Finding the Minimum-Weight k-Path

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.