Document Open Access Logo

Unique perfect matchings and proof nets

Author Lê Thành Dung Nguyen

Thumbnail PDF


  • Filesize: 0.54 MB
  • 20 pages

Document Identifiers

Author Details

Lê Thành Dung Nguyen
  • Département d'informatique, École normale supérieure, Paris Sciences et Lettres , 45 rue d'Ulm, 75005 Paris, France, Université Paris 13, Sorbonne Paris Cité, LIPN, CNRS , 99 avenue Jean-Baptiste Clément, 93430 Villetaneuse, France

Cite AsGet BibTex

Lê Thành Dung Nguyen. Unique perfect matchings and proof nets. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 25:1-25:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


This paper establishes a bridge between linear logic and mainstream graph theory, building previous work by Retoré (2003). We show that the problem of correctness for MLL+Mix proof nets is equivalent to the problem of uniqueness of a perfect matching. By applying matching theory, we obtain new results for MLL+Mix proof nets: a linear-time correctness criterion, a quasi-linear sequentialization algorithm, and a characterization of the sub-polynomial complexity of the correctness problem. We also use graph algorithms to compute the dependency relation of Bagnol et al. (2015) and the kingdom ordering of Bellin (1997), and relate them to the notion of blossom which is central to combinatorial maximum matching algorithms.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Mathematics of computing → Matchings and factors
  • Mathematics of computing → Graph algorithms
  • correctness criteria
  • matching algorithms


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


  1. Marc Bagnol, Amina Doumane, and Alexis Saurin. On the dependencies of logical rules. In FOSSACS, 18th International Conference on Foundations of Software Science and Computation Structures, 2015. Google Scholar
  2. Jørgen Bang-Jensen and Gregory Gutin. Digraphs. Theory, algorithms and applications. 2nd ed. London: Springer, 2nd ed. edition, 2009. Google Scholar
  3. David A. Mix Barrington. Quasipolynomial size circuit classes. In [1992] Proceedings of the Seventh Annual Structure in Complexity Theory Conference, pages 86-93, 1992. Google Scholar
  4. Gianluigi Bellin. Subnets of proof-nets in multiplicative linear logic with MIX. Mathematical Structures in Computer Science, 7(6):663-669, 1997. Google Scholar
  5. Gianluigi Bellin and Jacques van de Wiele. Subnets of Proof-nets in MLL-. In Proceedings of the Workshop on Advances in Linear Logic, pages 249-270, New York, NY, USA, 1995. Cambridge University Press. Google Scholar
  6. Ashok K. Chandra, Larry Stockmeyer, and Uzi Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13(2):423-439, 1984. Google Scholar
  7. Vincent Danos. La Logique Linéaire appliquée à l'étude de divers processus de normalisation (principalement du Lambda-calcul). PhD thesis, Université Paris-Diderot – Paris VII, 1990. Google Scholar
  8. Vincent Danos and Laurent Regnier. The structure of multiplicatives. Archive for Mathematical Logic, 28(3):181-203, 1989. Google Scholar
  9. Jack Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17(0):449-467, 1965. Google Scholar
  10. Arnaud Fleury and Christian Retoré. The mix rule. Mathematical Structures in Computer Science, 4(2):273-285, 1994. 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. Harold N. Gabow and Robert Endre Tarjan. A linear-time algorithm for a special case of disjoint set union. Journal of Computer and System Sciences, 30(2):209-221, apr 1985. Google Scholar
  13. Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1-101, jan 1987. Google Scholar
  14. Stefano Guerrini. A linear algorithm for MLL proof net correctness and sequentialization. Theoretical Computer Science, 412(20):1958-1978, apr 2011. Google Scholar
  15. Thanh Minh Hoang, Meena Mahajan, and Thomas Thierauf. On the bipartite unique perfect matching problem. In Automata, Languages and Programming, 33rd International Colloquium, ICALP 2006, Venice, Italy, July 10-14, 2006, Proceedings, Part I, pages 453-464, 2006. Google Scholar
  16. Jacob Holm, Eva Rotenberg, and Mikkel Thorup. Dynamic bridge-finding in Õ(log² n) amortized time. In Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, Proceedings, pages 35-52. Society for Industrial and Applied Mathematics, jan 2018. Google Scholar
  17. Paulin Jacobé de Naurois and Virgile Mogbil. Correctness of linear logic proof structures is NL-complete. Theoretical Computer Science, 412(20):1941-1957, apr 2011. Google Scholar
  18. Anton Kotzig. Z teórie konečných grafov s lineárnym faktorom. II. Matematicko-fyzikálny časopis, 09(3):136-159, 1959. 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 Foundations of Software Technology and Theoretical Computer Science, Fifth Conference, New Delhi, India, December 16-18, 1985, Proceedings, pages 496-503, 1985. Google Scholar
  20. László Lovász. On determinants, matchings, and random algorithms. In Fundamentals of Computation Theory, pages 565-574, 1979. Google Scholar
  21. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. Google Scholar
  22. Andrzej S. Murawski and C.-H. Luke Ong. Fast verification of MLL proof nets via IMLL. ACM Transactions on Computational Logic, 7(3):473-498, 2006. Google Scholar
  23. Lê Thành Dũng Nguyễn. On the complexity of finding cycles in proof nets. Extended abstract for the workshop Developments in Implicit Computational Complexity 2018. Google Scholar
  24. Michele Pagani. Visible acyclic differential nets, Part I: Semantics. Annals of Pure and Applied Logic, 163(3):238-265, 2012. Google Scholar
  25. Michael O. Rabin and Vijay V. Vazirani. Maximum matchings in general graphs through randomization. Journal of Algorithms, 10(4):557-567, dec 1989. Google Scholar
  26. Christian Retoré. Handsome proof-nets: R&B-graphs, perfect matchings and series-parallel graphs. Research Report, INRIA, 1999. Google Scholar
  27. Christian Retoré. Handsome proof-nets: perfect matchings and cographs. Theoretical Computer Science, 294(3):473-488, 2003. Google Scholar
  28. Ola Svensson and Jakub Tarnawski. The matching problem in general graphs is in quasi-NC. In Chris Umans, editor, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017, pages 696-707. IEEE Computer Society, 2017. Google Scholar
  29. Stefan Szeider. Finding paths in graphs avoiding forbidden transitions. Discrete Applied Mathematics, 126(2-3):261-273, 2003. Google Scholar
  30. Stefan Szeider. On theorems equivalent with Kotzig’s result on graphs with unique 1-factors. Ars Combinatoria, 73:53-64, 2004. Google Scholar
  31. Robert Endre Tarjan. Data Structures and Network Algorithms. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1983. 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