Structural properties of greedoids

Abstract

Greedoids were introduced by the authors as generalizations of matroids providing a framework for the greedy algorithm. In this paper they are studied from a structural aspect. Definitions of basic matroid-theoretical concepts such as rank and closure can be generalized to greedoids, even though they loose some of their fundamental properties. The rank function of a greedoid is only “locally” submodular. The closure operator is not monotone but possesses a (relaxed) Steinitz—McLane exchange property. We define two classes of subsets, called rank-feasible and closure-feasible, so that the rank and closure behave nicely for them. In particular, restricted to rank-feasible sets the rank function is submodular. Finally we show that Rado’s theorem on independent transversals of subsets of matroids remains valid for feasible transversals of certain sets of greedoids.