Algorithmica

, Volume 63, Issue 4, pp 733–762

When LP Is the Cure for Your Matching Woes: Improved Bounds for Stochastic Matchings

  • Nikhil Bansal
  • Anupam Gupta
  • Jian Li
  • Julián Mestre
  • Viswanath Nagarajan
  • Atri Rudra
Article

Abstract

Consider a random graph model where each possible edge e is present independently with some probability pe. Given these probabilities, we want to build a large/heavy matching in the randomly generated graph. However, the only way we can find out whether an edge is present or not is to query it, and if the edge is indeed present in the graph, we are forced to add it to our matching. Further, each vertex i is allowed to be queried at most ti times. How should we adaptively query the edges to maximize the expected weight of the matching? We consider several matching problems in this general framework (some of which arise in kidney exchanges and online dating, and others arise in modeling online advertisements); we give LP-rounding based constant-factor approximation algorithms for these problems. Our main results are the following:
  • We give a 4 approximation for weighted stochastic matching on general graphs, and a 3 approximation on bipartite graphs. This answers an open question from Chen et al. (ICALP’09, LNCS, vol. 5555, pp. 266–278, [2009]).

  • We introduce a generalization of the stochastic online matching problem (Feldman et al. in FOCS’09, pp. 117–126, [2009]) that also models preference-uncertainty and timeouts of buyers, and give a constant factor approximation algorithm.

