The Polymatroid Representation of a Greedoid, and Associated Galois Connections

TL;DR

This paper explores the relationship between greedoids and polymatroids, establishing conditions for their representations and dualities, and introduces new tools for analyzing interval greedoids.

Contribution

It proves that local poset greedoids solvable by greedy algorithms have polymatroid representations and introduces a Galois connection between greedoid flats and polymatroid closed sets.

Findings

Every local poset greedoid with correct greedy solutions has a polymatroid representation.

A Galois injection links greedoid flats to polymatroid closed sets.

Identifies a subclass of polymatroid greedoids with maximum representation.

Abstract

A greedoid is a generalization of a matroid allowing for more flexible analyses and modeling of combinatorial optimization problems. However, these structures decimate many matroid properties contributing to their pervasive nature. A polymatroid greedoid [KL85a] presents an interesting middle ground, so we further develop this class. First we prove every local poset greedoid for which the greedy algorithm correctly solves linear optimizations over its basic words must have a polymatroid representation. For this, we use relationships between the lattices of greedoid flats and closed sets of a polymatroid to generalize concepts in [KL85a]. Then, we show our generalization induces a Galois injection between the greedoid flats and closed sets of a representation. Finally, we apply this duality to identify a subclass of polymatroid greedoids with a maximum representation, giving a partial answer to an open problem of [KL85a]. As technical tools for our analyses, we introduce optimism and the Forking Lemma for interval greedoids. Both are pervasive in our work, and are of independent interest.