On the sizes of BDDs and ZDDs representing matroids

TL;DR

This work tackles the challenge of compactly representing matroids using binary decision diagrams (BDDs) and zero-suppressed BDDs (ZDDs), and analyzes how representation size scales with matroid structure. It introduces width bounds tied to the connectivity function $\lambda_M$ and pathwidth $pw(M)$, and ties these results to minor counts and strongly pigeonhole classes. The authors establish structural relations among eight diagram variants (bases vs independent sets, BDDs vs ZDDs, and duals), prove major bounds for widths under various matroid families, and show how optimal orders can dramatically reduce width via pathwidth. As a practical outcome, they show how ZDDs enable efficient rank queries in $O(|E|)$ time, providing a feasible oracle-based toolkit for matroid computations.

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.

Figures (11)

• Figure 1.1: An undirected graph $G$
• Figure 1.2: Relations of the sizes between different decision diagrams
• Figure 1.3: Upper bounds on the $i$th width depending only on $\mathop{\mathrm{\mathit{\lambda}}}\nolimits\lparen E_{\preceq, i}\rparen$
• Figure 2.1: A binary decision tree for $\lbrace\emptyset, \lbrace1\rbrace, \lbrace1, 2\rbrace, \lbrace1, 2, 3\rbrace\rbrace$
• Figure 2.2: Node sharing

Theorems & Definitions (42)

• Example 1.1: $\mathbb{F}$-representable matroid
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Lemma 3.2
• proof