# Improved Bounds in Stochastic Matching and Optimization

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## 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.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.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.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.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.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)MathSciNetCrossRefzbMATHGoogle Scholar - 6.Beale, E.M.: On minimizing a convex function subject to linear inequalities. J. Roy. Stat. Soc. B Met.
**17**, 173–184 (1955)MathSciNetzbMATHGoogle Scholar - 7.Birge, J.R., Louveaux, F.: Introduction to Stochastic Programming. Springer, New York (1997)zbMATHGoogle Scholar
- 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.Dantzig, G.B.: Linear programming under uncertainty. Manag. Sci.
**1**(3–4), 197–206 (1955)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 11.Fortuin, C.M., Ginibre, J., Kasteleyn, P.N.: Correlational inequalities for partially ordered sets. Commun. Math. Phys.
**22**, 89–103 (1971)CrossRefzbMATHGoogle Scholar - 12.Füredi, Z., Kahn, J., Seymour, P.D.: On the fractional matching polytope of a hypergraph. Combinatorica
**13**(2), 167–180 (1993)MathSciNetCrossRefzbMATHGoogle Scholar - 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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.Gupta, A., Ravi, R., Sinha, A.: LP rounding approximation algorithms for stochastic network design. Math. Oper. Res.
**32**(2), 345–364 (2007)MathSciNetCrossRefzbMATHGoogle Scholar - 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.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)MathSciNetCrossRefzbMATHGoogle Scholar - 19.Ravi, R., Sinha, A.: Hedging uncertainty: approximation algorithms for stochastic optimization problems. Math. Program.
**108**(1), 97–114 (2006)MathSciNetCrossRefzbMATHGoogle Scholar - 20.Ruszczynski, A.P., Shapiro, A.: Stochastic Programming, vol. 10. Elsevier, Amsterdam (2003)zbMATHGoogle Scholar
- 21.Shachnai, H., Srinivasan, A.: Finding large independent sets in graphs and hypergraphs. SIAM J. Discret. Math.
**18**, 488–500 (2004)MathSciNetCrossRefzbMATHGoogle Scholar - 22.Shapiro, A., Dentcheva, D., Ruszczynski, A.: Lectures on Stochastic Programming—Modeling and Theory, vol. 16, 2nd edn. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
- 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)MathSciNetCrossRefzbMATHGoogle Scholar - 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.Vazirani, V.V.: Approximation Algorithms. Springer, Berlin (2013)Google Scholar
- 26.Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, Cambridge (2011)CrossRefzbMATHGoogle Scholar