The Polymatroid Representation of a Greedoid, and Associated Galois Connections

2024-11-22

arXiv.org

AI Summary

This paper resolves a long-standing open problem posed by Korte and Lovász (1985): providing a necessary and sufficient combinatorial characterization of polymatroid greedoids—the structural framework underlying greedy algorithms. To bridge the theoretical–practical gap in identifying when greedy algorithms apply, the authors establish an intrinsic characterization: a set system is a polymatroid greedoid if and only if it is *optimistic interval* and its family of cores is intersection-closed. “Optimism” is a newly introduced structural property generalizing the Birkhoff–Edmonds theory of closed-set lattices. Moreover, the authors reveal that polynomial representability of greedoids fundamentally arises from a Galois connection between the flattening lattice of greedy matroids and their closed-set lattice. This work fully characterizes the class of greedoids admitting polynomial matroid representations, constructs an order-preserving mapping framework, and unifies and deepens foundational theories of submodular functions, lattice theory, and greedy algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

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.

Problem

• Characterizes polymatroid greedoid using combinatorial attributes.
• Resolves central open question from Korte and Lovász.
• Introduces optimism property to approximate matroid properties.

Innovation

• Polymatroid representation characterizes greedoids
• Optimistic interval greedoid kernels intersect
• Galois connection reveals polymatroid equivalence