The Polymatroid Steiner Problems

摘要

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S is contained in V in a weighted graph G = (V, E, c), c : E → R~+. In this paper we consider a generalization of the Steiner tree problem, so called Polymatroid Steiner Problem, in which a polymatroid P = P( V) is defined on V and the Steiner tree is required to span at least one base of P (in particular, there may be a single base S is contained in V). This formulation is motivated by the following application in sensor networks - given a set of sensors S = {s_1, ... , s_k}, each sensor s_i can choose to monitor only a single target from a subset of targets X_i, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X_1 ∪ ... ∪ X_k. The Polymatroid Steiner Problem generalizes many known Steiner tree problem formulations including the group and covering Steiner tree problems. We show that this problem can be solved with the polylogarithmic approximation ratio by a generalization of the combinatorial algorithm of Chekuri et. al. [7]. We also define the Polymatroid directed Steiner problem which asks for a minimum cost arborescence connecting a given root to a base of a polymatroid P defined on the terminal set S. We show that this problem can be approximately solved by algorithms generalizing methods of Charikar et al [6].