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)


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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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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