Two-stage stochastic matching with application to ride hailing
Pages 2862 - 2877
Abstract
We study a two-stage stochastic matching problem motivated in part by applications in online marketplaces used for ride hailing. Using a randomized primal-dual algorithm applied to a family of "balancing" convex programs, we obtain the optimal 3/4 competitive ratio against the optimum offline benchmark. These balancing convex programs offer a natural generalization of the matching skeleton by Goel et al. (2012) and may be of independent interest. Switching to the more precise benchmark of optimum online, we exploit connections to submodular optimization and use a factor-revealing program to improve the 3/4 ratio to (1 − 1/e + 1/e2) ≈ 0.767 for the unweighted and 0.761 for the weighted case. We also show it is NP-hard to obtain an FPTAS with respect to this benchmark.
References
[1]
Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online vertex-weighted bipartite matching and single-bid budgeted allocations. In Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pages 1253--1264. SIAM, 2011.
[2]
Shipra Agrawal and Nikhil R Devanur. Fast algorithms for online stochastic convex programming. In Proceedings of the twenty-sixth annual ACM-SIAM symposium on Discrete algorithms, pages 1405--1424. SIAM, 2014.
[3]
Nima Anari, Rad Niazadeh, Amin Saberi, and Ali Shameli. Nearly optimal pricing algorithms for production constrained and laminar bayesian selection. In Proceedings of the 2019 ACM Conference on Economics and Computation, pages 91--92, 2019.
[4]
John R Birge and Francois Louveaux. Introduction to stochastic programming. Springer Science & Business Media, 2011.
[5]
Niv Buchbinder, Kamal Jain, and Joseph Seffi Naor. Online primal-dual algorithms for maximizing adauctions revenue. In European Symposium on Algorithms, pages 253--264. Springer, 2007.
[6]
Gruia Calinescu, Chandra Chekuri, Martin Pál, and Jan Vondrák. Maximizing a monotone submodular function subject to a matroid constraint. SIAM Journal on Computing, 40(6):1740--1766, 2011.
[7]
Moses Charikar, Chandra Chekuri, and Martin Pál. Sampling bounds for stochastic optimization. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 257--269. Springer, 2005.
[8]
Nikhil R Devanur and Thomas P Hayes. The adwords problem: online keyword matching with budgeted bidders under random permutations. In Proceedings of the 10th ACM conference on Electronic commerce, pages 71--78, 2009.
[9]
Nikhil R Devanur and Kamal Jain. Online matching with concave returns. In Proceedings of the forty-fourth annual ACM symposium on Theory of computing, pages 137--144, 2012.
[10]
Nikhil R Devanur, Kamal Jain, Balasubramanian Sivan, and Christopher A Wilkens. Near optimal online algorithms and fast approximation algorithms for resource allocation problems. In Proceedings of the 12th ACM conference on Electronic commerce, pages 29--38, 2011.
[11]
Nikhil R Devanur, Kamal Jain, and Robert D Kleinberg. Randomized primal-dual analysis of ranking for online bipartite matching. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 101--107. SIAM, 2013.
[12]
Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17:449--467, 1965.
[13]
Bruno Escoffier, Laurent Gourvès, Jérôme Monnot, and Olivier Spanjaard. Two-stage stochastic matching and spanning tree problems: Polynomial instances and approximation. European Journal of Operational Research, 205(1):19--30, 2010.
[14]
Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM (JACM), 45(4):634--652, 1998.
[15]
Jon Feldman, Nitish Korula, Vahab Mirrokni, Shanmugavelayutham Muthukrishnan, and Martin Pál. Online ad assignment with free disposal. In International workshop on internet and network economics, pages 374--385. Springer, 2009a.
[16]
Jon Feldman, Aranyak Mehta, Vahab Mirrokni, and Shan Muthukrishnan. Online stochastic matching: Beating 1−1/e. In 2009 50th Annual IEEE Symposium on Foundations of Computer Science, pages 117--126. IEEE, 2009b.
[17]
Jon Feldman, Monika Henzinger, Nitish Korula, Vahab S Mirrokni, and Cliff Stein. Online stochastic packing applied to display ad allocation. In European Symposium on Algorithms, pages 182--194. Springer, 2010.
[18]
Yiding Feng, Rad Niazadeh, and Amin Saberi. Two-stage matching and pricing with applications to ride hailing. Available at SSRN, 2020.
[19]
Tibor Gallai. Maximale systeme unabhangiger kanten. Magyar Tud. Akad. Mat. Kutato Int. Kozl., 9:401--413, 1964.
[20]
Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, and Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. Journal of the ACM (JACM), 53(3):324--360, 2006.
[21]
Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. In Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms, pages 468--485. Society for Industrial and Applied Mathematics, 2012.
[22]
Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang, and Xue Zhu. How to match when all vertices arrive online. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 17--29, 2018.
[23]
Nicole Immorlica, David Karger, Maria Minkoff, and Vahab S Mirrokni. 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, pages 691--700. Society for Industrial and Applied Mathematics, 2004.
