Valuated matroids: a new look at the greedy algorithm

On Exchange Axioms for Valuated Matroids and Valuated Delta-Matroids

Two further equivalent axioms are given for valuations of a matroid. Let M = (V,B) be a matroid on a finite set V with the family of bases B. For ω : B → R the following three conditions are equivalent: (V1) ∀B,B′ ∈ B, ∀u ∈ B −B′,∃v ∈ B′ −B: ω(B) + ω(B′) ≤ ω(B − u+ v) + ω(B′ + u− v); (V2) ∀B,B′ ∈ B with B 6= B′, ∃u ∈ B −B′,∃v ∈ B′ −B: ω(B) + ω(B′) ≤ ω(B − u+ v) + ω(B′ + u− v); (V3) ∀B,B′ ∈ B, ∀...

A new look at infinite matroids

It has recently been shown that, contrary to common belief, infinite matroids can be axiomatized in a way very similar to finite matroids. This should make it possible now to extend much of the theory of finite matroids to infinite ones: an aim that had previously been thought to be unattainable, because the popular additional ‘finitary’ axiom believed to be necessary clearly spoils duality. We...

The greedy algorithm for matroids

Two Algorithms for Valuated ∆-matroids

Two algorithms are proposed for computing the maximum degree of a principal minor of specified order of a skew-symmetric rational function matrix. The algorithms are developed in the framework of valuated ∆matroid of Dress and Wenzel, and are valid also for valuated ∆-matroids in general.

The Greedy Algorithm and Coxeter Matroids

The notion of matroid has been generalized to Coxeter matroid by Gelfand and Serganova. To each pair (W, P) consisting of a finite irreducible Coxeter group W and parabolic subgroup P is associated a collection of objects called Coxeter matroids. The (ordinary) matroids are a special case, the case W = An (isomorphic to the symmetric group Symn+1) and P a maximal parabolic subgroup. The main re...