The Polymatroid Steiner Problems

The Steiner tree problem asks for a minimum cost tree spanning a given set of terminals S ⊆ 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 ⊆ V ). This formulation is motivated by the following application in sensor networks – given a set of sensors S = {s1, . . . , sk}, each sensor si can choose to monitor only a single target from a subset of targets Xi, find minimum cost tree spanning a set of sensors capable of monitoring the set of all targets X = X1 ∪ . . .∪Xk. 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].