Finding the Minimum-Weight k-Path

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 Õ(2poly(k)Mn) = O∗(2kM). If the weights are reals in [1,M ], we provide a (1+ε)-approximation which has a running time of Õ(2poly(k)n(log logM + 1/ε)). For the more general problem of k-tree, in which we wish to find a minimumweight 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 Õ(2poly(k)Mn) and a (1 + ε)-approximate solution of running time Õ(2poly(k)n(log logM + 1/ε)). All of the above algorithms are randomized with a polynomially-small error probability.