Stochastic Online Metric Matching
We study the minimum-cost metric perfect matching problem under online i.i.d arrivals. We are given a fixed metric with a server at each of the points, and then requests arrive online, each drawn independently from a known probability distribution over the points. Each request has to be matched to a free server, with cost equal to the distance. The goal is to minimize the expected total cost of the matching.
Such stochastic arrival models have been widely studied for the maximization variants of the online matching problem; however, the only known result for the minimization problem is a tight O(log n)-competitiveness for the random-order arrival model. This is in contrast with the adversarial model, where an optimal competitive ratio of O(log n) has long been conjectured and remains a tantalizing open question.
In this paper, we show that the i.i.d model admits substantially better algorithms: our main result is an O((log log log n)^2)-competitive algorithm in this model, implying a strict separation between the i.i.d model and the adversarial and random order models. Along the way we give a 9-competitive algorithm for the line and tree metrics - the first O(1)-competitive algorithm for any non-trivial arrival model for these much-studied metrics.
stochastic
online
online matching
metric matching
Theory of computation~Online algorithms
Mathematics of computing~Matchings and factors
67:1-67:14
Track A: Algorithms, Complexity and Games
A full version of the paper is available at https://arxiv.org/abs/1904.09284.
Anupam
Gupta
Anupam Gupta
Carnegie Mellon University, Pittsburgh, PA, USA
http://www.cs.cmu.edu/~anupamg/
Supported in part by NSF awards CCF-1536002, CCF-1540541, and CCF-1617790, and the Indo-US Joint Center for Algorithms Under Uncertainty.
Guru
Guruganesh
Guru Guruganesh
Google Research, United States
Binghui
Peng
Binghui Peng
Tsinghua University, China
David
Wajc
David Wajc
Carnegie Mellon University, Pittsburgh, PA, USA
http://www.cs.cmu.edu/~dwajc/
Supported in part by NSF grants CCF-1618280, CCF-1814603, CCF-1527110, NSF CAREER award CCF-1750808 and a Sloan Research Fellowship.
10.4230/LIPIcs.ICALP.2019.67
Gagan Aggarwal, Gagan Goel, Chinmay Karande, and Aranyak Mehta. Online vertex-weighted bipartite matching and single-bid budgeted allocations. In Proceedings of the 22nd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1253-1264, 2011.
Antonios Antoniadis, Neal Barcelo, Michael Nugent, Kirk Pruhs, and Michele Scquizzato. A o (n)-Competitive Deterministic Algorithm for Online Matching on a Line. In Proceedings of the 12th Workshop on Approximation and Online Algorithms (WAOA), pages 11-22, 2014.
Bahman Bahmani and Michael Kapralov. Improved bounds for online stochastic matching. In Proceedings of the 18th Annual European Symposium on Algorithms (ESA), pages 170-181. Springer, 2010.
Nikhil Bansal, Niv Buchbinder, Anupam Gupta, and Joseph Seffi Naor. An O(logĀ² k)-competitive algorithm for metric bipartite matching. In Proceedings of the 15th Annual European Symposium on Algorithms (ESA), pages 522-533, 2007.
Daniel Berend and Aryeh Kontorovich. A sharp estimate of the binomial mean absolute deviation with applications. Statistics &Probability Letters, 83(4):1254-1259, 2013.
Brian Brubach, Karthik Abinav Sankararaman, Aravind Srinivasan, and Pan Xu. New Algorithms, Better Bounds, and a Novel Model for Online Stochastic Matching. In Proceedings of the 24th Annual European Symposium on Algorithms (ESA), pages 24:1-24:16, 2016.
Minjun Chang, Dorit S Hochbaum, Quico Spaen, and Mark Velednitsky. DISPATCH: an optimally-competitive algorithm for maximum online perfect bipartite matching with iid arrivals. In Proceedings of the 16th Workshop on Approximation and Online Algorithms (WAOA), pages 149-164, 2018.
Ilan Reuven Cohen and David Wajc. Randomized Online Matching in Regular Graphs. In Proceedings of the 29th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 960-979, 2018.
Sina Dehghani, Soheil Ehsani, MohammadTaghi Hajiaghayi, Vahid Liaghat, and Saeed Seddighin. Stochastic k-Server: How Should Uber Work? In Proceedings of the 44th International Colloquium on Automata, Languages and Programming (ICALP), pages 126:1-126:14, 2017.
Nikhil R Devanur, Balasubramanian Sivan, and Yossi Azar. Asymptotically optimal algorithm for stochastic adwords. In Proceedings of the 13th ACM Conference on Electronic Commerce (EC), pages 388-404, 2012.
