Steiner trees and polyhedra Original Research Article

In this paper we study the dominant of the Steiner tree polytope. We introduce a new class of valid inequalities that generalizes the so-called odd hole, wheel, bipartite, anti-hole and Steiner partition inequalities introduced by Chopra and Rao (Math. Programming 64 (1994) 209–229, 231–246), and we give sufficient conditions for these inequalities to define facets. We describe some procedures that permit to construct facets from known ones for the dominant of the Steiner tree polytope and the closely related Steiner connected subgraph polytope. Using these methods we give a counterexample to a conjecture of Chopra and Rao on the dominant of the Steiner tree polytope on 2-trees. We also describe the dominant of the Steiner tree polytope and the Steiner connected subgraph polytope on special classes of graphs. In particular, we show that if the underlying graph is series–parallel and the terminals satisfy certain conditions, then both polyhedra are given by the trivial inequalities and the Steiner partition inequalities.