# Attenuate Locally, Win Globally: Attenuation-Based Frameworks for Online Stochastic Matching with Timeouts

## Abstract

Online matching problems have garnered significant attention in recent years due to numerous applications in e-commerce, online advertisements, ride-sharing, etc. Many of them capture the uncertainty in the real world by including stochasticity in both the arrival and matching processes. The online stochastic matching with timeouts problem introduced by Bansal et al. (*Algorithmica*, 2012) models matching markets (e.g., E-Bay, Amazon). Buyers arrive from an independent and identically distributed (i.i.d.) known distribution on buyer profiles and can be shown a list of items one at a time. Each buyer has some probability of purchasing each item and a limit (timeout) on the number of items they can be shown. Bansal et al. (*Algorithmica*, 2012) gave a 0.12-competitive algorithm which was improved by Adamczyk et al. (*ESA*, 2015) to 0.24. We present several online attenuation frameworks that use an algorithm for offline stochastic matching as a black box. On the upper bound side, we show that one framework, combined with a black-box adapted from Bansal et al. (*Algorithmica*, 2012), yields an online algorithm which nearly doubles the ratio to 0.46. Additionally, our attenuation frameworks extend to the more general setting of fractional arrival rates for online vertices. On the lower bound side, we show that no algorithm can achieve a ratio better than 0.632 using the standard LP for this problem. This framework has a high potential for further improvements since new algorithms for offline stochastic matching can directly improve the ratio for the online problem. Our online frameworks also have the potential for a variety of extensions. For example, we introduce a natural generalization: online stochastic matching with two-sided timeouts in which both online and offline vertices have timeouts. Our frameworks provide the first algorithm for this problem achieving a ratio of 0.30. We once again use the algorithm of Bansal et al. (*Algorithmica*, 2012) as a black-box and plug it into one of our frameworks.

## Keywords

Online algorithms Randomized algorithms Bipartite matching## Notes

### Acknowledgements

We wish to thank the anonymous referees for their generous comments, especially regarding the extension to fractional arrival rates.

