Document Open Access Logo

Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings

Authors Salwa Faour, Marc Fuchs, Fabian Kuhn

Thumbnail PDF


  • Filesize: 0.76 MB
  • 20 pages

Document Identifiers

Author Details

Salwa Faour
  • University of Freiburg, Germany
Marc Fuchs
  • University of Freiburg, Germany
Fabian Kuhn
  • University of Freiburg, Germany

Cite AsGet BibTex

Salwa Faour, Marc Fuchs, and Fabian Kuhn. Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings. In 25th International Conference on Principles of Distributed Systems (OPODIS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 217, pp. 17:1-17:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


We provide CONGEST model algorithms for approximating the minimum weighted vertex cover and the maximum weighted matching problem. For bipartite graphs, we show that a (1+ε)-approximate weighted vertex cover can be computed deterministically in poly((log n)/ε) rounds. This generalizes a corresponding result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS '20]. Moreover, we show that in general weighted graph families that are closed under taking subgraphs and in which we can compute an independent set of weight at least λ⋅ w(V) (where w(V) denotes the total weight of all nodes) in polylogarithmic time in the CONGEST model, one can compute a (2-2λ +ε)-approximate weighted vertex cover in poly((log n)/ε) rounds in the CONGEST model. Our result in particular implies that in graphs of arboricity a, one can compute a (2-1/a+ε)-approximate weighted vertex cover problem in poly((log n)/ε) rounds in the CONGEST model. For maximum weighted matchings, we show that a (1-ε)-approximate solution can be computed deterministically in time 2^{O(1/ε)}⋅ polylog n in the CONGEST model. We also provide a randomized algorithm that with arbitrarily good constant probability succeeds in computing a (1-ε)-approximate weighted matching in time 2^{O(1/ε)}⋅ polylog(Δ W)⋅ log^* n, where W denotes the ratio between the largest and the smallest edge weight. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie; SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17], who gave 2^{O(1/ε)}⋅ log n and 2^{O(1/ε)}⋅ (logΔ)/(log logΔ)-round randomized approximations for the unweighted matching problem. Finally, we show that even in the LOCAL model and in bipartite graphs of degree ≤ 3, if ε < ε₀ for some constant ε₀ > 0, then computing a (1+ε)-approximation for the unweighted minimum vertex cover problem requires Ω((log n)/ε) rounds. This generalizes a result of [Göös, Suomela; DISC '12], who showed that computing a (1+ε₀)-approximation in such graphs requires Ω(log n) rounds.

Subject Classification

ACM Subject Classification
  • Theory of computation → Distributed algorithms
  • distributed graph algorithms
  • minimum weighted vertex cover
  • maximum weighted matching
  • distributed optimization
  • CONGEST model