Hossein Esfandiari, Nitish Korula, and Vahab S. Mirrokni. Online Allocation with Traffic Spikes: Mixing Adversarial and Stochastic Models. In Proceedings of the 16th ACM Conference on Economics and Computation (EC), pages 169-186, 2015.
Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences, 69(3):485-497, 2004.
Jon Feldman, Aranyak Mehta, Vahab Mirrokni, and S Muthukrishnan. Online stochastic matching: Beating 1-1/e. In Proceedings of the 50th Symposium on Foundations of Computer Science (FOCS), pages 117-126, 2009.
Buddhima Gamlath, Michael Kapralov, Andreas Maggiori, Ola Svensson, and David Wajc. Online Matching with General Arrivals. arXiv preprint arXiv:1904.08255, 2019.
Naveen Garg, Anupam Gupta, Stefano Leonardi, and Piotr Sankowski. Stochastic analyses for online combinatorial optimization problems. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 942-951, 2008.
Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 982-991, 2008.
Fabrizio Grandoni, Anupam Gupta, Stefano Leonardi, Pauli Miettinen, Piotr Sankowski, and Mohit Singh. Set covering with our eyes closed. SIAM Journal on Computing (SICOMP), 42(3):808-830, 2013.
Anupam Gupta and Kevin Lewi. The online metric matching problem for doubling metrics. In Proceedings of the 39th International Colloquium on Automata, Languages and Programming (ICALP), pages 424-435, 2012.
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 Symposium on Theory of Computing (STOC), pages 17-29, 2018.
Zhiyi Huang, Binghui Peng, Zhihao Gavin Tang, Runzhou Tao, Xiaowei Wu, and Yuhao Zhang. Tight Competitive Ratios of Classic Matching Algorithms in the Fully Online Model. In Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2875-2886, 2019.
Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals. In Proceedings of the 45th International Colloquium on Automata, Languages and Programming (ICALP), pages 1070-1081, 2018.
Bala Kalyanasundaram and Kirk Pruhs. Online weighted matching. Journal of Algorithms, 14(3):478-488, 1993.
Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 587-596, 2011.
Richard M Karp, Umesh V Vazirani, and Vijay V Vazirani. An optimal algorithm for on-line bipartite matching. In Proceedings of the 22nd Annual ACM Symposium on Theory of Computing (STOC), pages 352-358, 1990.
Samir Khuller, Stephen G Mitchell, and Vijay V Vazirani. On-line algorithms for weighted bipartite matching and stable marriages. Theoretical Computer Science (TCS), 127(2):255-267, 1994.
Mohammad Mahdian, Hamid Nazerzadeh, and Amin Saberi. Allocating online advertisement space with unreliable estimates. In Proceedings of the 8th ACM Conference on Electronic Commerce (EC), pages 288-294, 2007.
Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: an approach based on strongly factor-revealing lps. In Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (STOC), pages 597-606, 2011.
Aranyak Mehta. Online matching and ad allocation. Foundations and Trendsregistered in Theoretical Computer Science, 8(4):265-368, 2013.
Aranyak Mehta, Amin Saberi, Umesh Vazirani, and Vijay Vazirani. Adwords and generalized online matching. Journal of the ACM (JACM), 54(5):22, 2007.
Adam Meyerson, Akash Nanavati, and Laura Poplawski. Randomized online algorithms for minimum metric bipartite matching. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 954-959, 2006.
Vahab S Mirrokni, Shayan Oveis Gharan, and Morteza Zadimoghaddam. Simultaneous approximations for adversarial and stochastic online budgeted allocation. In Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1690-1701, 2012.
Michael Mitzenmacher and Eli Upfal. Probability and computing: Randomized algorithms and probabilistic analysis. Cambridge university press, 2005.
Joseph Seffi Naor and David Wajc. Near-optimum online ad allocation for targeted advertising. ACM Transactions on Economics and Computation (TEAC), 6(3-4):16, 2018.
Krati Nayyar and Sharath Raghvendra. An input sensitive online algorithm for the metric bipartite matching problem. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 505-515, 2017.
Sharath Raghvendra. A robust and optimal online algorithm for minimum metric bipartite matching. In Proceedings of the 19th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), volume 60, 2016.
Sharath Raghvendra. Optimal Analysis of an Online Algorithm for the Bipartite Matching Problem on a Line. In Proceedings of the 34th Symposium on Computational geometry (SoCG), pages 67:1-67:14, 2018.
Anupam Gupta, Guru Guruganesh, Binghui Peng, and David Wajc
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