## References

- 1.Adamczyk, M., Grandoni, F., Mukherjee, J.: Improved approximation algorithms for stochastic matching. In: Bansal, N., Finocchi, I. (eds.) Algorithms—ESA: 23rd Annual European Symposium, Patras, Greece, September 14–16, 2015, Proceedings (Berlin, Heidelberg, 2015), pp. 1–12. Springer, Berlin (2015)Google Scholar
- 2.Agrawal, S., Devanur, N.R.: Fast algorithms for online stochastic convex programming. In: Proceedings of the 26th Annual ACM–SIAM Symposium on Discrete Algorithms (Philadelphia, PA, USA, 2015), SODA’15, pp. 1405–1424. Society for Industrial and Applied Mathematics, New YorkGoogle Scholar
- 3.Agrawal, S., Wang, Z., Ye, Y.: A dynamic near-optimal algorithm for online linear programming. Oper. Res.
**62**(4), 876–890 (2014)MathSciNetCrossRefzbMATHGoogle Scholar - 4.Alaei, S., Hajiaghayi, M., Liaghat, V.: Online prophet-inequality matching with applications to ad allocation. In: Proceedings of the 13th ACM Conference on Electronic Commerce (New York, NY, USA, 2012), EC’12, pp. 18–35. ACM, New YorkGoogle Scholar
- 5.Alaei, S., Hajiaghayi, M., Liaghat, V.: The online stochastic generalized assignment problem. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques: 16th International Workshop, APPROX 2013, and 17th International Workshop, RANDOM 2013, Berkeley, CA, USA, August 21–23: Proceedings (Berlin, Heidelberg, 2013), pp. 11–25. Springer, Berlin (2013)Google Scholar
- 6.Bahmani, B., Kapralov, M.: Improved bounds for online stochastic matching. In: Algorithms—ESA 2010, pp. 170–181. Springer, Berlin (2010)Google Scholar
- 7.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: Algorithms—ESA 2010: 18th Annual European Symposium, Liverpool, UK, September 6–8, Proceedings, Part II (Berlin, Heidelberg, 2010), pp. 218–229. Springer, Berlin (2010)Google Scholar
- 8.Baveja, A., Chavan, A., Nikiforov, A., Srinivasan, A., Xu, P.: Improved bounds in stochastic matching and optimization. In: APPROX-RANDOM 2015, LIPIcs-Leibniz International Proceedings in Informatics, vol. 40, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik (2015)Google Scholar
- 9.Brubach, B., Sankararaman, K.A., Srinivasan, A., Xu, P.: New algorithms, better bounds, and a novel model for online stochastic matching. In: European Symposium on Algorithms (ESA) (2016)Google Scholar
- 10.Brubach, B., Sankararaman, K.A., Srinivasan, A., Xu, P.: Attenuate locally, win globally: an attenuation-based framework for online stochastic matching with timeouts. In: Proceedings of the 16th Conference on Autonomous Agents and Multiagent Systems, International Foundation for Autonomous Agents and Multiagent Systems, pp. 1223–1231 (2017)Google Scholar
- 11.Buchbinder, N., Jain, K., Naor, J.S.: Online primal-dual algorithms for maximizing ad-auctions revenue. In: Algorithms—ESA 2007: 15th Annual European Symposium, Eilat, Israel, October 8–10, Proceedings (Berlin, Heidelberg, 2007), pp. 253–264. Springer, Berlin (2007)Google Scholar
- 12.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 5555, pp. 266–278 (2009)Google Scholar
- 13.Devanur, N.R., Hayes, T.P.: The adwords problem: online keyword matching with budgeted bidders under random permutations. In: Proceedings of the 10th ACM Conference on Electronic Commerce (2009), pp. 71–78. ACM, New YorkGoogle Scholar
- 14.Devanur, N.R., Jain, K.: Online matching with concave returns. In: Proceedings of the 44th Annual ACM Symposium on Theory of Computing, pp. 137–144. ACM, New York (2012)Google Scholar
- 15.Devanur, N.R., Jain, K., Sivan, B., Wilkens, C.A.: Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In: Proceedings of the 12th ACM Conference on Electronic Commerce (New York, NY, USA, 2011), EC’11, pp. 29–38. ACM, New YorkGoogle Scholar
- 16.Devanur, N.R., Sivan, B., Azar, Y.: Asymptotically optimal algorithm for stochastic adwords. In: Proceedings of the 13th ACM Conference on Electronic Commerce, EC’12 (2012)Google Scholar
- 17.Feldman, J., Henzinger, M., Korula, N., Mirrokni, V.S., Stein, C.: Online stochastic packing applied to display ad allocation. In: Proceedings of the 18th Annual European Conference on Algorithms: Part I (Berlin, Heidelberg, 2010), ESA’10, pp. 182–194. Springer, BerlinGoogle Scholar
- 18.Feldman, J., Mehta, A., Mirrokni, V., Muthukrishnan, S.: Online stochastic matching: Beating 1-1/e. In: FOCS’09. 50th Annual IEEE Symposium on Foundations of Computer Science, 2009, pp. 117–126. IEEE, New York (2009)Google Scholar
- 19.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 - 20.Gupta, A., Nagarajan, V.: A stochastic probing problem with applications. In: Integer Programming and Combinatorial Optimization: 16th International Conference, IPCO 2013, Valparaíso, Chile, March 18–20: Proceedings (Berlin, Heidelberg, 2013), pp. 205–216. Springer, Berlin (2013)Google Scholar
- 21.Haeupler, B., Mirrokni, V.S., Zadimoghaddam, M.: Online stochastic weighted matching: improved approximation algorithms. In: Internet and Network Economics, vol. 7090 of Lecture Notes in Computer Science, pp. 170–181. Springer, Berlin (2011)Google Scholar
- 22.Jaillet, P., Lu, X.: Online stochastic matching: new algorithms with better bounds. Math. Oper. Res.
**39**(3), 624–646 (2013)MathSciNetCrossRefzbMATHGoogle Scholar - 23.Kalyanasundaram, B., Pruhs, K.R.: An optimal deterministic algorithm for online \(b\)-matching. Theoret. Comput. Sci.
**233**, 2000 (2000)MathSciNetCrossRefzbMATHGoogle Scholar - 24.Karp, R.M., Vazirani, U.V., Vazirani, V.V.: An optimal algorithm for on-line bipartite matching. In: Proceedings of the 22nd Annual ACM Symposium on Theory of Computing, pp. 352–358. ACM, New York (1990)Google Scholar
- 25.Ma, W.: Improvements and generalizations of stochastic knapsack and multi-armed bandit approximation algorithms. In: Proceedings of the 25th Annual ACM–SIAM Symposium on Discrete Algorithms, pp. 1154–1163. Society for Industrial and Applied Mathematics, New York (2014)Google Scholar
- 26.Manshadi, V.H., Gharan, S.O., Saberi, A.: Online stochastic matching: online actions based on offline statistics. Math. Oper. Res.
**37**(4), 559–573 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 27.Mehta, A.: Online matching and ad allocation. Found. Trends Theor. Comput. Sci.
**8**(4), 265–368 (2012)MathSciNetCrossRefzbMATHGoogle Scholar - 28.Mehta, A., Saberi, A., Vazirani, U., Vazirani, V.: Adwords and generalized online matching. J. ACM
**54**, 5 (2007)MathSciNetCrossRefzbMATHGoogle Scholar