Document Open Access Logo

Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals

Authors Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2018.79.pdf
  • Filesize: 0.88 MB
  • 14 pages

Document Identifiers

Author Details

Zhiyi Huang
  • Department of Computer Sicence, The University of Hong Kong, Hong Kong
Zhihao Gavin Tang
  • Department of Computer Sicence, The University of Hong Kong, Hong Kong
Xiaowei Wu
  • Department of Computing, The Hong Kong Polytechnic University, Hong Kong
Yuhao Zhang
  • Department of Computer Sicence, The University of Hong Kong, Hong Kong

Cite AsGet BibTex

Zhiyi Huang, Zhihao Gavin Tang, Xiaowei Wu, and Yuhao Zhang. Online Vertex-Weighted Bipartite Matching: Beating 1-1/e with Random Arrivals. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 79:1-79:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)
https://doi.org/10.4230/LIPIcs.ICALP.2018.79

Abstract

We introduce a weighted version of the ranking algorithm by Karp et al. (STOC 1990), and prove a competitive ratio of 0.6534 for the vertex-weighted online bipartite matching problem when online vertices arrive in random order. Our result shows that random arrivals help beating the 1-1/e barrier even in the vertex-weighted case. We build on the randomized primal-dual framework by Devanur et al. (SODA 2013) and design a two dimensional gain sharing function, which depends not only on the rank of the offline vertex, but also on the arrival time of the online vertex. To our knowledge, this is the first competitive ratio strictly larger than 1-1/e for an online bipartite matching problem achieved under the randomized primal-dual framework. Our algorithm has a natural interpretation that offline vertices offer a larger portion of their weights to the online vertices as time goes by, and each online vertex matches the neighbor with the highest offer at its arrival.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Online algorithms
  • Theory of computation → Linear programming
Keywords
  • Vertex Weighted
  • Online Bipartite Matching
  • Randomized Primal-Dual

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Melika Abolhassani, T.-H. Hubert Chan, Fei Chen, Hossein Esfandiari, MohammadTaghi Hajiaghayi, Hamid Mahini, and Xiaowei Wu. Beating ratio 0.5 for weighted oblivious matching problems. In ESA, volume 57 of LIPIcs, pages 3:1-3:18. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  2. 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, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 1253-1264, 2011. URL: http://dx.doi.org/10.1137/1.9781611973082.95.
  3. Jonathan Aronson, Martin Dyer, Alan Frieze, and Stephen Suen. Randomized greedy matching. ii. Random Struct. Algorithms, 6(1):55-73, 1995. URL: http://dx.doi.org/10.1002/rsa.3240060107.
  4. Bahman Bahmani and Michael Kapralov. Improved bounds for online stochastic matching. In ESA (1), volume 6346 of Lecture Notes in Computer Science, pages 170-181. Springer, 2010. Google Scholar
  5. Benjamin Birnbaum and Claire Mathieu. On-line bipartite matching made simple. ACM SIGACT News, 39(1):80-87, 2008. Google Scholar
  6. Brian Brubach, Karthik Abinav Sankararaman, Aravind Srinivasan, and Pan Xu. New algorithms, better bounds, and a novel model for online stochastic matching. In ESA, volume 57 of LIPIcs, pages 24:1-24:16. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2016. Google Scholar
  7. Niv Buchbinder, Kamal Jain, and Joseph Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In ESA, volume 4698 of Lecture Notes in Computer Science, pages 253-264. Springer, 2007. Google Scholar
  8. Niv Buchbinder, Kamal Jain, and Mohit Singh. Secretary problems via linear programming. Math. Oper. Res., 39(1):190-206, 2014. Google Scholar
  9. T.-H. Hubert Chan, Fei Chen, Xiaowei Wu, and Zhichao Zhao. Ranking on arbitrary graphs: Rematch via continuous lp with monotone and boundary condition constraints. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1112-1122. SIAM, 2014. URL: http://dx.doi.org/10.1137/1.9781611973402.82.
  10. Ilan Reuven Cohen and David Wajc. Randomized online matching in regular graphs. In SODA, pages 960-979. SIAM, 2018. Google Scholar
  11. Nikhil R. Devanur and Kamal Jain. Online matching with concave returns. In STOC, pages 137-144. ACM, 2012. Google Scholar
  12. Nikhil R. Devanur, Kamal Jain, and Robert D. Kleinberg. Randomized primal-dual analysis of RANKING for online bipartite matching. In SODA, pages 101-107. SIAM, 2013. Google Scholar
  13. Jon Feldman, Aranyak Mehta, Vahab S. Mirrokni, and S. Muthukrishnan. Online stochastic matching: Beating 1-1/e. In FOCS, pages 117-126. IEEE Computer Society, 2009. Google Scholar
  14. Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In SODA, pages 982-991, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347189.
  15. Bernhard Haeupler, Vahab S. Mirrokni, and Morteza Zadimoghaddam. Online stochastic weighted matching: Improved approximation algorithms. In WINE, volume 7090 of Lecture Notes in Computer Science, pages 170-181. Springer, 2011. Google Scholar
  16. Zhiyi Huang, Ning Kang, Zhihao Gavin Tang, Xiaowei Wu, Yuhao Zhang, and Xue Zhu. How to match when all vertices arrive online. CoRR (to appear in STOC 2018), abs/1802.03905, 2018. URL: http://arxiv.org/abs/1802.03905.
  17. Patrick Jaillet and Xin Lu. Online stochastic matching: New algorithms with better bounds. Math. Oper. Res., 39(3):624-646, 2014. Google Scholar
  18. Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 587-596, 2011. URL: http://dx.doi.org/10.1145/1993636.1993715.
  19. 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, May 13-17, 1990, Baltimore, Maryland, USA, pages 352-358, 1990. URL: http://dx.doi.org/10.1145/100216.100262.
  20. Thomas Kesselheim, Klaus Radke, Andreas Tönnis, and Berthold Vöcking. An optimal online algorithm for weighted bipartite matching and extensions to combinatorial auctions. In ESA, volume 8125 of Lecture Notes in Computer Science, pages 589-600. Springer, 2013. Google Scholar
  21. Mohammad Mahdian and Qiqi Yan. Online bipartite matching with random arrivals: an approach based on strongly factor-revealing LPs. In Proceedings of the 43rd ACM Symposium on Theory of Computing, STOC 2011, San Jose, CA, USA, 6-8 June 2011, pages 597-606, 2011. URL: http://dx.doi.org/10.1145/1993636.1993716.
  22. Vahideh H. Manshadi, Shayan Oveis Gharan, and Amin Saberi. Online stochastic matching: Online actions based on offline statistics. Math. Oper. Res., 37(4):559-573, 2012. Google Scholar
  23. Aranyak Mehta, Amin Saberi, Umesh V. Vazirani, and Vijay V. Vazirani. Adwords and generalized online matching. J. ACM, 54(5):22, 2007. Google Scholar
  24. Yajun Wang and Sam Chiu-wai Wong. Two-sided online bipartite matching and vertex cover: Beating the greedy algorithm. In ICALP (1), volume 9134 of Lecture Notes in Computer Science, pages 1070-1081. Springer, 2015. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail