Document Open Access Logo

Distributed Maximum Matching Verification in CONGEST

Authors Mohamad Ahmadi, Fabian Kuhn

PDF
Thumbnail PDF

File

LIPIcs.DISC.2020.37.pdf
  • Filesize: 0.61 MB
  • 18 pages

Document Identifiers

Author Details

Mohamad Ahmadi
  • University of Freiburg, Germany
Fabian Kuhn
  • University of Freiburg, Germany

Cite AsGet BibTex

Mohamad Ahmadi and Fabian Kuhn. Distributed Maximum Matching Verification in CONGEST. In 34th International Symposium on Distributed Computing (DISC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 179, pp. 37:1-37:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.DISC.2020.37

Abstract

We study the maximum cardinality matching problem in a standard distributed setting, where the nodes V of a given n-node network graph G = (V,E) communicate over the edges E in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of G can send an O(log n)-bit message to each of its neighbors. We show that for every graph G and a matching M of G, there is a randomized CONGEST algorithm to verify M being a maximum matching of G in time O(|M|) and disprove it in time O(D + 𝓁), where D is the diameter of G and 𝓁 is the length of a shortest augmenting path. We hope that our algorithm constitutes a significant step towards developing a CONGEST algorithm to compute a maximum matching in time Õ(s^*), where s^* is the size of a maximum matching.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
Keywords
  • distributed matching
  • distributed graph algorithms
  • augmenting paths

