Multi-rooted Greedy Approximation of Directed Steiner Trees with Applications

We present a greedy algorithm for the directed Steiner tree problem (DST), where any tree rooted at any (uncovered) terminal can be a candidate for greedy choice. It will be shown that the algorithm, running in polynomial time for any constant l, outputs a directed Steiner tree of cost no larger than 2(l-1)(ln n + 1) times the cost of the minimum l-restricted Steiner tree. We derive from this result that 1) DST for a class of graphs, including quasi-bipartite graphs, in which the length of paths induced by Steiner vertices is bounded by some constant can be approximated within a factor of O(log n), and 2) the tree cover problem on directed graphs can also be approximated within a factor of O(log n).