Compact Representations of Matroids Using Binary Decision Diagrams and Zero-Suppressed Binary Decision Diagrams

Core Concepts

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. The focus is on analyzing the sizes of BDDs and ZDDs for representing matroids.

Abstract

The paper makes the following key contributions: Comparison of different variations of BDDs and ZDDs for representing matroids: The size of the ZDD Z(B(M)) and the BDD B(I(M)) are never greater than the BDD B(B(M)). The BDDs B(B(M)) and B(B(M*)) have the same size. The ZDDs Z(I(M)), Z(B(M)), and the BDD B(I(M*)) all have the same size. Upper bounds on the width (size) of BDDs and ZDDs for representing matroids: For free matroids, the width is at most 1. For uniform matroids, the width is at most λ(E⪯,i) + 1. For partition and nested matroids, the width is at most 2λ(E⪯,i). For transversal and laminar matroids, the width is unbounded. Improved upper bounds for partition and nested matroids using the pathwidth of the matroid: The width is at most pathwidth(M) + 1 for a certain total order on the ground set. Implementation of a rank oracle using the ZDD structure. The results provide insights into the compact representation of matroids using BDDs and ZDDs, and reveal new strongly pigeonhole classes of matroids.

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Key Insights Distilled From

On the sizes of BDDs and ZDDs representing matroids

by Hiromi Emoto... at arxiv.org 04-24-2024

https://arxiv.org/pdf/2404.14670.pdf

Deeper Inquiries

How can the insights from this work on BDD/ZDD representations of matroids be extended to other combinatorial structures beyond matroids

The insights gained from the study on Binary Decision Diagrams (BDDs) and Zero-Suppressed Binary Decision Diagrams (ZDDs) representing matroids can be extended to various other combinatorial structures beyond matroids. One way to extend these insights is by applying similar techniques to represent and analyze other combinatorial structures that involve subsets and their relationships. For example, problems in graph theory, network optimization, and database management often involve subsets and their properties. By adapting the concepts of BDDs and ZDDs to these areas, it may be possible to create more efficient data structures for representing and manipulating subsets in these contexts. Additionally, the idea of using decision diagrams to represent complex structures can be applied to problems in artificial intelligence, machine learning, and computational biology. Decision diagrams have been used in these fields to represent and reason about complex systems efficiently. By leveraging the principles behind BDDs and ZDDs, researchers can potentially develop new data structures and algorithms for solving problems in these domains.

What are the practical implications of the upper bounds on BDD/ZDD widths for specific matroid classes

The upper bounds on BDD/ZDD widths for specific matroid classes have practical implications in various real-world applications. Algorithm Design: The insights gained from the upper bounds can be used to design more efficient algorithms for solving combinatorial optimization problems related to matroids. By understanding the limitations imposed by the widths of BDDs and ZDDs, researchers and practitioners can develop algorithms that are optimized for specific matroid classes, leading to faster and more scalable solutions. Resource Optimization: In practical applications where computational resources are limited, such as in embedded systems or resource-constrained environments, the upper bounds on BDD/ZDD widths can help in optimizing resource utilization. By knowing the maximum width of the decision diagrams, developers can allocate resources more effectively and ensure that the algorithms run within the available constraints. Modeling Complex Systems: The insights from the upper bounds can also be used in modeling and analyzing complex systems that can be represented as matroids. By understanding the structural limitations imposed by the widths of BDDs and ZDDs, researchers can gain deeper insights into the behavior and properties of these systems, leading to better decision-making and problem-solving strategies.

How can these insights be leveraged in real-world applications

The techniques used to derive the pathwidth-based upper bounds on BDD/ZDD widths for specific matroid classes can be generalized to other parameters of the matroid structure beyond pathwidth. Rank Functions: Similar to pathwidth, the rank function of a matroid plays a crucial role in determining the structure and properties of the matroid. By analyzing the relationship between the rank function and the widths of BDDs and ZDDs, researchers can derive upper bounds based on the rank of the matroid. Circuit Rank: The concept of circuit rank in matroid theory is another important parameter that characterizes the matroid's structure. By studying the circuit rank and its relationship with decision diagram widths, it is possible to establish upper bounds that depend on the circuit rank of the matroid. Closure Operations: Closure operations in matroid theory, such as closure under union or intersection, can also influence the structure of decision diagrams. By exploring how different closure operations impact the widths of BDDs and ZDDs, researchers can generalize the techniques to derive upper bounds based on various closure properties of the matroid.