Dual greedy polyhedra, choice functions, and abstract convex geometries

Abstract

We consider a system of linear inequalities and its associated polyhedron for which we can maximize any linear objective function by finding tight inequalities at an optimal solution in a greedy way. We call such a system of inequalities a dual greedy system and its associated polyhedron a dual greedy polyhedron. Such dual greedy systems have been considered by Faigle and Kern, and Krüger for antichains of partially ordered sets, and by Kashiwabara and Okamoto for extreme points of abstract convex geometries. Faigle and Kern also considered dual greedy systems in a more general framework than antichains. A related dual greedy algorithm was proposed by Frank for a class of lattice polyhedra. In the present paper we show relationships among dual greedy systems, substitutable choice functions, and abstract convex geometries. We also examine the submodularity and facial structures of the dual greedy polyhedra determined by dual greedy systems. Furthermore, we consider an extension of the class of dual greedy polyhedra.