Document Open Access Logo

Scalable Auction Algorithms for Bipartite Maximum Matching Problems

Authors Quanquan C. Liu , Yiduo Ke , Samir Khuller

Thumbnail PDF


  • Filesize: 0.91 MB
  • 24 pages

Document Identifiers

Author Details

Quanquan C. Liu
  • Northwestern University, Evanston, IL, USA
Yiduo Ke
  • Northwestern University, Evanston, IL, USA
Samir Khuller
  • Northwestern University, Evanston, IL, USA

Cite AsGet BibTex

Quanquan C. Liu, Yiduo Ke, and Samir Khuller. Scalable Auction Algorithms for Bipartite Maximum Matching Problems. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 275, pp. 28:1-28:24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)


Bipartite maximum matching and its variants are well-studied problems under various models of computation with the vast majority of approaches centering around various methods to find and eliminate augmenting paths. Beginning with the seminal papers of Demange, Gale and Sotomayor [DGS86] and Bertsekas [Ber81], bipartite maximum matching problems have also been studied in the context of auction algorithms. These algorithms model the maximum matching problem as an auction where one side of the bipartite graph consists of bidders and the other side consists of items; as such, these algorithms offer a very different approach to solving this problem that do not use classical methods. Dobzinski, Nisan and Oren [DNO14] demonstrated the utility of such algorithms in distributed, interactive settings by providing a simple and elegant O(log n/ε²) round maximum cardinality bipartite matching (MCM) algorithm that has small round and communication complexity and gives a (1-ε)-approximation for any (not necessarily constant) ε > 0. They leave as an open problem whether an auction algorithm, with similar guarantees, can be found for the maximum weighted bipartite matching (MWM) problem. Very recently, Assadi, Liu, and Tarjan [ALT21] extended the utility of auction algorithms for MCM into the semi-streaming and massively parallel computation (MPC) models, by cleverly using maximal matching as a subroutine, to give a new auction algorithm that uses O(1/ε²) rounds and achieves the state-of-the-art bipartite MCM results in the streaming and MPC settings. In this paper, we give new auction algorithms for maximum weighted bipartite matching (MWM) and maximum cardinality bipartite b-matching (MCbM). Our algorithms run in O(log n/ε⁸) and O(log n/ε²) rounds, respectively, in the distributed setting. We show that our MWM algorithm can be implemented in the distributed, interactive setting using O(log² n) and O(log n) bit messages, respectively, directly answering the open question posed by Demange, Gale and Sotomayor [DNO14]. Furthermore, we implement our algorithms in a variety of other models including the the semi-streaming model, the shared-memory work-depth model, and the massively parallel computation model. Our semi-streaming MWM algorithm uses O(1/ε⁸) passes in O(n log n ⋅ log(1/ε)) space and our MCbM algorithm runs in O(1/ε²) passes using O((∑_{i ∈ L} b_i + |R|) log(1/ε)) space (where parameters b_i represent the degree constraints on the b-matching and L and R represent the left and right side of the bipartite graph, respectively). Both of these algorithms improves exponentially the dependence on ε in the space complexity in the semi-streaming model against the best-known algorithms for these problems, in addition to improvements in round complexity for MCbM. Finally, our algorithms eliminate the large polylogarithmic dependence on n in depth and number of rounds in the work-depth and massively parallel computation models, respectively, improving on previous results which have large polylogarithmic dependence on n (and exponential dependence on ε in the MPC model).

Subject Classification

ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • auction algorithms
  • maximum weight bipartite matching
  • maximum b-matching
  • distributed blackboard model
  • parallel work-depth model
  • streaming model
  • massively parallel computation model


