Lehman's theorem and the directed Steiner tree problem

In the directed Steiner tree problem, we are given a digraph, nonnegative arc weights, a subset of vertices called terminals, and a special terminal called the root. The goal is to compute a minimum weight directed tree that connects each terminal to the root. We study the classical directed cut linear programming (LP) formulation which has a variable for every arc, and a constraint for every cut that separates a terminal from the root. For what instances is the directed cut LP integral? In this paper we demonstrate how the celebrated theorem of Lehman [Math. Program., 17 (1979), pp. 403-417] on minimally nonideal clutters provides a framework for deriving answers to this question. Specifically, we show that this framework yields short proofs of the optimum arborescences theorem and the integrality result for series-parallel digraphs. Furthermore, we use this framework to show that the directed cut linear program is integral for digraphs that are acyclic and have at most two nonterminal vertices.