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


  1. M. Ahmadi and F. Kuhn. Distributed maximum matching verification in CONGEST. In Proc. 34th Symp. on Distributed Computing (DISC), pages 37:1-37:18, 2020. Google Scholar
  2. M. Ahmadi, F. Kuhn, and R. Oshman. Distributed approximate maximum matching in the CONGEST model. In Proc. 32nd Symp. on Distr. Computing (DISC), pages 6:1-6:17, 2018. Google Scholar
  3. 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
  4. M. Åstrand, P. Floréen, V. Polishchuk, J. Rybicki, J. Suomela, and J. Uitto. A local 2-approximation algorithm for the vertex cover problem. In Proc. 23rd Symp. on Distributed Computing (DISC), pages 191-205, 2009. Google Scholar
  5. M. Åstrand and J. Suomela. Fast distributed approximation algorithms for vertex cover and set cover in anonymous networks. In Proc. 22nd ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pages 294-302, 2010. Google Scholar
  6. B. Awerbuch and D. Peleg. Sparse partitions. In Proc. 31st IEEE Symp. on Foundations of Computer Science (FOCS), pages 503-513, 1990. Google Scholar
  7. R. Bar-Yehuda, K. Censor-Hillel, M. Ghaffari, and G. Schwartzman. Distributed approximation of maximum independent set and maximum matching. In Proc. 36th ACM Symp. on Principles of Distributed Computing (PODC), full version:, pages 165-174, 2017.
  8. R. Bar-Yehuda, K. Censor-Hillel, Y. Maus, S. Pai, and S. V. Pemmaraju. Distributed approximation on power graphs. In Proc. 39th ACM Symp. on Principles of Distributed Computing (PODC), pages 501-510, 2020. Google Scholar
  9. R. Bar-Yehuda, K. Censor-Hillel, and G. Schwartzman. A distributed (2+ε)-approximation for vertex cover in O(logΔ/ε log log Δ) rounds. In Proc. 35th ACM Symp. on Principles of Distributed Computing (PODC), pages 3-8, 2016. Google Scholar
  10. R. Ben-Basat, G. Even, Kawarabayashi K, and G. Schwartzman. A deterministic distributed 2-approximation for weighted vertex cover in O(log N log Δ /log²logΔ) rounds. In Proc. 25th Coll. on Structural Information and Comm. Complexity (SIROCCO), pages 226-236, 2018. Google Scholar
  11. R. Ben-Basat, G. Even, K. Kawarabayashi, and G. Schwartzman. Optimal distributed covering algorithms. In Proc. 33rd Symp. on Distributed Computing (DISC), pages 5:1-5:15, 2019. Google Scholar
  12. Ran Ben-Basat, Ken ichi Kawarabayashi, and Gregory Schwartzman. Parameterized distributed algorithms. In Proc. 33rd In. Symp. on Distributed Computing (DISC), pages 6:1-6:16, 2019. Google Scholar
  13. K. Censor-Hillel, S. Khoury, and A. Paz. Quadratic and near-quadratic lower bounds for the CONGEST model. In Proc. 31st Symp. on Distr. Computing (DISC), pages 10:1-10:16, 2017. Google Scholar
  14. A. Czygrinow and M. Hańćkowiak. Distributed algorithm for better approximation of the maximum matching. In 9th Annual Int. Computing and Combinatorics Conf. (COCOON), pages 242-251, 2003. Google Scholar
  15. A. Czygrinow, M. Hańćkowiak, and E. Szymanska. A fast distributed algorithm for approximating the maximum matching. In Proceedings of 12th Annual European Symp. on Algorithms (ESA), pages 252-263, 2004. Google Scholar
  16. R. Diestel. Graph Theory, chapter 2.1. Springer, Berlin, 3rd edition, 2005. Google Scholar
  17. I. Dinur and S. Safra. On the hardness of approximating vertex cover. Annals of Mathematics, 162(1):439-485, 2005. Google Scholar
  18. 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
  19. J. Edmonds. Paths, trees, and flowers. Canadian Journal of Mathematics, 17(3):449-467, 1965. Google Scholar
  20. J. Egerváry. Matrixok kombinatorius tulajdonsáairól [On combinatorial properties of matrices]. Matematikai és Fizikai Lapok (in Hungarian), 38(16-28), 1931. Google Scholar
  21. G. Even, M. Medina, and D. Ron. Distributed maximum matching in bounded degree graphs. In Proceedings of the 2015 Int. Conf. on Distributed Computing and Networking (ICDCN), pages 18:1-18:10, 2015. Google Scholar
  22. S. Faour, M. Fuchs, and F. Kuhn. Distributed CONGEST approximation of weighted vertex covers and matchings. CoRR, abs/2111.10577, 2021. URL:
  23. S. Faour and F. Kuhn. Approximating bipartite minimum vertex cover in the CONGEST model. In Proc. 24th Conf. on Principles of Distr. Systems (OPODIS), pages 29:1-29:16, 2020. Google Scholar
  24. U. Feige, Y. Mansour, and R.~E. Schapire. Learning and inference in the presence of corrupted inputs. In Proc. 28th Conf. on Learning Theory (COLT), pages 637-657, 2015. Google Scholar
  25. M. Fischer. Improved deterministic distributed matching via rounding. In Proc. 31st Symp. on Distributed Computing (DISC), pages 17:1-17:15, 2017. Google Scholar
  26. M. Fischer, S. Mitrovic, and J. Uitto. Deterministic (1+ε)-approximate maximum matching with poly(1/ε) passes in the semi-streaming model. CoRR, abs/2106.04179, 2021. URL:
