A Practical Greedy Approximation for the Directed Steiner Tree Problem

The Directed Steiner Tree (DST) NP-hard problem asks, considering a directed weighted graph with n nodes and m arcs, a node r called root and a set of k nodes X called terminals, for a minimum cost directed tree rooted at r spanning X. The best known polynomial approximation ratio for DST is a O(k(epsilon))-approximation greedy algorithm. However, a much faster k-approximation, returning the shortest paths from r to X, is generally used in practice. We give in this paper a new O(root k)-approximation greedy algorithm called GreedyFLAC(Delta), derived from a new fast k-approximation algorithm called GreedyFLAC running in time at most O(nmk(2)). We provide computational results to show that, GreedyFLAC rivals the running time of the fast k-approximation and returns solution with smaller cost in practice.