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.


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.


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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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