The natural matroid of an integer polymatroid

TL;DR

This paper explores the natural matroid associated with integer polymatroids, providing new characterizations and insights into their structure through bases, circuits, and cyclic flats.

Contribution

It introduces novel characterizations of integer polymatroids using their bases, circuits, and cyclic flats, expanding understanding beyond existing methods.

Findings

New characterizations of integer polymatroids using bases and circuits

Insights into the structure of cyclic flats and their ranks

Enhanced understanding of the natural matroid's role in integer polymatroids

Abstract

The natural matroid of an integer polymatroid was introduced to show that a simple construction of integer polymatroids from matroids yields all integer polymatroids. As we illustrate, the natural matroid can shed much more light on integer polymatroids. We focus on characterizations of integer polymatroids using their bases, their circuits, and their cyclic flats along with the rank of each cyclic flat and each element; we offer some new characterizations and insights into known characterizations.