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(2^kpoly(k)Mn^ω) = O*(2^kM). If the weights are reals in [1,M], we provide a (1 + ε)-approximation which has a running time of Õ(2^kpoly(k) n^ω (log log M + 1/ε)). 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 Õ(2 ^kpoly(k)Mn³) and a (1 + ε)-approximate solution of running time Õ(2^kpoly(k)n³(log log M + 1/ε)). All of the above algorithms are randomized with a polynomially-small error probability.