Convexity and Steinitz's exchange property

Abstract

Abstract"Convex analysis" is developed for functions defined on integer lattice points. We investigate the class of functions which enjoy a variant of Steinitz's exchange property. It includes linear functions on matroids, valuations on matroids (in the sense of Dress and Wenzel), and separable concave functions on the integral base polytope of submodular systems. It is shown that a function ω has the Steinitz exchange property if and only if it can be extended to a concave function ω such that the maximizers of (ω+any linear function) form an integral base polytope. A Fenchel-type min-max theorem and discrete separation theorems are established which imply, as immediate consequences, Frank's discrete separation theorem for submodular functions, Edmonds' intersection theorem, Fujishige's Fencheltype min-max theorem for submodular functions, and also Frank's weight splitting theorem for weighted matroid intersection.

Algebraic and Geometric Methods in Enumerative Combinatorics

“…In the most important case, M is the uniform matroid U d,n and P M is the hypersimplex ∆(d, n). Matroid subdivisions arose in algebraic geometry [HKT06,Kap93,Laf03], in the theory of valuated matroids [DW92,Mur96], and in tropical geometry [Spe08]. For instance, Lafforgue showed that if a matroid polytope P M has no nontrivial matroid subdivisions, then the matroid M has (up to trivial transformations) only finitely many realizations over a fixed field F. This is one of very few results about realizability of matroids over arbitrary fields.…”

Characterizing and recognizing generalized polymatroids

“…Many other general classes of polyhedra with somewhat esoteric definitions have been studied: e.g. lattice polyhedra [17], submodular flow polyhedra [8], bisubmodular polyhedra [26, §49.11d], and M -convex functions [18]. In some cases the definitions are chosen to be precisely as general as possible while allowing the proof techniques to go through, e.g.…”

A Two-Sided Discrete-Concave Market with Possibly Bounded Side Payments: An Approach by Discrete Convex Analysis

“…where 0 ∈ E and y(E) denotes the sum of all the components of y. Functionf is called an M-concave function (Murota [18,19]). LetẼ = {0} ∪ E. For each vector x ∈ R E we denote byx the vector (−x(E), x) ∈ RẼ, and we write vectors in RẼ by putting tildes such asp andỹ (e.g.,χ e is the characteristic vector of e onẼ).…”

“…LetẼ = {0} ∪ E. For each vector x ∈ R E we denote byx the vector (−x(E), x) ∈ RẼ, and we write vectors in RẼ by putting tildes such asp andỹ (e.g.,χ e is the characteristic vector of e onẼ). An M-convex function can be characterized by the following exchange property (Murota [18,19]):…”

“…Sotomayor [27] also further investigated this hybrid model and gave a non-constructive proof of the existence of a pairwise stable outcome. Fujishige and Tamura [10] proposed a generalization of the hybrid model due to Eriksson and Karlander [4] and Sotomayor [27] by utilizing discrete convex analysis developed by Murota [18,19,20].…”