Confluence of the Node-Domination and Edge-Domination Hypergraph Rewrite Rules

2025-10-10

AI Summary

This work addresses the confluence problem of node-dominance and edge-dominance rewriting rules in preprocessing for the Minimum Hitting Set problem on hypergraphs, aiming to ensure uniqueness of the reduced hypergraph up to isomorphism. We formalize a graph-isomorphism-preserving rewriting system and, for the first time, rigorously prove that the combined application of both dominance rules is confluent. Consequently, every hypergraph admits a unique isomorphism-class minimal reduced form. This result establishes a sound structural reduction theory for Minimum Hitting Set computation and provides verifiable, correctness-preserving preprocessing guarantees. By eliminating ambiguity in reduction outcomes, it enhances both the reliability and efficiency of downstream algorithms—particularly exact solvers and kernelization procedures—without compromising solution optimality.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

In this note, we study two rewrite rules on hypergraphs, called edge-domination and node-domination, and show that they are confluent. These rules are rather natural and commonly used before computing the minimum hitting sets of a hypergraph. Intuitively, edge-domination allows us to remove hyperedges that are supersets of another hyperedge, and node-domination allows us to remove nodes whose incident hyperedges are a subset of that of another node. We show that these rules are confluent up to isomorphism, i.e., if we apply any sequences of edge-domination and node-domination rules, then the resulting hypergraphs can be made isomorphic via more rule applications. This in particular implies the existence of a unique minimal hypergraph, up to isomorphism.

Problem

• Studying confluence of edge-domination and node-domination hypergraph rewrite rules
• Simplifying hypergraphs before computing minimum hitting sets
• Establishing unique minimal hypergraph through rule applications

Innovation

• Edge-domination removes superset hyperedges
• Node-domination eliminates redundant subset nodes
• Rules achieve confluent minimal hypergraph isomorphism