  27. K.-T. Foerster, J. H. Korhonen, J. Rybicki, and S. Schmid. Brief announcement: Does preprocessing help under congestion? In Proc. 38th ACM Symp. on Principles of Distributed Computing (PODC), pages 259-261, 2019. Google Scholar
  28. M. Ghaffari, C. Jin, and D. Nilis. A massively parallel algorithm for minimum weight vertex cover. In Proc. 32nd ACM Symp. on Parallelism in Algorithms and Architectures (SPAA), pages 259-268, 2020. Google Scholar
  29. M. Ghaffari and F. Kuhn. Deterministic distributed vertex coloring: Simpler, faster, and without network decomposition. In Proc. 62nd IEEE Symp. on Foundations of Computer Science (FOCS), 2021. Google Scholar
  30. M. Ghaffari, F. Kuhn, and Y. Maus. On the complexity of local distributed graph problems. In Proc. 39th ACM Symp. on Theory of Computing (STOC), pages 784-797, 2017. Google Scholar
  31. M. Ghaffari, F. Kuhn, Y. Maus, and J. Uitto. Deterministic distributed edge-coloring with fewer colors. In Proc. 50th ACM Symp. on Theory of Comp. (STOC), pages 418-430, 2018. Google Scholar
  32. M. Göös and J. Suomela. No sublogarithmic-time approximation scheme for bipartite vertex cover. Distributed Computing, 27(6):435-443, 2014. Google Scholar
  33. F. Grandoni, J. Könemann, and A. Panconesi. Distributed weighted vertex cover via maximal matchings. ACM Trans. Algorithms, 5(1):6:1-6:12, 2008. Google Scholar
  34. F. Grandoni, J. Könemann, A. Panconesi, and M. Sozio. A primal-dual bicriteria distributed algorithm for capacitated vertex cover. SIAM J. Comput., 38(3):825-840, 2008. Google Scholar
  35. D. G. Harris. Distributed local approximation algorithms for maximum matching in graphs and hypergraphs. SIAM J. Computing, 49(4):711-746, 2020. Google Scholar
  36. J. Håstad. Some optimal inapproximability results. J. of the ACM, 48(4):798-859, 2001. Google Scholar
  37. D. Hochbaum, editor. Approximation Algorithms for NP-hard Problems. PWS Publishing Company, 1997. Google Scholar
  38. J.H. Hoepman, S. Kutten, and Z. Lotker. Efficient distributed weighted matchings on trees. In Proc 13th Coll. on Structural Inf. and Comm. Complexity (SIROCCO), pages 115-129, 2006. Google Scholar
  39. J. E. Hopcroft and R. M. Karp. An n^5/2 algorithm for maximum matchings in bipartite graphs. SIAM J. Comput., 2(4):225-231, 1973. Google Scholar
  40. 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
  41. G. Karakostas. A better approximation ratio for the vertex cover problem. ACM Trans. Algorithms, 5(4):41:1-41:8, 2009. Google Scholar
  42. D. Kőnig. Gráfok és mátrixok. Matematikai és Fizikai Lapok, 38:116-119, 1931. Google Scholar
  43. S. Khot and O. Regev. Vertex cover might be hard to approximate to within 2-epsilon. J. Comput. Syst. Sci., 74(3):335-349, 2008. Google Scholar
  44. S. Khuller, U. Vishkin, and N. E. Young. A primal-dual parallel approximation technique applied to weighted set and vertex covers. J. Algorithms, 17(2):280-289, 1994. Google Scholar
  45. C. Koufogiannakis and N. E. Young. Distributed algorithms for covering, packing and maximum weighted matching. Distributed Computing, 24(1):45-63, 2011. Google Scholar
  46. F. Kuhn, T. Moscibroda, and R. Wattenhofer. What cannot be computed locally! In Proc. 23rd ACM Symp. on Principles of Distributed Computing (PODC), pages 300-309, 2004. Google Scholar
  47. F. Kuhn, T. Moscibroda, and R. Wattenhofer. The price of being near-sighted. In Proceedings of 17th Symp. on Discrete Algorithms (SODA), pages 980-989, 2006. Google Scholar
  48. N. Linial and M. Saks. Low diameter graph decompositions. Combinatorica, 13(4):441-454, 1993. Google Scholar
  49. Z. Lotker, B. Patt-Shamir, and S. Pettie. Improved distributed approximate matching. J. ACM, 62(5):38:1-38:17, 2015. Google Scholar
  50. Z. Lotker, B. Patt-Shamir, and A. Rosén. Distributed approximate matching. SIAM Journal on Computing, 39(2):445-460, 2009. Google Scholar
  51. G. L. Miller, R. Peng, and S. C. Xu. Parallel graph decompositions using random shifts. In Proc. 25th ACM Symp. on Parallelism in Alg. and Arch. (SPAA), pages 196-203, 2013. Google Scholar
  52. T. Nieberg. Local, distributed weighted matching on general and wireless topologies. In Proc. Joint Workshop on Foundations of Mobile Computing (DIALM-POMC), pages 87-92, 2008. Google Scholar
  53. D. Peleg. Distributed Computing: A Locality-Sensitive Approach. SIAM, 2000. Google Scholar
  54. V. Rozhoň and M. Ghaffari. Polylogarithmic-time deterministic network decomposition and distributed derandomization. In Proc. 52nd ACM Symp. on Theory of Computing (STOC), pages 350-363, 2020. Google Scholar
  55. S. Schmid and J. Suomela. Exploiting locality in distributed SDN control. In Proc. 2nd ACM SIGCOMM Works. on Hot Topics in Software Defined Netw. (HotSDN), pages 121-126, 2013. Google Scholar
  56. M. Wattenhofer and R. Wattenhofer. Distributed weighted matching. In Proc. 18th Conf. on Distributed Computing (DISC), pages 335-348, 2004. 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