Advertisement

Finding the Minimum-Weight k-Path

  • Avinatan Hassidim
  • Orgad Keller
  • Moshe Lewenstein
  • Liam Roditty
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8037)

Abstract

Given a weighted n-vertex graph G with integer edge-weights taken from a range [ − M,M], we show that the minimum-weight simple path visiting k vertices can be found in time \(\tilde{O}(2^k \mathrm{poly}(k) M n^\omega) = O^*(2^k M)\). If the weights are reals in [1,M], we provide a (1 + ε)-approximation which has a running time of \(\tilde{O}(2^k \mathrm{poly}(k) n^\omega(\log\log M + 1/\varepsilon))\). For the more general problem of k-tree, in which we wish to find a minimum-weight copy of a k-node tree T in a given weighted graph G, under the same restrictions on edge weights respectively, we give an exact solution of running time \(\tilde{O}(2^k \mathrm{poly}(k) M n^3) \) and a (1 + ε)-approximate solution of running time \(\tilde{O}(2^k \mathrm{poly}(k) n^3(\log\log M + 1/\varepsilon))\). All of the above algorithms are randomized with a polynomially-small error probability.

Keywords

Approximation Algorithm Edge Weight Weighted Graph Hamiltonian Path Main Loop 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Abasi, H., Bshouty, N.H.: A simple algorithm for undirected hamiltonicity. Electronic Colloquium on Computational Complexity (ECCC) 20, 12 (2013)Google Scholar
  2. 2.
    Alon, N., Yuster, R., Zwick, U.: Color-coding. J. ACM 42(4), 844–856 (1995)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Björklund, A.: Determinant sums for undirected hamiltonicity. In: FOCS, pp. 173–182. IEEE Computer Society (2010)Google Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Narrow sieves for parameterized paths and packings. CoRR, abs/1007.1161 (2010)Google Scholar
  5. 5.
    Chen, J., Lu, S., Sze, S.-H., Zhang, F.: Improved algorithms for path, matching, and packing problems. In: Bansal, N., Pruhs, K., Stein, C. (eds.) SODA, pp. 298–307. SIAM (2007)Google Scholar
  6. 6.
    Cygan, M., Gabow, H.N., Sankowski, P.: Algorithmic applications of baur-strassen’s theorem: Shortest cycles, diameter and matchings. In: FOCS, pp. 531–540. IEEE Computer Society (2012)Google Scholar
  7. 7.
    Ergün, F., Sinha, R.K., Zhang, L.: An improved fptas for restricted shortest path. Inf. Process. Lett. 83(5), 287–291 (2002)CrossRefGoogle Scholar
  8. 8.
    Kneis, J., Mölle, D., Richter, S., Rossmanith, P.: Divide-and-color. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 58–67. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    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
  10. 10.
    Koutis, I., Williams, R.: Limits and applications of group algebras for parameterized problems. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 653–664. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  11. 11.
    Monien, B.: How to find long paths efficiently. Annals of Discrete Mathematics 25, 239–254 (1985)MathSciNetGoogle Scholar
  12. 12.
    Williams, R.: Finding paths of length k in o*(2k) time. Inf. Process. Lett. 109(6), 315–318 (2009)MATHCrossRefGoogle Scholar
  13. 13.
    Williams, V.V.: Multiplying matrices faster than coppersmith-winograd. In: Karloff, H.J., Pitassi, T. (eds.) STOC, pp. 887–898. ACM (2012)Google Scholar
  14. 14.
    Zwick, U.: All pairs shortest paths using bridging sets and rectangular matrix multiplication. J. ACM 49(3), 289–317 (2002)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Avinatan Hassidim
    • 1
  • Orgad Keller
    • 1
  • Moshe Lewenstein
    • 1
  • Liam Roditty
    • 1
  1. 1.Department of Computer ScienceBar-Ilan UniversityRamat-GanIsrael

Personalised recommendations