Approximation Algorithms for Directed Steiner Tree Problems

The Steiner tree problem which is known to be NP-Complete is the following. Given a weighted undirected graph G = (V;E), and a set X V of terminals, the objective is to nd a tree of minimum cost which connects all the terminals. If the graph is directed, in addition to X, we are given a root r 2 V , and the objective is to nd a minimum cost arborescence which connects the root to each of the terminals. In the Generalized Steiner tree problem, we are given a set X of pairs of vertices, and the goal is to nd a subgraph of minimum cost such that each pair in X is connected. In the undirected case, constant factor algorithms are known for both the versions [11, 14, 17, 1, 15], but essentially no approximation algorithms were known for these problems in the directed case, other than the trivial O(k)-approximations. We obtain the rst non-trivial approximation algorithms for both problems in general directed graphs. For the Directed Steiner tree problem, we design a family of algorithms that achieve an approximation ratio of O(k ) in time O(kn ) for any xed > 0, where k is the number of terminals. For the Directed Generalized Steiner tree problem, we give an algorithm that achieves an approximation ratio of O(k log k), where k is the number of pairs of vertices that are to be connected. Related problems including the Group Steiner tree problem, the Node Weighted Steiner tree problem and several others can be reduced in an approximation preserving fashion to the problems we solve, giving the rst non-trivial approximations to those as well. For the Directed Steiner tree problem, a result similar to ours has been obtained independently in [6] by To-yat Cheung, Zuo Dai and Ming Li of the Department of Computer Science, City University of Hong Kong. Supported by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Supported by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation. Supported by ARO Grant DAAH04-95-1-0121 and NSF Grant CCR9304971 Supported by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.