Data Reductions for the Strong Maximum Independent Set Problem in Hypergraphs

2026-02-11

AI Summary

This study addresses the problem of finding a strong maximum independent set in hypergraphs—defined as a vertex set containing at most one vertex from each hyperedge—which arises in applications such as the construction of perfect minimal hash functions. The work introduces, for the first time, nine specialized data reduction rules grounded in structural properties of hypergraphs, serving as a preprocessing phase to substantially shrink instance sizes. Empirical evaluation demonstrates that this approach reduces instances to an average of 22% of their original size within just 6.76 seconds of preprocessing time. When integrated with the state-of-the-art exact solver, the method yields an average speedup of 3.84×, with peak acceleration reaching 53×, and successfully solves a previously intractable instance.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

This work addresses the well-known Maximum Independent Set problem in the context of hypergraphs. While this problem has been extensively studied on graphs, we focus on its strong extension to hypergraphs, where edges may connect any number of vertices. A set of vertices in a hypergraph is strongly independent if there is at most one vertex per edge in the set. One application for this problem is to find perfect minimal hash functions. We propose nine new data reduction rules specifically designed for this problem. Our reduction routine can serve as a preprocessing step for any solver. We analyze the impact on the size of the reduced instances and the performance of several subsequent solvers when combined with this preprocessing. Our results demonstrate a significant reduction in instance size and improvements in running time for subsequent solvers. The preprocessing routine reduces instances, on average, to 22% of their original size in 6.76 seconds. When combining our reduction preprocessing with the best-performing exact solver, we observe an average speedup of 3.84x over not using the reduction rules. In some cases, we can achieve speedups of up to 53x. Additionally, one more instance becomes solvable by a method when combined with our preprocessing.

Problem

• Strong Maximum Independent Set
• Hypergraphs
• Data Reduction
• Perfect Minimal Hash Functions

Innovation

• strong maximum independent set
• hypergraphs
• data reduction rules
• preprocessing
• perfect minimal hash functions