Finding the minimum-weight k-path

  • Authors:

  • Avinatan Hassidim;Orgad Keller;Moshe Lewenstein;Liam Roditty

  • Affiliations:

  • Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel;Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel;Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel;Department of Computer Science, Bar-Ilan University, Ramat-Gan, Israel

  • Venue:
  • WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
  • Year:
  • 2013

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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.