Document Open Access Logo

Algebraic Representations of Unique Bipartite Perfect Matching

Author Gal Beniamini

Thumbnail PDF


  • Filesize: 0.86 MB
  • 17 pages

Document Identifiers

Author Details

Gal Beniamini
  • The Hebrew University of Jerusalem, Israel

Cite AsGet BibTex

Gal Beniamini. Algebraic Representations of Unique Bipartite Perfect Matching. In 47th International Symposium on Mathematical Foundations of Computer Science (MFCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 241, pp. 16:1-16:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We obtain complete characterizations of the Unique Bipartite Perfect Matching function, and of its Boolean dual, using multilinear polynomials over the reals. Building on previous results [Beniamini, 2020; Beniamini and Nisan, 2021], we show that, surprisingly, the dual description is sparse and has low 𝓁₁-norm - only exponential in Θ(n log n), and this result extends even to other families of matching-related functions. Our approach relies on the Möbius numbers in the matching-covered lattice, and a key ingredient in our proof is Möbius' inversion formula. These polynomial representations yield complexity-theoretic results. For instance, we show that unique bipartite matching is evasive for classical decision trees, and nearly evasive even for generalized query models. We also obtain a tight Θ(n log n) bound on the log-rank of the associated two-party communication task.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Matchings and factors
  • Theory of computation → Communication complexity
  • Theory of computation → Oracles and decision trees
  • Bipartite Perfect Matching
  • Boolean Functions
  • Partially Ordered Sets


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


  1. Scott Aaronson, Shalev Ben-David, Robin Kothari, Shravas Rao, and Avishay Tal. Degree vs. approximate degree and quantum implications of Huang’s sensitivity theorem. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, 2021. Google Scholar
  2. Gal Beniamini. The approximate degree of bipartite perfect matching. arXiv preprint, 2020. URL:
  3. Gal Beniamini and Noam Nisan. Bipartite perfect matching as a real polynomial. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, 2021. Google Scholar
  4. Anna Bernasconi and Bruno Codenotti. Spectral analysis of boolean functions as a graph eigenvalue problem. IEEE transactions on computers, 48(3):345-351, 1999. Google Scholar
  5. Louis J Billera and Aravamuthan Sarangarajan. The combinatorics of permutation polytopes. In Formal power series and algebraic combinatorics, volume 24, pages 1-23, 1994. Google Scholar
  6. Harry Buhrman and Ronald De Wolf. Complexity measures and decision tree complexity: a survey. Theoretical Computer Science, 288(1):21-43, 2002. Google Scholar
  7. Mark Bun and Justin Thaler. Guest column: Approximate degree in classical and quantum computing. ACM SIGACT News, 51(4):48-72, 2021. Google Scholar
  8. Martin Dietzfelbinger, Juraj Hromkovič, and Georg Schnitger. A comparison of two lower-bound methods for communication complexity. Theoretical Computer Science, 168(1):39-51, 1996. Google Scholar
  9. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of mathematics, 17:449-467, 1965. Google Scholar
  10. Stephen Fenner, Rohit Gurjar, and Thomas Thierauf. Bipartite perfect matching is in Quasi-NC. SIAM Journal on Computing, 50(3):STOC16-218, 2019. Google Scholar
  11. Harold N Gabow, Haim Kaplan, and Robert E Tarjan. Unique maximum matching algorithms. Journal of Algorithms, 40(2):159-183, 2001. Google Scholar
  12. Martin Charles Golumbic, Tirza Hirst, and Moshe Lewenstein. Uniquely restricted matchings. Algorithmica, 31(2):139-154, 2001. Google Scholar
  13. Gábor Hetyei. Rectangular configurations which can be covered by 2×1 rectangles. Pécsi Tan. Foisk. Közl, 8:351-367, 1964. Google Scholar
  14. Thanh Minh Hoang, Meena Mahajan, and Thomas Thierauf. On the bipartite unique perfect matching problem. In International Colloquium on Automata, Languages, and Programming, pages 453-464. Springer, 2006. Google Scholar
  15. John E Hopcroft and Richard M Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on computing, 2(4):225-231, 1973. Google Scholar
  16. Hao Huang. Induced subgraphs of hypercubes and a proof of the sensitivity conjecture. Annals of Mathematics, 190(3):949-955, 2019. Google Scholar
  17. Jeff Kahn, Michael Saks, and Dean Sturtevant. A topological approach to evasiveness. Combinatorica, 4(4):297-306, 1984. Google Scholar
  18. Alexander Knop, Shachar Lovett, Sam McGuire, and Weiqiang Yuan. Log-rank and lifting for AND-functions. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, pages 197-208, 2021. Google Scholar
  19. Dexter Kozen, Umesh V Vazirani, and Vijay V Vazirani. NC algorithms for comparability graphs, interval graphs, and testing for unique perfect matching. In International Conference on Foundations of Software Technology and Theoretical Computer Science, pages 496-503. Springer, 1985. Google Scholar
  20. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, USA, 1996. Google Scholar
  21. László Lovász. On determinants, matchings, and random algorithms. In FCT, volume 79, pages 565-574, 1979. Google Scholar
  22. László Lovász and Michael Saks. Lattices, Möbius functions and communications complexity. In [Proceedings 1988] 29th Annual Symposium on Foundations of Computer Science, pages 81-90. IEEE Computer Society, 1988. Google Scholar
  23. Laszlo Lovasz and Neal E Young. Lecture notes on evasiveness of graph properties. arXiv preprint cs/0205031, 2002. Google Scholar
  24. Kurt Mehlhorn and Erik M Schmidt. Las vegas is better than determinism in VLSI and distributed computing. In Proceedings of the fourteenth annual ACM symposium on Theory of computing, pages 330-337, 1982. Google Scholar
  25. 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
  26. Noam Nisan and Avi Wigderson. On rank vs. communication complexity. Combinatorica, 15(4):557-565, 1995. Google Scholar
  27. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  28. M.D. Plummer and L. Lovász. Matching Theory. North-Holland Mathematics Studies. Elsevier Science, 1986. Google Scholar
  29. Alexander A Sherstov. Algorithmic polynomials. SIAM Journal on Computing, 49(6):1173-1231, 2020. Google Scholar
  30. Richard P. Stanley. Enumerative Combinatorics: Volume 1. Cambridge University Press, New York, NY, USA, 2nd edition, 2011. Google Scholar
  31. Andrew Chi-Chih Yao. Monotone bipartite graph properties are evasive. SIAM Journal on Computing, 17(3):517-520, 1988. Google Scholar
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