Approximation algorithms for directed Steiner problems

We give the first non-trivial approximation algorithms for the Steiner tree problem and the generalized Steiner network problem on general directed graphs. These problems have several applications in network design and multicast routing. For both problems, the best ratios known before our work were the trivial O(k)-approximations. For the directed Steiner tree problem, we design a family of algorithms that achieves an approximation ratio of i(i-1)k^1/i in time O(nⁱk²ⁱ) for any fixed i>1, where k is the number of terminals. Thus, an O(k^ε) approximation ratio can be achieved in polynomial time for any fixed ε>0. Setting i = log k, we obtain an O(log² k) approximation ratio in quasi-polynomial time. For the directed generalized Steiner network problem, we give an algorithm that achieves an approximation ratio of O(k^2/3 log^1/3 k), where k is the number of pairs of vertices that are to be connected. Related problems including the group Steiner tree problem, the set TSP problem and several others in both directed and undirected graphs can be reduced in an approximation preserving fashion to the directed Steiner tree problem. Thus we obtain the first non-trivial approximations to those as well. All these problems are known be as hard as set cover to approximate.