Polynomial-time approximability of the k-Sink Location problem

A dynamic network N = (G, c, τ, S) where G = (V,E) is a graph, integers τ (e) and c(e) represent, for each edge e ∈ E, the time required to traverse edge e and its nonnegative capacity, and the set S ⊆ V is a set of sources. In the k-Sink Location problem, one is given as input a dynamic network N where every source u ∈ S is given a nonnegative supply value σ(u). The task is then to find a set of sinks X = {x1, . . . , xk} in G that minimizes the routing time of all supply to X. Note that, in the case where G is an undirected graph, the optimal position of the sinks in X needs not be at vertices, and can be located along edges. Hoppe and Tardos[6] showed that, given an instance of k-Sink Location and a set of k vertices X ⊆ V , one can find an optimal routing scheme of all the supply in G to X in polynomial time, in the case where graph G is directed. Note that when G is directed, this suffices to obtain polynomial-time solvability of the k-Sink Location problem, since any optimal position will be located at vertices of G. However, the computational complexity of the k-Sink Location problem on general undirected graphs is still open. In this paper, we show that the k-Sink Location problem admits a fully polynomial-time approximation scheme (FPTAS) for every fixed k, and that the problem is W [1]-hard when parameterized by k.