Document Open Access Logo

Optimal Online Bipartite Matching in Degree-2 Graphs

Authors Amey Bhangale , Arghya Chakraborty , Prahladh Harsha

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2025.13.pdf
  • Filesize: 2.71 MB
  • 19 pages

Document Identifiers

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
Keywords
  • Online Algorithm
  • Bipartite matching

Metrics

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

Abstract

Online bipartite matching is a classical problem in online algorithms and we know that both the deterministic fractional and randomized integral online matchings achieve the same competitive ratio of 1-1/e. In this work, we study classes of graphs where the online degree is restricted to 2. As expected, one can achieve a competitive ratio of better than 1-1/e in both the deterministic fractional and randomized integral cases, but surprisingly, these ratios are not the same. It was already known that for fractional matching, a 0.75 competitive ratio algorithm is optimal. We show that the folklore Half-Half algorithm achieves a competitive ratio of η ≈ 0.717772… and more surprisingly, show that this is optimal by giving a matching lower-bound. This yields a separation between the two problems: deterministic fractional and randomized integral, showing that it is impossible to obtain a perfect rounding scheme.

Cite As Get BibTex

Amey Bhangale, Arghya Chakraborty, and Prahladh Harsha. Optimal Online Bipartite Matching in Degree-2 Graphs. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 13:1-13:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.13

Author Details

Amey Bhangale
  • University of California, Riverside, CA, USA
Arghya Chakraborty
  • Tata Institute of Fundamental Research, Mumbai, India
Prahladh Harsha
  • Tata Institute of Fundamental Research, Mumbai, India

Funding

Research of the second and third authors supported by the Department of Atomic Energy, Government of India, under Project Identification No. RTI4001.
  • Bhangale, Amey: Supported by Hellman Fellowship award & NSF CAREER award 2440882.
  • Chakraborty, Arghya: Supported in part by Prof. Mrinal Kumar’s SERB grant CRG/2023/006433.
  • Harsha, Prahladh: Supported in part by the Google India Research Award.

