劣モジュラ・システムに関連する基多面体のすべての面の特徴づけ

Abstract

For a distributive lattice D ⫋ 2^E and a submodular function f on D with φ ∈ D and f(φ) = 0, the pair (D,f) is called a submodular system and, when E ∈ D, the polyhedron given by B(f) = {x | x ∈ R^E, ∀X ∈De : x(X) ≦ f(X), x(E) =f(E)} is called the base polyhedron associated with (D, f)・ We examine the structure of the base polyhedron B(f) and give a characterization of all the faces of B(f). Faces of B(f) are made correspond one-to-one to certain sublattices of D, so that the collection D of all such sublattices of D is anti-isomorphic with the collection F of all the nonempty faces of B(f). Here, D and Fare considered as posets relative to set inclusion. The incidence relation among faces, dimensions of faces, and extreme points and extreme rays of faces are given based on the structure of the sublattices in D. These include as special cases recent results on (1) a poset structure of a polymatroid extreme point and connected components by Bixby, Cunningham and Topkis, (2) extreme rays of a cone determined by a distributive lattice by Tomizawa, and (3) adjacency for polymatroid extreme points by Topkis. Moreover, given a sublattice D_1 of D_2 on which f is modular, F(D_1) = {x | x ∈ B(f), ∀X ∈D_1 : x(X) = f(X) } is a nonempty face of B(f) and there uniquely exists a sublattice D_2 in D which corresponds to the face F(D_1). We show a theorem which characterizes the relationship between D_1 and D_2. D_2 is considered as a closure of D_1 and this closure operation is closely related to the concept of maximal skeleton recently considered by Nakamura and Iri. Algorithmic aspects of these characterizations are also discussed.