In this paper, we address the problem of finding a representation of a subtree distance, which is an extension of a tree metric. We show that a minimal representation is uniquely determined by a given subtree distance, and give an O(n2) time algorithm that finds such a representation, where n is the size of the ground set. Since a lower bound of the problem is Ω(n2), our algorithm achieves the optimal time complexity.

A dual greedy polyhedron is defined by a system of linear inequalities, where the right-hand sides are given by a submodular function and the coefficients matrix is given by the incidence vectors of antichains of a rooted forest. Faigle and Kern introduced this concept and showed that a dual greedy algorithm works for the linear program over dual greedy polyhedra. In this paper, we show that a dual greedy polyhedron is the isomorphic image of an ordinary submodular polyhedron under the Mobius function of the underlying rooted forest. This observation enables us to reduce linear optimization problems over dual greedy polyhedra to those over ordinary submodular polyhedra. We show a new max-min theorem for intersection of two dual greedy polyhedra as well.