Metrics

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

Related Versions

References

  1. M. Ahmadi and F. Kuhn. Distributed maximum matching verification in congest, 2020. URL: http://arxiv.org/abs/2002.07649.
  2. M. Ahmadi, F. Kuhn, and R. Oshman. Distributed approximate maximum matching in the CONGEST model. In Proc. 32nd Symp. on Distributed Computing (DISC), 2018. Google Scholar
  3. N. Alon, L. Babai, and A. Itai. A fast and simple randomized parallel algorithm for the maximal independent set problem. Journal of Algorithms, 7(4):567-583, 1986. Google Scholar
  4. H. Arfaoui, P. Fraigniaud, and A. Pelc. Local decision and verification with bounded-size outputs. In Symposium on Self-Stabilizing Systems. Springer, 2013. Google Scholar
  5. S. Assadi, M. Bateni, A. Bernstein, V. Mirrokni, and C. Stein. Coresets meet EDCS: Algorithms for matching and vertex cover on massive graphs. In Proc. 30th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1616-1635, 2019. Google Scholar
  6. N. Bachrach, K. Censor-Hillel, M. Dory, Y. Efron, D. Leitersdorf, and A. Paz. Hardness of distributed optimization. CoRR, abs/1905.10284, 2019. Google Scholar
  7. A. Balliu, S. J. Hirvonen, D. Olivetti, M. Rabie, and J. Suomela. Lower bounds for maximal matchings and maximal independent sets. In Proc. 60th IEEE Symp. on Foundations of Computer Science (FOCS), pages 481-497, 2019. Google Scholar
  8. R. Bar-Yehuda, K. Censor-Hillel, M. Ghaffari, and G. Schwartzman. Distributed approximation of maximum independent set and maximum matching. In Proceedings of the ACM Symposium on Principles of Distributed Computing (PODC), pages 165-174, 2017. Google Scholar
  9. L. Barenboim, M. Elkin, S. Pettie, and J. Schneider. The locality of distributed symmetry breaking. In Proceedings of 53th Symposium on Foundations of Computer Science (FOCS), 2012. Google Scholar
  10. R. Ben-Basat, K. Kawarabayashi, and G. Schwartzman. Parameterized Distributed Algorithms. In 33rd International Symposium on Distributed Computing (DISC 2019), Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, 2019. Google Scholar
  11. K. Censor-Hillel, S. Khoury, and A. Paz. Quadratic and near-quadratic lower bounds for the CONGEST model. In Proc. 31st Symp. on Distributed Computing (DISC), pages 10:1-10:16, 2017. Google Scholar
  12. A. Czumaj, J. Lacki, A. Madry, S. Mitrovic, K. Onak, and P. Sankowski. ound compression for parallel matching algorithms. In Proc. 50th ACM Symp. on Theory of Computing (STOC), pages 471-484, 2018. Google Scholar
  13. A. Czygrinow and M. Hańćkowiak. Distributed algorithm for better approximation of the maximum matching. In 9th Annual International Computing and Combinatorics Conference (COCOON), pages 242-251, 2003. Google Scholar
  14. A. Czygrinow, M. Hańćkowiak, and E. Szymanska. A fast distributed algorithm for approximating the maximum matching. In Proceedings of 12th Annual European Symposium on Algorithms (ESA), pages 252-263, 2004. Google Scholar
  15. A. Das Sarma, S. Holzer, L. Kor, A. Korman, D. Nanongkai, G. Pandurangan, D. Peleg, and R. Wattenhofer. Distributed verification and hardness of distributed approximation. In Proc. 43rd Symp. on Theory of Computing (STOC), pages 363-372, 2011. Google Scholar
  16. J. Edmonds. Maximum matching and a polyhedron with 0, l-vertices. J. of Res. the Nat. Bureau of Standards, 69 B:125-130, 1965. Google Scholar
  17. J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17(3):449-467, 1965. Google Scholar
  18. Y. Emek, S. Kutten, and R. Wattenhofer. Online matching: haste makes waste! In Proc. 48th Symp. on Theory of Computing (STOC), pages 333-344, 2016. Google Scholar
  19. G. Even, M. Medina, and D. Ron. Distributed maximum matching in bounded degree graphs. In Proceedings of the 2015 International Conference on Distributed Computing and Networking (ICDCN), pages 18:1-18:10, 2015. Google Scholar
  20. L. Feuilloley and P. Fraigniaud. Survey of distributed decision. Bulletin of the EATCS, 2016. Google Scholar
  21. M. Fischer. Improved deterministic distributed matching via rounding. In Proc. 31st Symp. on Distributed Computing (DISC), pages 17:1-17:15, 2017. Google Scholar
  22. M. Fischer, M. Ghaffari, and F. Kuhn. Deterministic distributed edge-coloring via hypergraph maximal matching. In Proceedings of 58th IEEE Annual Symposium on Foundations of Computer Science (FOCS), pages 180-191, 2017. Google Scholar
  23. T. Fischer, A. V. Goldberg, D. J. Haglin, and S. Plotkin. Approximating matchings in parallel. Information Processing Letters, 46(3):115-118, 1993. Google Scholar
  24. M. Ghaffari, D. G. Harris, and F. Kuhn. On derandomizing local distributed algorithms. In Proc. 59th Symp. on Foundations of Computer Science (FOCS), pages 662-673, 2018. Google Scholar
  25. M. Ghaffari and J. Uitto. Sparsifying distributed algorithms with ramifications in massively parallel computation and centralized local computation. In Proc. 30th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1636-1653, 2019. Google Scholar
  26. M. Hańćkowiak, M. Karoński, and A. Panconesi. On the distributed complexity of computing maximal matchings. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 219-225, 1998. Google Scholar
  27. M. Hańćkowiak, M. Karoński, and A. Panconesi. A faster distributed algorithm for computing maximal matchings deterministically. In Proceedings of the Eighteenth Annual ACM Symposium on Principles of Distributed Computing (PODC), pages 219-228, 1999. Google Scholar
  28. J.H. Hoepman, S. Kutten, and Z. Lotker. Efficient distributed weighted matchings on trees. In Proceedings of 13th International Colloquium on Structural Information and Communication Complexity (SIROCCO), pages 115-129, 2006. Google Scholar
  29. J. E. Hopcroft and R. M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM Journal on Computing, 1973. Google Scholar
  30. A. Israeli and A. Itai. A fast and simple randomized parallel algorithm for maximal matching. Inf. Process. Lett., 22(2):77-80, 1986. Google Scholar
  31. R. M. Karp, E. Upfal, and A. Widgerson. Constructing a perfect matching is in random NC. In Proc. 17th ACM Symp. on Theory of Computing (STOC), pages 22-32, 1985. Google Scholar
  32. R. M. Karp, U. V. Vazirani, and V. V. Vazirani. An optimal algorithm for on-line bipartite matching. In Proc. 22nd ACM Symp. on Theory of Computing (STOC), pages 352-358, 1990. Google Scholar
  33. L. Kor, A. Korman, and D. Peleg. Tight Bounds For Distributed MST Verification. In 28th International Symposium on Theoretical Aspects of Computer Science (STACS11), 2011. Google Scholar
  34. F. Kuhn, T. Moscibroda, and R. Wattenhofer. Local computation: Lower and upper bounds. J. of the ACM, 63(2), 2016. Google Scholar
  35. N. Linial. Locality in distributed graph algorithms. SIAM Journal on Computing, 21(1):193-201, 1992. Google Scholar
  36. Z. Lotker, B. Patt-Shamir, and S. Pettie. Improved distributed approximate matching. In Proceedings of the 20th Annual ACM Symposium on Parallelism in Algorithms and Architectures (SPAA), pages 129-136, 2008. Google Scholar
  37. Z. Lotker, B. Patt-Shamir, and A. Rosén. Distributed approximate matching. SIAM Journal on Computing, 39(2):445-460, 2009. Google Scholar
  38. L. Lovász and M.D. Plummer. Matching Theory. North-Holand, 1986. Google Scholar
  39. M. Luby. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 15:1036-1053, 1986. Google Scholar
  40. Y. Mansour and S. Vardi. A local computation approximation scheme to maximum matching. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, pages 260-273, 2013. Google Scholar
  41. A. McGregor. Finding graph matchings in data streams. In Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, pages 170-181, 2005. Google Scholar
  42. T. Nieberg. Local, distributed weighted matching on general and wireless topologies. In Proceedings of the DIALM-POMC Joint Workshop on Foundations of Mobile Computing, pages 87-92, 2008. Google Scholar
  43. A. Panconesi and R. Rizzi. Some simple distributed algorithms for sparse networks. Distributed Computing, 14(2):97-100, 2001. Google Scholar
  44. D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. Google Scholar
  45. V. V. Vazirani. A simplification of the mv matching algorithm and its proof. CoRR, abs/1210.4594, 2012. Google Scholar
  46. M. Wattenhofer and R. Wattenhofer. Distributed weighted matching. In Proc. 18th Conf. on Distributed Computing (DISC), pages 335-348, 2004. Google Scholar
  47. Y. Yoshida, M. Yamamoto, and H. Ito. An improved constant-time approximation algorithm for maximum matchings. In Proc. 41st ACM Symp. on Theory of Computing (STOC), pages 225-234, 2009. 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
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact