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.

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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

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