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


  1. Kook Jin Ahn and Sudipto Guha. Linear programming in the semi-streaming model with application to the maximum matching problem. In Proceedings of the 38th International Conference on Automata, Languages and Programming - Volume Part II, ICALP'11, pages 526-538, Berlin, Heidelberg, 2011. Springer-Verlag. Google Scholar
  2. Kook Jin Ahn and Sudipto Guha. Access to data and number of iterations: Dual primal algorithms for maximum matching under resource constraints. ACM Trans. Parallel Comput., 4(4), January 2018. URL:
  3. Sepehr Assadi. A two-pass (conditional) lower bound for semi-streaming maximum matching. In Joseph (Seffi) Naor and Niv Buchbinder, editors, Proceedings of the 2022 ACM-SIAM Symposium on Discrete Algorithms, SODA 2022, Virtual Conference / Alexandria, VA, USA, January 9 - 12, 2022, pages 708-742. SIAM, 2022. URL:
  4. Sepehr Assadi, Arun Jambulapati, Yujia Jin, Aaron Sidford, and Kevin Tian. Semi-streaming bipartite matching in fewer passes and optimal space. In Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 627-669. SIAM, 2022. Google Scholar
  5. Sepehr Assadi, Gillat Kol, Raghuvansh R. Saxena, and Huacheng Yu. Multi-pass graph streaming lower bounds for cycle counting, max-cut, matching size, and other problems. In 2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), pages 354-364, 2020. URL:
  6. Sepehr Assadi, S. Cliff Liu, and Robert E. Tarjan. An Auction Algorithm for Bipartite Matching in Streaming and Massively Parallel Computation Models, pages 165-171. SIAM, 2021. URL:
  7. Aaron Bernstein, Aditi Dudeja, and Zachary Langley. A framework for dynamic matching in weighted graphs. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC 2021, pages 668-681, New York, NY, USA, 2021. Association for Computing Machinery. URL:
  8. Dimitri P Bertsekas. A new algorithm for the assignment problem. Mathematical Programming, 21(1):152-171, 1981. Google Scholar
  9. Guy E. Blelloch, Jeremy T. Fineman, and Julian Shun. Greedy sequential maximal independent set and matching are parallel on average. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 308-317, 2012. Google Scholar
  10. Gabrielle Demange, David Gale, and Marilda Sotomayor. Multi-item auctions. Journal of political economy, 94(4):863-872, 1986. Google Scholar
  11. Shahar Dobzinski, Noam Nisan, and Sigal Oren. Economic efficiency requires interaction. In Proceedings of the Forty-Sixth Annual ACM Symposium on Theory of Computing, STOC '14, pages 233-242, New York, NY, USA, 2014. Association for Computing Machinery. URL:
  12. Jack Edmonds. Maximum matching and a polyhedron with 0, 1-vertices. Journal of Research of the National Bureau of Standards B, 69:125-130, 1965. Google Scholar
  13. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17:449-467, 1965. Google Scholar
  14. Manuela Fischer, Slobodan Mitrović, and Jara Uitto. Deterministic (1+ε)-approximate maximum matching with poly(1/ε) passes in the semi-streaming model and beyond. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 248-260, New York, NY, USA, 2022. Association for Computing Machinery. URL:
  15. Buddhima Gamlath, Sagar Kale, Slobodan Mitrovic, and Ola Svensson. Weighted matchings via unweighted augmentations. In Proceedings of the 2019 ACM Symposium on Principles of Distributed Computing, PODC '19, pages 491-500, New York, NY, USA, 2019. Association for Computing Machinery. URL:
  16. Mohsen Ghaffari, Christoph Grunau, and Slobodan Mitrović. Massively parallel algorithms for b-matching. In ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), 2022. Google Scholar
  17. Ashish Goel, Michael Kapralov, and Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching, pages 468-485. SIAM, 2012. URL:
  18. Manoj Gupta and Richard Peng. Fully dynamic (1+ e)-approximate matchings. In FOCS, pages 548-557. IEEE Computer Society, 2013. Google Scholar
  19. Nicholas J. A. Harvey. Algebraic structures and algorithms for matching and matroid problems. 2006 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS'06), pages 531-542, 2006. Google Scholar
  20. John E. Hopcroft and Richard M. Karp. A n5/2 algorithm for maximum matchings in bipartite. In 12th Annual Symposium on Switching and Automata Theory (swat 1971), pages 122-125, 1971. URL:
  21. Shang-En Huang and Hsin-Hao Su. (1-ε)-approximate maximum weighted matching in poly(1/ε, log n) time in the distributed and parallel settings. CoRR, abs/2212.14425, 2022. URL:
  22. Arun Jambulapati, Yang P. Liu, and Aaron Sidford. Parallel reachability in almost linear work and square root depth. In 2019 IEEE 60th Annual Symposium on Foundations of Computer Science (FOCS), pages 1664-1686, 2019. URL:
  23. Michael Kapralov. Better bounds for matchings in the streaming model. In Proceedings of the twenty-fourth annual ACM-SIAM symposium on Discrete algorithms, pages 1679-1697. SIAM, 2013. Google Scholar
  24. Christian Konrad, Peter Robinson, and Viktor Zamaraev. Robust lower bounds for graph problems in the blackboard model of communication. CoRR, abs/2103.07027, 2021. URL:
  25. D. König. Über graphen und ihre anwendung auf determinantentheorie und mengenlehre. Mathematische Annalen, 77:453-465, 1916. URL:
  26. S Cliff Liu, Zhao Song, and Hengjie Zhang. Breaking the n-pass barrier: A streaming algorithm for maximum weight bipartite matching. arXiv preprint, 2020. URL:
  27. Yang P. Liu and Aaron Sidford. Faster Energy Maximization for Faster Maximum Flow, pages 803-814. Association for Computing Machinery, New York, NY, USA, 2020. URL:
  28. Zvi Lotker, Boaz Patt-Shamir, and Seth Pettie. Improved distributed approximate matching. J. ACM, 62(5), November 2015. URL:
  29. Aleksander Madry. Navigating central path with electrical flows: From flows to matchings, and back. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science, pages 253-262, 2013. URL:
  30. Silvio Micali and Vijay V. Vazirani. An o(√|v| ⋅ |e|) algorithm for finding maximum matching in general graphs. In 21st Annual Symposium on Foundations of Computer Science (sfcs 1980), pages 17-27, 1980. URL:
  31. M. Mucha and P. Sankowski. Maximum matchings via gaussian elimination. In 45th Annual IEEE Symposium on Foundations of Computer Science, pages 248-255, 2004. URL:
  32. Noam Nisan. The demand query model for bipartite matching. In Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 592-599. SIAM, 2021. Google Scholar
  33. Daniel Stubbs and Virginia Vassilevska Williams. Metatheorems for dynamic weighted matching. In Christos H. Papadimitriou, editor, 8th Innovations in Theoretical Computer Science Conference, ITCS 2017, January 9-11, 2017, Berkeley, CA, USA, volume 67 of LIPIcs, pages 58:1-58:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2017. URL:
  34. Da Wei Zheng and Monika Henzinger. Multiplicative auction algorithm for approximate maximum weight bipartite matching. arXiv preprint, 2023. URL:
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail