Dual greedy polyhedra, choice functions, and abstract convex geometries

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...

Greedy Fans: A geometric approach to dual greedy algorithms

The purpose of this paper is to understand greedily solvable linear programs in a geometric way. Such linear programs have recently been considered by Queyranne, Spieksma and Tardella, Faigle and Kern, and Krüger for antichains of posets, and by Frank for a class of lattice polyhedra, and by Kashiwabara and Okamoto for extreme points of abstract convex geometries. Our guiding principle is that ...

Clustering on antimatroids and convex geometries

The clustering problem as a problem of set function optimization with constraints is considered. The behavior of quasi-concave functions on antimatroids and on convex geometries is investigated. The duality of these two set function optimizations is proved. The greedy type Chain algorithm, which allows to find an optimal cluster, both as the “most distant” group on antimatroids and as a dense c...

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

Bisubmodular polyhedra, simplicial divisions, and discrete convexity

We consider a class of integer-valued discrete convex functions, called BS-convex functions, defined on integer lattices whose affinity domains are sets of integral points of integral bisubmodular polyhedra. We examine discrete structures of BSconvex functions and give a characterization of BS-convex functions in terms of their convex conjugate functions by means of (discordant) Freudenthal sim...