Structural properties of greedoids

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. Dedicated to Paul Erdős on his seventieth birthday Supported by the joint research project “Algorithmic Aspects of Combinatorial Optimization” of the Hungarian Academy of Sciences (Magyar Tudományos Akadémia) and the German Research Association (Deutsche Forschungsgemeinschaft, SFB 21).