frac{\zeta(3)}{4}+o(1)\).

Supported in part by NSF grant CCR-0105548 and ITR grant CCR-0122581 (The ALADDIN project).

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

Access this chapter

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Similar content being viewed by others

References

  1. Alon, N.: A note on network reliability. In: Aldous, D., et al. (eds.) Discrete Probability and Algorithms. IMA Volumes in mathematics and its applications, vol. 72, pp. 11–14. Springer, Heidelberg (1995)

    Google Scholar 

  2. Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, Heidelberg (1997)

    MATH  Google Scholar 

  3. Flaxman., A.D., Frieze, A.M., Krivelevich, M.: On the random 2-stage minimum spanning tree (2004) (Preprint)

    Google Scholar 

  4. Frieze, A.M.: On the value of a random minimum spanning tree problem. Discrete Applied Mathematics 10, 47–56 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anupam, G.: Personal Communication (2004)

    Google Scholar 

  6. Gupta, A., Ravi, R., Sinha, A.: Boosted Sampling: Approximation Algorithms for Stochastic Optimization. In: Proceedings of the thirty-sixth annual ACM symposium on Theory of computing (2004)

    Google Scholar 

  7. Gupta, A., Ravi, R., Sinha, A.: An edge in time saves nine: LP rounding Approximation Algorithms for Stochastic Network Design. In: The Proceedings of 45th Annual IEEE Symposium on Foundations of Computer Science (2004)

    Google Scholar 

  8. Feige, U.: A threshold of ln n for approximating set cover. J. ACM 45, 634–652 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  9. Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.S.: On the Costs and Benefits of Procastination: Approximation Algorithms for Stochastic Combinatorial Optimization Problems. In: Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms, pp. 691–700 (2004)

    Google Scholar 

  10. Karger, D., Klein, P., Tarjan, R.: A randomized linear-time algorithm to find minimum spanning trees. Journal of the ACM 42(2), 321–328 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  11. Magnanti, T.L., Wolsey, L.A.: Optimal Trees. In: Ball, M.O., et al. (eds.) Handbook in OR and MS, vol. 7, pp. 503–615 (1995)

    Google Scholar 

  12. Ravi, R., Sinha, A.: Hedging Uncertainty: Approximation Algorithms for Stochastic Optimization Problems. In: Proceedings of the 10th International Conference on Integer Programming and Combinatorial Optimization, IPCO (2004)

    Google Scholar 

  13. Raz, R., Safra, S.: A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In: Proceedings of the twenty-ninth annual ACM symposium on Theory of computing, pp. 475–484 (1999)

    Google Scholar 

  14. Shapiro, A.: Monte Carlo sampling approach to stochastic programming. In: Proceedings of 2003 MODE-SMAI Conference Pau, France, March 27-29, pp. 65–73 (2003)

    Google Scholar 

  15. Shmoys, D.B., Swamy, C.: Stochastic Optimization is (Almost) as easy as Deterministic Optimization. In: Proceedings of 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), Rome, Italy, October 17 - 19 (2004)

    Google Scholar 

  16. van der Vlerk, M.H.: Stochastic programming bibliography. World Wide Web (1996-2003), http://mally.eco.rug.nl/spbib.html

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Dhamdhere, K., Ravi, R., Singh, M. (2005). On Two-Stage Stochastic Minimum Spanning Trees. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_24

Download citation

  • DOI: https://doi.org/10.1007/11496915_24

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-26199-5

  • Online ISBN: 978-3-540-32102-6

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics