The Polymatroid Representation of a Greedoid, and Associated Galois Connections

The greedoid is a significant abstraction of the matroid allowing for a more flexible analysis of structures in which the greedy algorithm "works." However, their diverse structure imposes difficulties towards their application in combinatorial optimization [Sze21]. In response, we revisit the polymatroid greedoid [KL85a] to characterize it by properties approximating those of matroids, by using the submodularity of its polymatroid representation in particular. Towards doing so, our main contribution is a full description of this class. Specifically, we show that a greedoid is a polymatroid greedoid if and only if it is an optimistic interval greedoid whose kernels are closed under intersection. This constitutes the first necessary and sufficient characterization of the polymatroid greedoid in terms of its combinatorial attributes, thereby resolving a central open question of Korte and Lov\'asz [KL85a]. Here, we introduce the optimism property to approximate properties of a matroid's continuations which are implied by the closure axioms of its span, which no longer hold for greedoids. And, because the kernels of an interval greedoid are in many ways an extension of a matroid's closed sets, our direction of necessity is a direct generalization of Birkhoff and Edmond's characterization of the meet in the lattice of a matroid's closed sets [Bir35, Edm03]. Towards achieving this result, our main technical insights arise from relating the lattice of flats of a polymatroid greedoid to that of the closed sets of its representation through order preserving mappings. Specifically, we will show the novel insight that the notion of polymatroid representation considered in [KL85a] is equivalent to the existence of a certain Galois connection. As a consequence, the representation of a greedoid via a polymatroid is an order theoretic concept in disguise.