References

  1. Susanne Albers and Sebastian Schubert. Online ad allocation in bounded-degree graphs. In Kristoffer Arnsfelt Hansen, Tracy Xiao Liu, and Azarakhsh Malekian, editors, Proc. 18th International Conference on Web and Internet Economics (WINE), volume 13778 of LNCS, pages 60-77. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22832-2_4.
  2. Susanne Albers and Sebastian Schubert. Tight bounds for online matching in bounded-degree graphs with vertex capacities. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, Proc. 30th Annual European Symp. of Algorithms (ESA), volume 244 of LIPIcs, pages 4:1-4:16. Schloss Dagstuhl, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.4.
  3. Yossi Azar, Ilan Reuven Cohen, and Alan Roytman. Online lower bounds via duality. In Philip N. Klein, editor, Proc. 28th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1038-1050, 2017. URL: https://doi.org/10.1137/1.9781611974782.66.
  4. Claude Berge. Two theorems in graph theory. PNAS, 43(9):842-844, 1957. URL: https://doi.org/10.1073/pnas.43.9.842.
  5. Guy Blanc and Moses Charikar. Multiway online correlated selection. In Jelani Nelson, editor, Proc. 63rd IEEE Symp. on Foundations of Comp. Science (FOCS), pages 1277-1284, 2022. URL: https://doi.org/10.1109/FOCS52979.2021.00124.
  6. Niv Buchbinder, Kamal Jain, and Joseph (Seffi) Naor. Online primal-dual algorithms for maximizing ad-auctions revenue. In Lars Arge, Michael Hoffmann, and Emo Welzl, editors, Proc. 15th Annual European Symp. of Algorithms (ESA), volume 4698 of LNCS, pages 253-264. Springer, 2007. URL: https://doi.org/10.1007/978-3-540-75520-3_24.
  7. Niv Buchbinder and Joseph (Seffi) Naor. The design of competitive online algorithms via a primal-dual approach. Found. Trends Theor. Comput. Sci., 3(2-3):93-263, 2009. URL: https://doi.org/10.1561/0400000024.
  8. Niv Buchbinder, Joseph (Seffi) Naor, and David Wajc. Lossless online rounding for online bipartite matching (despite its impossibility). In Nikhil Bansal and Viswanath Nagarajan, editors, Proc. 34th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 2030-2068, 2023. URL: https://doi.org/10.1137/1.9781611977554.CH78.
  9. Deeparnab Chakrabarty and Gagan Goel. On the approximability of budgeted allocations and improved lower bounds for submodular welfare maximization and GAP. SIAM J. Comput., 39(6):2189-2211, 2010. (Preliminary version in 49th FOCS, 2008). URL: https://doi.org/10.1137/080735503.
  10. Denis Xavier Charles, Max Chickering, Nikhil R. Devanur, Kamal Jain, and Manan Sanghi. Fast algorithms for finding matchings in lopsided bipartite graphs with applications to display ads. In David C. Parkes, Chrysanthos Dellarocas, and Moshe Tennenholtz, editors, Proc. 11th ACM Conf. Electronic Commerce (EC), pages 121-128. ACM, 2010. URL: https://doi.org/10.1145/1807342.1807362.
  11. Ilan Reuven Cohen and Binghui Peng. Primal-dual schemes for online matching in bounded degree graphs. In Inge Li Gørtz, Martin Farach-Colton, Simon J. Puglisi, and Grzegorz Herman, editors, Proc. 31st Annual European Symp. of Algorithms (ESA), volume 274 of LIPIcs, pages 35:1-35:17. Schloss Dagstuhl, 2023. URL: https://doi.org/10.4230/LIPICS.ESA.2023.35.
  12. Ilan Reuven Cohen and David Wajc. Randomized online matching in regular graphs. In Artur Czumaj, editor, Proc. 29th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 960-979, 2018. URL: https://doi.org/10.1137/1.9781611975031.62.
  13. Nikhil R. Devanur and Thomas P. Hayes. The adwords problem: online keyword matching with budgeted bidders under random permutations. In John Chuang, Lance Fortnow, and Pearl Pu, editors, Proc. 10th ACM Conf. on Electronic Commerce (EC), pages 71-78. ACM, 2009. URL: https://doi.org/10.1145/1566374.1566384.
  14. Nikhil R. Devanur, Kamal Jain, and Robert D. Kleinberg. Randomized primal-dual analysis of RANKING for online bipartite matching. In Sanjeev Khanna, editor, Proc. 24th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 101-107, 2013. URL: https://doi.org/10.1137/1.9781611973105.7.
  15. Tomer Ezra, Michal Feldman, Nick Gravin, and Zhihao Gavin Tang. Online stochastic max-weight matching: Prophet inequality for vertex and edge arrival models. In Péter Biró, Jason D. Hartline, Michael Ostrovsky, and Ariel D. Procaccia, editors, Proc. 21st ACM Conf. Economics and Comput. (EC), pages 769-787. ACM, 2020. URL: https://doi.org/10.1145/3391403.3399513.