Keywords

Stochastic optimization Stochastic packing Online dating Dependent rounding 

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References

  1. 1.
    Adamczyk, M.: Greedy algorithm for stochastic matching is a 2-approximation. arXiv:1007.3036 (2010)
  2. 2.
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley-Interscience, New York (2008) MATHCrossRefGoogle Scholar
  3. 3.
    Bahmani, B., Kapralov, M.: Improved bounds for online stochastic matching. In: Proceedings of the 18th European Symposium on Algorithms. LNCS, vol. 6346, pp. 170–181. Springer, Berlin (2010) Google Scholar
  4. 4.
    Bansal, N., Gupta, A., Li, J., Mestre, J., Nagarajan, V., Rudra, A.: When LP is the cure for your matching woes: improved bounds for stochastic matchings. In: Proceedings of the 18th European Symposium on Algorithms. LNCS, vol. 6346, pp. 218–229. Springer, Berlin (2010) Google Scholar
  5. 5.
    Bansal, N., Korula, N., Nagarajan, V., Srinivasan, A.: On k-column sparse packing programs. In: Proceedings of the 14th Conference on Integer Programming and Combinatorial Optimization. LNCS, vol. 6080, pp. 369–382 (2010) Google Scholar
  6. 6.
    Bhattacharya, S., Goel, G., Gollapudi, S., Munagala, K.: Budget constrained auctions with heterogeneous items. In: Proceedings of the 41st ACM Symposium on Theory of Computing, pp. 379–388. ACM, New York (2010) Google Scholar
  7. 7.
    Birnbaum, B.E., Mathieu, C.: On-line bipartite matching made simple. SIGACT News 39(1), 80–87 (2008) CrossRefGoogle Scholar
  8. 8.
    Carr, R., Vempala, S.: Randomized metarounding. Random Struct. Algorithms 20(3), 343–352 (2002). Probabilistic methods in combinatorial optimization MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chen, N., Immorlica, N., Karlin, A.R., Mahdian, M., Rudra, A.: Approximating matches made in heaven. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming. LNCS, vol. 5555, pp. 266–278 (2009) CrossRefGoogle Scholar
  10. 10.
    Dean, B.C., Goemans, M.X., Vondrák, J.: Adaptivity and approximation for stochastic packing problems. In: Proceedings of the 16th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 395–404 (2005) Google Scholar
  11. 11.
    Dean, B.C., Goemans, M.X., Vondrák, J.: Approximating the stochastic knapsack problem: the benefit of adaptivity. Math. Oper. Res. 33(4), 945–964 (2008) MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Feldman, J., Mehta, A., Mirrokni, V.S., Muthukrishnan, S.: Online stochastic matching: beating 1−1/e. In: Proceedings of the 50th Annual Symposium on Foundations of Computer Science, pp. 117–126. IEEE, New York (2009) CrossRefGoogle Scholar
  13. 13.
    Füredi, Z., Kahn, J., Seymour, P.: On the fractional matching polytope of a hypergraph. Combinatorica 13(2), 167–180 (1993) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. J. ACM 53(3), 360 (2006) MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goel, G., Mehta, A.: Online budgeted matching in random input models with applications to adwords. In: Proceedings of the 19th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 982–991 (2008) Google Scholar
  16. 16.
    Guha, S., Munagala, K.: Approximation algorithms for partial-information based stochastic control with Markovian rewards. In: Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science, pp. 483–493 (2007) Google Scholar
  17. 17.
    Guha, S., Munagala, K.: Multi-armed bandits with metric switching costs. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming. LNCS, vol. 5555, pp. 496–507 (2009) CrossRefGoogle Scholar
  18. 18.
    Gupta, A., Krishnaswamy, R., Molinaro, M., Ravi, R.: Approximation algorithms for correlated knapsacks and non-martingale bandits. CoRR, abs/1102.3749 (2011) Google Scholar
  19. 19.
    Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: Proceedings of the 36th ACM Symposium on Theory of Computing, pp. 417–426. ACM, New York (2004) Google Scholar
  20. 20.
    Hazan, E., Safra, S., Schwartz, O.: On the complexity of approximating k-set packing. Comput. Complex. 15(1), 20–39 (2006) MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Hurkens, C., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with an application to the worst-case ratio of heuristics for packing problems. SIAM J. Discrete Math. 2(1), 68–72 (1989) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Kalyanasundaram, B., Pruhs, K.: Online weighted matching. J. Algorithms 14(3), 478–488 (1993) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for online bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pp. 352–358. ACM, New York (1990) Google Scholar
  24. 24.
    Katriel, I., Kenyon-Mathieu, C., Upfal, E.: Commitment under uncertainty: two-stage stochastic matching problems. Theor. Comput. Sci. 408(2–3), 213–223 (2008) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Mahdian, M., Nazerzadeh, H., Saberi, A.: Allocating online advertisement space with unreliable estimates. In: Proceedings of the 8th ACM Conference on Electronic Commerce, pp. 288–294 (2007) CrossRefGoogle Scholar
  26. 26.
    Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: Online actions based on offline statistics. In: Proceedings of the 22th Annual ACM–SIAM Symposium on Discrete Algorithms (2011) Google Scholar
  27. 27.
    Mehta, A., Saberi, A., Vazirani, U.V., Vazirani, V.V.: Adwords and generalized on-line matching. In: Proceedings of the 46th Annual Symposium on Foundations of Computer Science, pp. 264–273 (2005) Google Scholar
  28. 28.
    Schrijver, A.: Combinatorial Optimization. Springer, Berlin (2003) MATHGoogle Scholar
  29. 29.
    Shachnai, H., Srinivasan, A.: Finding large independent sets in graphs and hypergraphs. SIAM J. Discrete Math. 18(3), 488–500 (2005) MathSciNetCrossRefGoogle Scholar
  30. 30.
    Shmoys, D., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM 53(6), 978–1012 (2006) MathSciNetCrossRefGoogle Scholar
  31. 31.
    Swamy, C., Shmoys, D.: Approximation algorithms for 2-stage stochastic optimization problems. SIGACT News 37(1), 46 (2006) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  • Nikhil Bansal
    • 1
  • Anupam Gupta
    • 2
  • Jian Li
    • 3
  • Julián Mestre
    • 4
  • Viswanath Nagarajan
    • 1
  • Atri Rudra
    • 5
  1. 1.IBM T.J. Watson Research CenterYorktown HeightsUSA
  2. 2.Computer Science DepartmentCarnegie Mellon UniversityPittsburghUSA
  3. 3.Computer Science DepartmentUniversity of MarylandCollege ParkUSA
  4. 4.School of Information TechnologiesUniversity of SydneySydneyAustralia
  5. 5.Department of Computer Science and EngineeringUniversity at Buffalo, SUNYBuffaloUSA

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