Advertisement

A Local 2-Approximation Algorithm for the Vertex Cover Problem

  • Matti Åstrand
  • Patrik Floréen
  • Valentin Polishchuk
  • Joel Rybicki
  • Jukka Suomela
  • Jara Uitto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5805)

Abstract

We present a distributed 2-approximation algorithm for the minimum vertex cover problem. The algorithm is deterministic, and it runs in \(({\it \Delta}+1)^2\) synchronous communication rounds, where \({\it \Delta}\) is the maximum degree of the graph. For \({\it \Delta}=3\), we give a 2-approximation algorithm also for the weighted version of the problem.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)CrossRefGoogle Scholar
  2. 2.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York (1979)MATHGoogle Scholar
  3. 3.
    Khot, S., Regev, O.: Vertex cover might be hard to approximate to within 2 − ε. Journal of Computer and System Sciences 74(3), 335–349 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Grandoni, F., Könemann, J., Panconesi, A.: Distributed weighted vertex cover via maximal matchings. ACM Transactions on Algorithms 5(1), 1–12 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Hańćkowiak, M., Karoński, M., Panconesi, A.: On the distributed complexity of computing maximal matchings. SIAM Journal on Discrete Mathematics 15(1), 41–57 (2001)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Panconesi, A., Rizzi, R.: Some simple distributed algorithms for sparse networks. Distributed Computing 14(2), 97–100 (2001)CrossRefGoogle Scholar
  7. 7.
    Linial, N.: Locality in distributed graph algorithms. SIAM Journal on Computing 21(1), 193–201 (1992)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Naor, M., Stockmeyer, L.: What can be computed locally? SIAM Journal on Computing 24(6), 1259–1277 (1995)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Suomela, J.: Survey of local algorithms (2009, manuscript), http://www.iki.fi/jukka.suomela/local-survey
  10. 10.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: What cannot be computed locally! In: Proc. 23rd Symposium on Principles of Distributed Computing (PODC), pp. 300–309. ACM Press, New York (2004)Google Scholar
  11. 11.
    Kuhn, F., Moscibroda, T., Wattenhofer, R.: The price of being near-sighted. In: Proc. 17th Symposium on Discrete Algorithms (SODA), pp. 980–989. ACM Press, New York (2006)Google Scholar
  12. 12.
    Hochbaum, D.S.: Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing 11(3), 555–556 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Moscibroda, T.: Locality, Scheduling, and Selfishness: Algorithmic Foundations of Highly Decentralized Networks. PhD thesis, ETH Zürich (2006)Google Scholar
  14. 14.
    Polishchuk, V., Suomela, J.: A simple local 3-approximation algorithm for vertex cover. Information Processing Letters 109(12), 642–645 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Czygrinow, A., Hańćkowiak, M., Wawrzyniak, W.: Fast distributed approximations in planar graphs. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 78–92. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Lenzen, C., Wattenhofer, R.: Leveraging Linial’s locality limit. In: Taubenfeld, G. (ed.) DISC 2008. LNCS, vol. 5218, pp. 394–407. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  17. 17.
    Awerbuch, B., Varghese, G.: Distributed program checking: a paradigm for building self-stabilizing distributed protocols. In: Proc. 32nd Symposium on Foundations of Computer Science (FOCS), pp. 258–267. IEEE Computer Society Press, Los Alamitos (1991)Google Scholar
  18. 18.
    Bar-Yehuda, R., Even, S.: A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms 2(2), 198–203 (1981)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Heidelberg (2003)MATHGoogle Scholar
  20. 20.
    Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2001)MATHGoogle Scholar
  21. 21.
    Papadimitriou, C.H., Steiglitz, K.: Combinatorial Optimization: Algorithms and Complexity. Dover Publications, New York (1998)MATHGoogle Scholar
  22. 22.
    Gonzalez, T.F.: A simple LP-free approximation algorithm for the minimum weight vertex cover problem. Information Processing Letters 54(3), 129–131 (1995)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Khuller, S., Vishkin, U., Young, N.: A primal-dual parallel approximation technique applied to weighted set and vertex covers. Journal of Algorithms 17(2), 280–289 (1994)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Hańćkowiak, M., Karoński, M., Panconesi, A.: On the distributed complexity of computing maximal matchings. In: Proc. 9th Symposium on Discrete Algorithms (SODA), pp. 219–225. SIAM, Philadelphia (1998)Google Scholar
  25. 25.
    Ramsey, F.P.: On a problem of formal logic. Proceedings of the London Mathematical Society 30, 264–286 (1930)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Floréen, P., Kaasinen, J., Kaski, P., Suomela, J.: An optimal local approximation algorithm for max-min linear programs. In: Proc. 21st Symposium on Parallelism in Algorithms and Architectures (SPAA). ACM Press, New York (2009)Google Scholar
  27. 27.
    Mayer, A., Naor, M., Stockmeyer, L.: Local computations on static and dynamic graphs. In: Proc. 3rd Israel Symposium on the Theory of Computing and Systems (ISTCS 1995), pp. 268–278. IEEE Computer Society Press, Los Alamitos (1995)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Matti Åstrand
    • 1
  • Patrik Floréen
    • 1
  • Valentin Polishchuk
    • 1
  • Joel Rybicki
    • 1
  • Jukka Suomela
    • 1
  • Jara Uitto
    • 1
  1. 1.Helsinki Institute for Information Technology HIITUniversity of HelsinkiFinland

Personalised recommendations