Online Algorithms for Maximum Cardinality Matching with Edge Arrivals

Authors Niv Buchbinder, Danny Segev, Yevgeny Tkach



PDF
Thumbnail PDF

File

LIPIcs.ESA.2017.22.pdf
  • Filesize: 484 kB
  • 14 pages

Document Identifiers

Author Details

Niv Buchbinder
Danny Segev
Yevgeny Tkach

Cite As Get BibTex

Niv Buchbinder, Danny Segev, and Yevgeny Tkach. Online Algorithms for Maximum Cardinality Matching with Edge Arrivals. In 25th Annual European Symposium on Algorithms (ESA 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 87, pp. 22:1-22:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017) https://doi.org/10.4230/LIPIcs.ESA.2017.22

Abstract

In the adversarial edge arrival model for maximum cardinality matching, edges of an unknown graph are revealed one-by-one in arbitrary order, and should be irrevocably accepted or rejected. Here, the goal of an online algorithm is to maximize the number of accepted edges while maintaining a feasible matching at any point in time. For this model, the standard greedy heuristic is 1/2-competitive, and on the other hand, no algorithm that outperforms this ratio is currently known, even for very simple graphs.

We present a clean Min-Index framework for devising a family of randomized algorithms, and provide a number of positive and negative results in this context. Among these results, we present a 5/9-competitive algorithm when the underlying graph is a forest, and prove that this ratio is best possible within the Min-Index framework. In addition, we prove a new general upper bound of 2/(3+1/phi^2) ~ 0.5914 on the competitiveness of any algorithm in the edge arrival model. Interestingly, this bound holds even for an easier model in which vertices (along with their adjacent edges) arrive online, and when the underlying graph is a tree of maximum degree at most 3.

Subject Classification

Keywords
  • Maximum matching
  • online algorithms
  • competitive analysis
  • primal-dual method

Metrics

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

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 22nd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1253-1264, 2011. Google Scholar
  2. Yossi Azar, Ilan Reuven Cohen, and Alan Roytman. Online lower bounds via duality. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 1038-1050, 2017. Google Scholar
  3. Niv Buchbinder, Kamal Jain, and Joseph Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In Proceedings of the 15th Annual European Symposium on Algorithms, pages 253-264, 2007. Google Scholar
  4. Ashish Chiplunkar, Sumedh Tirodkar, and Sundar Vishwanathan. On randomized algorithms for matching in the online preemptive model. In Proceedings of the 23rd Annual European Symposium on Algorithms, pages 325-336, 2015. Google Scholar
  5. Nikhil R. Devanur and Thomas P. Hayes. The adwords problem: online keyword matching with budgeted bidders under random permutations. In Proceedings 10th ACM Conference on Electronic Commerce, pages 71-78, 2009. Google Scholar
  6. Nikhil R. Devanur and Kamal Jain. Online matching with concave returns. In Proceedings of the 44th ACM Symposium on Theory of Computing, pages 137-144, 2012. Google Scholar
  7. Nikhil R. Devanur, Kamal Jain, and Robert D. Kleinberg. Randomized primal-dual analysis of RANKING for online bipartite matching. In Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 101-107, 2013. Google Scholar
  8. Leah Epstein, Asaf Levin, Danny Segev, and Oren Weimann. Improved bounds for online preemptive matching. In Proceedings of the 30th International Symposium on Theoretical Aspects of Computer Science, pages 389-399, 2013. Google Scholar
  9. Jon Feldman, Aranyak Mehta, Vahab S. Mirrokni, and S. Muthukrishnan. Online stochastic matching: Beating 1-1/e. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pages 117-126, 2009. Google Scholar
  10. 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, pages 982-991, 2008. Google Scholar
  11. Guru Prashanth Guruganesh and Sahil Singla. Online matroid intersection: Beating half for random arrival. Preprint, arXiv:1512.06271, 2015. Google Scholar
  12. Bala Kalyanasundaram and Kirk Pruhs. An optimal deterministic algorithm for online b-matching. Theoretical Computer Science, 233(1-2):319-325, 2000. Google Scholar
  13. Chinmay Karande, Aranyak Mehta, and Pushkar Tripathi. Online bipartite matching with unknown distributions. In Proceedings of the 43rd ACM Symposium on Theory of Computing, pages 587-596, 2011. Google Scholar
  14. 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, pages 352-358, 1990. Google Scholar
  15. 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, pages 597-606, 2011. Google Scholar
  16. 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. Google Scholar
  17. Aranyak Mehta. Online matching and ad allocation. Foundations and Trends in Theoretical Computer Science, 8(4):265-368, 2013. Google Scholar
  18. Aranyak Mehta, Amin Saberi, Umesh V. Vazirani, and Vijay V. Vazirani. Adwords and generalized online matching. Journal of the ACM, 54(5):22, 2007. Google Scholar
  19. Sumedh Tirodkar and Sundar Vishwanathan. Maximum matching on trees in the online preemptive and the incremental dynamic graph models. Preprint, arXiv:1612.05419, 2016. Google Scholar
  20. Yajun Wang and Sam Chiu-wai Wong. Two-sided online bipartite matching and vertex cover: Beating the greedy algorithm. In Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming, pages 1070-1081, 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