  16. Matthew Fahrbach, Zhiyi Huang, Runzhou Tao, and Morteza Zadimoghaddam. Edge-weighted online bipartite matching. J. ACM, 69(6):45:1-45:35, 2022. (Preliminary version in 61st FOCS, 2020). URL: https://doi.org/10.1145/3556971.
  17. Jon Feldman, Monika Henzinger, Nitish Korula, Vahab S. Mirrokni, and Clifford Stein. Online stochastic packing applied to display ad allocation. In Mark de Berg and Ulrich Meyer, editors, Proc. 18th Annual European Symp. of Algorithms (ESA), Part I, volume 6346 of LNCS, pages 182-194. Springer, 2010. URL: https://doi.org/10.1007/978-3-642-15775-2_16.
  18. Jon Feldman, Nitish Korula, Vahab S. Mirrokni, S. Muthukrishnan, and Martin Pál. Online ad assignment with free disposal. In Stefano Leonardi, editor, Proc. 5th International Workshop on Internet and Network Economics (WINE), volume 5929 of LNCS, pages 374-385. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-10841-9_34.
  19. Jon Feldman, Aranyak Mehta, Vahab S. Mirrokni, and S. Muthukrishnan. Online stochastic matching: Beating 1-(1)/(e). In Daniel A. Spielman, editor, Proc. 50th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 117-126, 2009. URL: https://doi.org/10.1109/FOCS.2009.72.
  20. Buddhima Gamlath, Michael Kapralov, Andreas Maggiori, Ola Svensson, and David Wajc. Online matching with general arrivals. In David Zuckerman, editor, Proc. 60th IEEE Symp. on Foundations of Comp. Science (FOCS), pages 26-37, 2019. URL: https://doi.org/10.1109/FOCS.2019.00011.
  21. Ruiquan Gao, Zhongtian He, Zhiyi Huang, Zipei Nie, Bijun Yuan, and Yan Zhong. Improved online correlated selection. In Nisheeth Vishnoi, editor, Proc. 62nd IEEE Symp. on Foundations of Comp. Science (FOCS), pages 1265-1276, 2021. URL: https://doi.org/10.1109/FOCS52979.2021.00123.
  22. Gagan Goel and Aranyak Mehta. Online budgeted matching in random input models with applications to adwords. In Shang-Hua Teng, editor, Proc. 19th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 982-991, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347189.
  23. Bernhard Haeupler, Vahab S. Mirrokni, and Morteza Zadimoghaddam. Online stochastic weighted matching: Improved approximation algorithms. In Ning Chen, Edith Elkind, and Elias Koutsoupias, editors, Proc. 7th International Workshop on Internet and Network Economics (WINE), volume 7090 of LNCS, pages 170-181. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-25510-6_15.
  24. Richard M. Karp, Umesh V. Vazirani, and Vijay V. Vazirani. An optimal algorithm for on-line bipartite matching. In Harriet Ortiz, editor, Proc. 22nd ACM Symp. on Theory of Computing (STOC), pages 352-358, 1990. URL: https://doi.org/10.1145/100216.100262.
  25. Harold W. Kuhn. The Hungarian method for the assignment problem. Naval Research Logistics Quarterly, 2(1-2):83-97, 1955. URL: https://doi.org/10.1002/nav.3800020109.
  26. Benny Lehmann, Daniel Lehmann, and Noam Nisan. Combinatorial auctions with decreasing marginal utilities. Games Econ. Behav., 55(2):270-296, 2006. URL: https://doi.org/10.1016/J.GEB.2005.02.006.
  27. Aranyak Mehta. Online matching and ad allocation. Found. Trends Theor. Comput. Sci., 8(4):265-368, 2013. URL: https://doi.org/10.1561/0400000057.
  28. Aranyak Mehta, Amin Saberi, Umesh V. Vazirani, and Vijay V. Vazirani. Adwords and generalized online matching. J. ACM, 54(5):22, 2007. (Preliminary version in 46th FOCS, 2005). URL: https://doi.org/10.1145/1284320.1284321.
  29. Joseph (Seffi) Naor and David Wajc. Near-optimum online ad allocation for targeted advertising. ACM Trans. Economics and Comput., 6(3-4):16:1-16:20, 2018. (Preliminary version in 16th EC, 2015). URL: https://doi.org/10.1145/3105447.
  30. Yongho Shin and Hyung-Chan An. Making three out of two: Three-way online correlated selection. In Hee-Kap Ahn and Kunihiko Sadakane, editors, Proc. 32nd International Symposium on Algorithms and Computation (ISAAC), volume 212 of LIPIcs, pages 49:1-49:17. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPICS.ISAAC.2021.49.
Any Issues?
X

Feedback on the Current Page

CAPTCHA

Thanks for your feedback!

Feedback submitted to Dagstuhl Publishing

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact