Commitment Under Uncertainty: Two-Stage Stochastic Matching Problems

  • Irit Katriel
  • Claire Kenyon-Mathieu
  • Eli Upfal
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)


We define and study two versions of the bipartite matching problem in the framework of two-stage stochastic optimization with recourse. In one version the uncertainty is in the second stage costs of the edges, in the other version the uncertainty is in the set of vertices that needs to be matched. We prove lower bounds, and analyze efficient strategies for both cases. These problems model real-life stochastic integral planning problems such as commodity trading, reservation systems and scheduling under uncertainty.


Minimum Span Tree Steiner Tree Robust Optimization Stochastic Optimization Maximum Match 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Birge, J., Louveaux, F.: Introduction to Stochastic Programming. Springer, Heidelberg (1997)zbMATHGoogle Scholar
  2. 2.
    Charikar, M., Chekuri, C., Pal, M.: Sampling bounds fpr stochastic optimization. In: APPROX-RANDOM, pp. 257–269 (2005)Google Scholar
  3. 3.
    Chlebík, M., Chlebíková, J.: Inapproximability results for bounded variants of optimization problems. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 27–38. Springer, Heidelberg (2003)Google Scholar
  4. 4.
    Dhamdhere, K., Goyal, V., Ravi, R., Singh, M.: How to pay, come what may: Approximation algorithms for demand-robust covering problems. In: FOCS, pp. 367–378 (2005)Google Scholar
  5. 5.
    Dhamdhere, K., Ravi, R., Singh, M.: On two-stage stochastic minimum spanning trees. In: Jünger, M., Kaibel, V. (eds.) Integer Programming and Combinatorial Optimization. LNCS, vol. 3509, pp. 321–334. Springer, Heidelberg (2005)Google Scholar
  6. 6.
    Dye, S., Stougie, L., Tomasgard, A.: The stochastic single resource service-provision problem. Naval Research Logistics 50, 257–269 (2003)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Elbassioni, K.M., Katriel, I., Kutz, M., Mahajan, M.: Simultaneous matchings. In: Deng, X., Du, D.-Z. (eds.) ISAAC 2005. LNCS, vol. 3827, pp. 106–115. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  8. 8.
    Flaxman, A.D., Frieze, A.M., Krivelevich, M.: On the random 2-stage minimum spanning tree. In: SODA, pp. 919–926 (2005)Google Scholar
  9. 9.
    Gupta, A., Pál, M.: Stochastic steiner trees without a root. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds.) ICALP 2005. LNCS, vol. 3580, pp. 1051–1063. Springer, Heidelberg (2005)Google Scholar
  10. 10.
    Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: STOC, pp. 417–426. ACM, New York (2004)Google Scholar
  11. 11.
    Gupta, A., Ravi, R., Sinha, A.: An edge in time saves nine: LP rounding approx. algorithms for stochastic network design. In: FOCS, pp. 218–227 (2004)Google Scholar
  12. 12.
    Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procratination: approximation algorithms for stochastic combinatorial optimization problems. In: SODA, pp. 691–700 (2004)Google Scholar
  13. 13.
    Katriel, I., Kenyon-Mathieu, C., Upfal, E.: Commitment under uncertainty: Two-stage stochastic matching problems. In: ECCC (2007),
  14. 14.
    Kenyon, C., Rémila, E.: A near-optimal solution to a two-dimensional cutting stock problem. Math. Oper. Res. 25(4), 645–656 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Kong, N., Schaefer, A.J.: A factor 1/2 approximation algorithm for two-stage stochastic matching problems. Eur. J. of Operational Research 172, 740–746 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. of Computing and System Sciences 43, 425–440 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Ravi, R., Sinha, A.: Hedging uncertainty: Approximation algorithms for stochastic optimization problems. In: Bienstock, D., Nemhauser, G.L. (eds.) Integer Programming and Combinatorial Optimization. LNCS, vol. 3064, pp. 101–115. Springer, Heidelberg (2004)Google Scholar
  18. 18.
    Raz, R., Safra, S.: A sub-constant error-prob. low-degree test, and a sub-constant error-prob. PCP characterization of NP. In: STOC, pp. 475–484 (1997)Google Scholar
  19. 19.
    Shmoys, D.B., Sozio, M.: Approximation algorithms for 2-stage stochastic scheduling problems. In: IPCO (2007)Google Scholar
  20. 20.
    Shmoys, D.B., Swamy, C.: The sample average approximation method for 2-stage stochastic optimization (2004)Google Scholar
  21. 21.
    Shmoys, D.B., Swamy, C.: Stochastic optimization is almost as easy as deterministic optimization. In: FOCS, pp. 228–237 (2004)Google Scholar
  22. 22.
    Swamy, C., Shmoys, D.B.: The sampling-based approximation algorithms for multi-stage stochastic optimization. In: FOCS, pp. 357–366 (2005)Google Scholar
  23. 23.
    Swamy, C., Shmoys, D.B.: Algorithms column: Approximation algorithms for 2-stage stochastic optimization problems. ACM SIGACT News 37(1), 33–46 (2006)CrossRefGoogle Scholar
  24. 24.
    Verweij, B., Ahmed, S., Kleywegt, A.J., Nemhauser, G., Shapiro, A.: The sample average approximation method applied to stochastic routing problems: a computational study. Comp. Optimization and Applications 24, 289–333 (2003)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Irit Katriel
    • 1
  • Claire Kenyon-Mathieu
    • 1
  • Eli Upfal
    • 1
  1. 1.Brown University 

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