On the sizes of BDDs and ZDDs representing matroids

TL;DR

This paper explores the potential of BDDs and ZDDs as compact data structures for representing matroids, providing size comparisons and upper bounds related to matroid minors and connectivity properties.

Contribution

It introduces the first systematic study of BDDs and ZDDs for matroids, establishing size bounds linked to matroid minors and connectivity measures.

Findings

Compared sizes of BDDs and ZDDs for different matroids.

Provided upper bounds on BDD/ZDD sizes based on matroid minors.

Linked size bounds to connectivity and pathwidth of matroids.

Abstract

Matroids are often represented as oracles since there are no unified and compact representations for general matroids. This paper initiates the study of binary decision diagrams (BDDs) and zero-suppressed binary decision diagrams (ZDDs) as relatively compact data structures for representing matroids in a computer. This study particularly focuses on the sizes of BDDs and ZDDs representing matroids. First, we compare the sizes of different variations of BDDs and ZDDs for a matroid. These comparisons involve concise transformations between specific decision diagrams. Second, we provide upper bounds on the size of BDDs and ZDDs for several classes of matroids. These bounds are closely related to the number of minors of the matroid and depend only on the connectivity function or pathwidth of the matroid, which deeply relates to the classes of matroids called strongly pigeonhole classes. In essence, these results indicate upper bounds on the number of minors for specific classes of matroids and new strongly pigeonhole classes.