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Matching and P2-Packing: Weighted Versions

  • Qilong Feng
  • Jianxin Wang
  • Jianer Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)

Abstract

Parameterized algorithms are presented for the weighted P 2-Packing problem, which is a generalization of the famous Graph Matching problem. The algorithms are based on the following new techniques and observations: (1) new study on structure relationship between graph matchings in general graphs and P 2-packings in bipartite graphs; (2) an effective graph bi-partitioning algorithm; and (3) a polynomial-time algorithm for a constrained weighted P 2-Packing problem in bipartite graphs. These techniques lead to randomized and deterministic parameterized algorithms that significantly improve the previous best upper bounds for the problem for both weighted and unweighted versions.

Keywords

Bipartite Graph Vehicle Route Problem Input Graph Splitting Function Auxiliary Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Bar-Yehuda, R., Halldórsson, M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. In: Proc. 13th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 732–741. ACM Press, New York (2002)Google Scholar
  2. 2.
    Bazgan, C., Hassin, R., Monnot, J.: Approximation algorithms for some vehicle routing problems. Discrete Appl. Math. 146, 27–42 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, J., Lu, S.: Improved parameterized set splitting algorithms: A probabilistic approach. Algorithmica 54(4), 472–489 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    De Bontridder, K., Halldórsson, B., Lenstra, J., Ravi, R., Stougie, L.: Approximation algorithms for the test cover problem. Math. Program, Ser. B 98, 477–491 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fellows, M., Heggernes, P., Rosamond, F., Sloper, C., Telle, J.A.: Exact algorithms for finding k disjoint triangles in an arbitrary graph. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds.) WG 2004. LNCS, vol. 3353, pp. 235–244. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  6. 6.
    Feng, Q., Liu, Y., Lu, S., Wang, J.: Improved deterministic algorithms for weighted matching and packing problems. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 211–220. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  7. 7.
    Gabow, H.: Data structures for weighted matching and nearest common ancestoers. In: Proc. 1st Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 434–443. ACM Press, New York (1990)Google Scholar
  8. 8.
    Fernau, H., Raible, D.: A parameterized perspective on packing paths of length two. Journal of Combinatorial Optimization 18(4), 319–341 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hassin, R., Rubinstein, S.: An approximation algorithm for maximum triangle packing. Discrete Appl. Math. 154, 971–979 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Hell, P., Kirkpatrick, D.: On the complexity of a generalized matching problem. In: Mitzenmacher, M. (ed.) Proc. 10th Annual ACM Symposium on Theory of Computing, pp. 240–245. ACM Press, New York (1978)Google Scholar
  11. 11.
    Hurkens, C., Schrijver, A.: On the size of systems of sets every t of which have an SDR, with application to worst case ratio of heuristics for packing problems. SIAM Journal on Discrete Mathmatics 2, 68–72 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kann, V.: Maximum bounded H-matching is MAX-SNP-complete. Information Processing Letters 49, 309–318 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Koutis, I.: Faster algebraic algorithms for path and packing problems. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 575–586. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  14. 14.
    Kirkpatrick, D.G., Hell, P.: On the complexity of general graph factor problems. SIAM Journal on Computing 12, 601–609 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    van Leeuwen, J.: Graph algorithms. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. A, pp. 525–631 (1990)Google Scholar
  16. 16.
    Monnot, J., Toulouse, S.: The P k partitioning problem and related problems in bipartite graphs. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds.) SOFSEM 2007. LNCS, vol. 4362, pp. 422–433. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  17. 17.
    Naor, M., Schulman, L., Srinivasan, A.: Splitters and near-optimal derandomization. In: Proc. 39th Annual Symposium on Foundatins of Computer Science, pp. 182–190. IEEE Press, New York (1995)Google Scholar
  18. 18.
    Prieto, E., Sloper, C.: Looking at the stars. Theoretical Computer Science 351, 437–445 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Wang, J., Feng, Q.: An O *(3.523k) parameterized algorithm for 3-set packing. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 82–93. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  20. 20.
    Wang, J., Ning, D., Feng, Q., Chen, J.: An improved parameterized algorithm for a generalized matching problem. In: Agrawal, M., Du, D.-Z., Duan, Z., Li, A. (eds.) TAMC 2008. LNCS, vol. 4978, pp. 212–222. Springer, Heidelberg (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Qilong Feng
    • 1
  • Jianxin Wang
    • 1
  • Jianer Chen
    • 1
    • 2
  1. 1.School of Information Science and EngineeringCentral South UniversityChangshaP.R. China
  2. 2.Department of Computer Science and EngineeringTexas A&M University College StationTexasUSA

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