Dual greedy polyhedra, choice functions, and abstract convex geometries

Dual greedy polyhedra, choice functions, and abstract convex geometries

We consider a system of linear inequalities and its associated polyhedron for which we can maximize any linear objective function by /nding 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 Kr1 uger for ...

Choice functions and abstract convex geometries *

A main aim of this paper is to make connections between two well – but up to now independently – developed theories, the theory of choice functions and the theory of closure operators. It is shown that the classes of ordinally rationalizable and path independent choice functions are related to the classes of distributive and anti-exchange closures.  1999 Elsevier Science B.V. All rights reserved.

A greedy algorithm for convex geometries

Convex geometries are closure spaces which satisfy anti-exchange property, and they are known as dual of antimatroids. We consider functions defined on the sets of the extreme points of a convex geometry. Faigle– Kern (1996) presented a greedy algorithm to linear programming problems for shellings of posets, and Krüger (2000) introduced b-submodular functions and proved that Faigle–Kern’s algor...

Möbius Functions on Rooted Forests and Their Applications to Faigle-Kern's Dual Greedy Polyhedra

Realization of abstract convex geometries by point configurations

The following classical example of convex geometries shows how they earned their name. Given a set of points X in Euclidean space R, one defines a closure operator on X as follows: for any Y ⊆ X, Y = convex hull(Y ) ∩ X. One easily verifies that such an operator satisfies the anti-exchange axiom. Thus, (X,−) is a convex geometry. Denote by Co(R, X) the closure lattice of this closure space, nam...