Rank axiom of modular supermatroids: A connection with directional DR submodular functions

TL;DR

This paper characterizes supermatroids on modular lattices using a rank axiom involving directional DR-submodular functions, extending matroid theory and enabling new optimization approaches.

Contribution

It provides the first equivalent characterization of supermatroids on modular lattices via a rank axiom with directional DR-submodular functions, and proves their strong exchange property.

Findings

Supermatroids on modular lattices can be characterized by a rank axiom involving directional DR-submodular functions.

The strong exchange property of supermatroids on modular lattices is established.

The relation between axioms of supermatroids on lower semimodular lattices is clarified.

Abstract

A matroid has been one of the most important combinatorial structures since it was introduced by Whitney as an abstraction of linear independence. As an important property of a matroid, it can be characterized by several different (but equivalent) axioms, such as the augmentation, the base exchange, or the rank axiom. A supermatroid is a generalization of a matroid defined on lattices. Here, the central question is whether a supermatroid can be characterized by several equivalent axioms similar to a matroid. Barnabei, Nicoletti, and Pezzoli characterized supermatroids on distributive lattices, and Fujishige, Koshevoy, and Sano generalized the results for cg-matroids (supermatroids on lower locally distributive lattices). In this study, we focus on modular lattices, which are an important superclass of distributive lattices, and provide equivalent characterizations of supermatroids on modular lattices. We characterize supermatroids on modular lattices using the rank axiom in which the rank function is a directional DR-submodular function, which is a generalization of a submodular function introduced by the authors. Using a characterization based on rank functions, we further prove the strong exchange property of a supermatroid, which has application in optimization. We also reveal the relation between the axioms of a supermatroid on lower semimodular lattices, which is a common superclass of a lower locally distributive lattice and a modular lattice.