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Algorithmica

, Volume 80, Issue 11, pp 3225–3252 | Cite as

Improved Bounds in Stochastic Matching and Optimization

  • Alok Baveja
  • Amit Chavan
  • Andrei Nikiforov
  • Aravind Srinivasan
  • Pan Xu
Article
  • 80 Downloads

Abstract

Real-world problems often have parameters that are uncertain during the optimization phase; stochastic optimization or stochastic programming is a key approach introduced by Beale and by Dantzig in the 1950s to address such uncertainty. Matching is a classical problem in combinatorial optimization. Modern stochastic versions of this problem model problems in kidney exchange, for instance. We improve upon the current-best approximation bound of 3.709 for stochastic matching due to Adamczyk et al. (in: Algorithms-ESA 2015, Springer, Berlin, 2015) to 3.224; we also present improvements on Bansal et al. (Algorithmica 63(4):733–762, 2012) for hypergraph matching and for relaxed versions of the problem. These results are obtained by improved analyses and/or algorithms for rounding linear-programming relaxations of these problems.

Keywords

Stochastic optimization Linear programming Approximation algorithms Randomized algorithms 

Notes

Acknowledgements

A preliminary version of this paper appears as part of a paper in the Proc. International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2015. We thank R. Ravi and the referees for their valuable comments regarding the details as well as context of this work. The research of the first author is partially supported in part by grants from British Council’s UKIERI program and the United States Department of Transportation’s UTRC Region II consortium (RF # 49198-25-26). The research of the fourth author is partially supported in part by NSF Awards CNS-1010789, CCF-1422569 and CCF-1749864, and by research awards from Adobe, Inc. The research of the last author is supported in part by NSF Awards CNS 1010789 and CCF 1422569.

References

  1. 1.
    Abolhassani, M., Esfandiari, H., Hajiaghayi, M., Mahini, H., Malec, D., Srinivasan, A.: Selling tomorrow’s bargains today. In: Proceedings of the 2015 International Conference on Autonomous Agents and Multiagent Systems, International Foundation for Autonomous Agents and Multiagent Systems, pp. 337–345 (2015)Google Scholar
  2. 2.
    Adamczyk, M., Grandoni, F., Mukherjee, J.: Improved approximation algorithms for stochastic matching. In: Bansal, N., Finocchi, I. (eds.) Algorithms—ESA 2015. Lecture Notes in Computer Science, vol. 9294. Springer, Berlin, Heidelberg (2015)Google Scholar
  3. 3.
    Alon, N., Spencer, J.H.: Wiley interscience series in discrete mathematics and optimization. In: The Probabilistic Method, vol. 6, pp. 85–96 (2008)Google Scholar
  4. 4.
    Bansal, N., Gupta, A., Nagarajan, V., Rudra, A.: When lp is the cure for your matching woes: approximating stochastic matchings. arXiv preprint arXiv:1003.0167v1 [cs DS] (2010)
  5. 5.
    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. Algorithmica 63(4), 733–762 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Beale, E.M.: On minimizing a convex function subject to linear inequalities. J. Roy. Stat. Soc. B Met. 17, 173–184 (1955)MathSciNetMATHGoogle Scholar
  7. 7.
    Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)MATHGoogle Scholar
  8. 8.
    Chen, N., Immorlica, N., Karlin, A.R., Mahdian, M., Rudra, A.: Approximating matches made in heaven. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) Automata, Languages and Programming. ICALP 2009. Lecture Notes in Computer Science, vol. 5555. Springer, Berlin, Heidelberg (2009)Google Scholar
  9. 9.
    Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci. 1(3–4), 197–206 (1955)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Fortuin, C.M., Ginibre, J., Kasteleyn, P.N.: Correlational inequalities for partially ordered sets. Commun. Math. Phys. 22, 89–103 (1971)CrossRefMATHGoogle Scholar
  12. 12.
    Füredi, Z., Kahn, J., Seymour, P.D.: On the fractional matching polytope of a hypergraph. Combinatorica 13(2), 167–180 (1993)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gandhi, R., Khuller, S., Parthasarathy, S., Srinivasan, A.: Dependent rounding and its applications to approximation algorithms. J. ACM (JACM) 53(3), 324–360 (2006)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Garg, N., Gupta, A., Leonardi, S., Sankowski, P.: Stochastic analyses for online combinatorial optimization problems. In: Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, pp. 942–951 (2008)Google Scholar
  15. 15.
    Gupta, A., Kumar, A.: A constant-factor approximation for stochastic steiner forest. In: Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing, pp 659–668. ACM (2009)Google Scholar
  16. 16.
    Gupta, A., Ravi, R., Sinha, A.: LP rounding approximation algorithms for stochastic network design. Math. Oper. Res. 32(2), 345–364 (2007)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Immorlica, N., Karger, D., Minkoff, M., Mirrokni, V.S.: On the costs and benefits of procrastination: approximation algorithms for stochastic combinatorial optimization problems. In: Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, pp 691–700 (2004)Google Scholar
  18. 18.
    Levi, R., Pál, M., Roundy, R.O., Shmoys, D.B.: Approximation algorithms for stochastic inventory control models. Math. Oper. Res. 32(2), 284–302 (2007)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ravi, R., Sinha, A.: Hedging uncertainty: approximation algorithms for stochastic optimization problems. Math. Program. 108(1), 97–114 (2006)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Ruszczynski, A.P., Shapiro, A.: Stochastic Programming, vol. 10. Elsevier, Amsterdam (2003)MATHGoogle Scholar
  21. 21.
    Shachnai, H., Srinivasan, A.: Finding large independent sets in graphs and hypergraphs. SIAM J. Discret. Math. 18, 488–500 (2004)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming—Modeling and Theory, vol. 16, 2nd edn. SIAM, Philadelphia (2014)MATHGoogle Scholar
  23. 23.
    Shmoys, D.B., Swamy, C.: An approximation scheme for stochastic linear programming and its application to stochastic integer programs. J. ACM (JACM) 53(6), 978–1012 (2006)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Srinivasan, A.: Approximation algorithms for stochastic and risk-averse optimization. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp 1305–1313. Society for Industrial and Applied Mathematics (2007)Google Scholar
  25. 25.
    Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2013)Google Scholar
  26. 26.
    Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • Alok Baveja
    • 1
  • Amit Chavan
    • 2
  • Andrei Nikiforov
    • 3
  • Aravind Srinivasan
    • 2
  • Pan Xu
    • 2
  1. 1.Department of Supply Chain Management, Rutgers Business SchoolRutgers, The State University of New JerseyPiscatawayUSA
  2. 2.Department of Computer ScienceUniversity of MarylandCollege ParkUSA
  3. 3.School of BusinessRutgers, The State University of New JerseyCamdenUSA

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