A greedy-algorithm characterization of valuated Δ-matroids

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, ∀...

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...

The greedy algorithm for strict cg-matroids

A matroid-like structure defined on a convex geometry, called a cg-matroid, is defined by S. Fujishige, G. A. Koshevoy, and Y. Sano in [6]. Strict cg-matroids are the special subclass of cg-matroids. In this paper, we show that the greedy algorithm works for strict cg-matroids with natural weightings, and also show that the greedy algorithm works for a hereditary system on a convex geometry wit...