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Fencheltype duality for matroid valuations,
, 1995
"... Abstract The weighted matroid intersection problem has recently been extended to the valuated matroid intersection problem: Given a pair of valuated matroids M i = (V, B i , ω i ) (i = 1, 2) defined on a common ground set V , find a common base B ∈ B 1 ∩ B 2 that maximizes ω 1 (B) + ω 2 (B). This p ..."
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Cited by 11 (10 self)
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). This paper develops a Fencheltype duality theory related to this problem with a view to establishing a convexity framework for nonlinear integer programming. A Fencheltype minmax theorem and a discrete separation theorem are given. Furthermore, the subdifferentials of matroid valuations are investigated.
A quasisymmetric function for matroids
, 2007
"... A new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant • defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients, • is a multivariate generating functio ..."
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Cited by 16 (2 self)
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function for integer weight vectors that give minimum total weight to a unique base of the matroid, • is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight, • behaves simply under matroid duality, • has a
Matroids over a ring
, 2013
"... We introduce the notion of a matroid M over a commutative ring R, assigning to every subset of the ground set an Rmodule according to some axioms. When R is a field, we recover matroids. When R = Z, and when R is a DVR, we get (structures which contain all the data of) quasiarithmetic matroids, ..."
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Cited by 1 (0 self)
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, and valuated matroids, respectively. More generally, whenever R is a Dedekind domain, we extend the usual properties and operations holding for matroids (e.g., duality), and we compute the TutteGrothendieck group of matroids over R.
Convexity and Steinitz’s Exchange Property
 ADVANCES IN MATHEMATICS, 124 (1996), 272–311
, 1997
"... “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 func ..."
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Cited by 64 (28 self)
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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 minmax theorem for submodular functions, and also Frank’s weight splitting theorem for weighted matroid