[24]
Patrick Jaillet and Xin Lu. Online stochastic matching: New algorithms with better bounds. Mathematics of Operations Research, 39(3):624--646, 2014.
[25]
Richard M Karp, Umesh V Vazirani, and Vijay V Vazirani. An optimal algorithm for on-line bipartite matching. In Proceedings of the twenty-second annual ACM symposium on Theory of computing, pages 352--358, 1990.
[26]
Irit Katriel, Claire Kenyon-Mathieu, and Eli Upfal. Commitment under uncertainty: Two-stage stochastic matching problems. Theoretical Computer Science, 408(2--3):213--223, 2008.
[27]
Nan Kong and Andrew J Schaefer. A factor 12 approximation algorithm for two-stage stochastic matching problems. European Journal of Operational Research, 172(3):740--746, 2006.
[28]
Euiwoong Lee and Sahil Singla. Maximum matching in the online batch-arrival model. In International Conference on Integer Programming and Combinatorial Optimization, pages 355--367. Springer, 2017.
[29]
Benny Lehmann, Daniel Lehmann, and Noam Nisan. Combinatorial auctions with decreasing marginal utilities. Games and Economic Behavior, 55(2):270--296, 2006.
[30]
Lyft. Matchmaking in Lyft line: Part 1, 2016. URL https://eng.lyft.com/matchmaking-in-lyft-line-9c2635fe62c4.
[31]
Vahideh H Manshadi, Shayan Oveis Gharan, and Amin Saberi. Online stochastic matching: Online actions based on offline statistics. Mathematics of Operations Research, 37(4):559--573, 2012.
[32]
Aranyak Mehta, Amin Saberi, Umesh Vazirani, and Vijay Vazirani. Adwords and generalized online matching. Journal of the ACM (JACM), 54(5):22--es, 2007.
[33]
Aryan Mokhtari, Hamed Hassani, and Amin Karbasi. Conditional gradient method for stochastic submodular maximization: Closing the gap. In International Conference on Artificial Intelligence and Statistics, pages 1886--1895, 2018.
[34]
John F Nash. The bargaining problem. Econometrica: Journal of the Econometric Society, pages 155--162, 1950.
[35]
John Rawls. A theory of justice. Cambridge, MA: p, University, 1971.
[36]
Alexander Schrijver. Combinatorial optimization: polyhedra and efficiency, volume 24. Springer Science & Business Media, 2003.
[37]
David B Shmoys and Mauro Sozio. Approximation algorithms for 2-stage stochastic scheduling problems. In International Conference on Integer Programming and Combinatorial Optimization, pages 145--157. Springer, 2007.
[38]
David B Shmoys and Chaitanya Swamy. Stochastic optimization is (almost) as easy as deterministic optimization. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 228--237. IEEE, 2004.
[39]
Maxim Sviridenko, Jan Vondrák, and Justin Ward. Optimal approximation for submodular and supermodular optimization with bounded curvature. Mathematics of Operations Research, 42(4):1197--1218, 2017.
[40]
Uber. Uber marketplace and matching, 2020. URL https://marketplace.uber.com/matching.
[41]
Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle model. In Proceedings of the fortieth annual ACM symposium on Theory of computing, pages 67--74. ACM, 2008.
[42]
Lingyu Zhang, Tao Hu, Yue Min, Guobin Wu, Junying Zhang, Pengcheng Feng, Pinghua Gong, and Jieping Ye. A taxi order dispatch model based on combinatorial optimization. In Proceedings of the 23rd ACM SIGKDD international conference on knowledge discovery and data mining, pages 2151--2159, 2017.
- Two-stage stochastic matching with application to ride hailing
Recommendations
Edge-Weighted Online Bipartite Matching
Online bipartite matching is one of the most fundamental problems in the online algorithms literature. Karp, Vazirani, and Vazirani (STOC 1990) gave an elegant algorithm for unweighted bipartite matching that achieves an optimal competitive ratio of 1-1/e ...
An optimal algorithm for on-line bipartite matching
STOC '90: Proceedings of the twenty-second annual ACM symposium on Theory of ComputingOnline bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs
STOC '11: Proceedings of the forty-third annual ACM symposium on Theory of computingIn a seminal paper, Karp, Vazirani, and Vazirani show that a simple ranking algorithm achieves a competitive ratio of 1-1/e for the online bipartite matching problem in the standard adversarial model, where the ratio of 1-1/e is also shown to be ...
Comments
Information & Contributors
Information
Published In
Sponsors
Publisher
Society for Industrial and Applied Mathematics
United States
Publication History
Published: 21 March 2021
Check for updates
Qualifiers
- Research-article
Conference
SODA '21
Sponsor:
Acceptance Rates
Overall Acceptance Rate 411 of 1,322 submissions, 31%
Contributors
Other Metrics
Bibliometrics & Citations
Bibliometrics
Article Metrics
- 0Total Citations
- 45Total Downloads
- Downloads (Last 12 months)4
- Downloads (Last 6 weeks)0
Reflects downloads up to 04 Feb 2025
Other Metrics
Citations
View Options
Login options
Check if you have access through your login credentials or your institution to get full access on this article.
Sign in