Greedoids: A Generalization of Matroids in Combinatorial Optimization

1. Introduction

Greedoids are a generalization of matroids that allow for more flexibility in modeling optimization problems. While matroids provide a powerful framework for problems such as finding spanning trees or maximizing independent sets, they impose strict independence and exchange properties that do not always hold in real-world applications. Greedoids relax some of these conditions, making them applicable to problems involving rooted trees, language models, and feasible sets in combinatorial optimization.

The term "greedoid" comes from the fact that greedy algorithms still work under certain conditions, similar to how they work for matroids. However, unlike matroids, greedoids allow for dependencies between elements that make them suitable for more general settings.


2. Definition of a Greedoid

A greedoid is a pair where:

• is a finite ground set.
• is a collection of subsets of , called feasible sets, satisfying the following axioms:

• (Accessibility Axiom): If and , then there exists an element such that . This means that every non-empty feasible set can be reduced step by step to the empty set.
• (Exchange Axiom for Greedoids): If and , then there exists an element such that is also in , provided that is accessible.

These axioms allow greedoids to model problems where elements must be added in a structured order, such as language generation or network construction.

3. Examples of Greedoids

• Matroids as a Special Case: Every matroid is a greedoid because the independent sets in a matroid satisfy the accessibility and exchange properties.
• Spanning Tree Greedoids: In a rooted graph, the collection of all subtrees that include the root and are connected forms a greedoid. Unlike matroids, cycles may exist in some feasible sets.
• Language Greedoids: If is a set of words, and consists of all prefixes of valid sentences in a language, then is a greedoid. This structure models syntax trees in computational linguistics.
• Path Greedoids in Graphs: Consider a directed graph with a fixed starting vertex. The collection of all paths starting at this vertex forms a greedoid.

4. Rank Function in Greedoids

The rank function of a greedoid assigns to each subset the size of the largest feasible subset contained in . Unlike in matroids, the rank function in greedoids may not satisfy submodularity.

A rank function satisfies:

• for all .
• If , then .

However, unlike matroids, the function may not always hold, meaning that greedoids lack a certain level of regularity.

5. Greedy Algorithms in Greedoids

One of the key motivations for defining greedoids is that greedy algorithms still work under certain conditions.

Theorem (Greedy Algorithm for Greedoids): If is a greedoid and is a weight function, then the greedy algorithm correctly finds a maximal feasible set of optimal weight, provided that feasible sets grow in a structured way.

This means that greedy algorithms can solve problems in:

• Rooted spanning trees: Finding the best tree structure in a network.
• Shortest path problems: Computing optimal paths in graphs.
• Scheduling problems: Selecting an optimal sequence of jobs.

6. Differences Between Greedoids and Matroids

While greedoids generalize matroids, they differ in key ways:

• Matroids require the hereditary property (every subset of an independent set is independent), but greedoids only require the accessibility property (every non-empty feasible set can be reduced step by step to the empty set).
• Matroids satisfy a strong exchange property, while greedoids allow more flexible dependence structures.
• Matroids arise naturally in vector spaces and graphs, whereas greedoids are useful in hierarchical and language-based structures.

7. Applications of Greedoids

Greedoids are used in various areas, including:

• Graph Theory: Modeling rooted spanning trees and directed acyclic graphs.
• Combinatorial Optimization: Finding optimal sequences of actions in decision problems.
• Computational Linguistics: Parsing languages and constructing syntax trees.
• AI and Machine Learning: Feature selection in datasets where variables have dependencies.
• Economics: Modeling decision-making processes with structured dependencies.

8. Conclusion

Greedoids extend the theory of matroids by allowing for more general structures where elements must be selected in a particular order. They provide a foundation for solving complex optimization problems that involve hierarchical dependencies. Unlike matroids, greedoids allow for more flexible feasible sets, making them useful in linguistics, graph theory, and artificial intelligence.

References

• Wikipedia: Greedoid
• Korte, B., Lovász, L., & Schrader, R. Greedoids (1991).
• Oxley, J. Matroid Theory (1992).