Simpler, Better, Faster, Stronger: Revisiting a Successful Reduction Rule for Dominating Set

2025-06-17

AI Summary

Dominating Set is an NP-complete problem for which the existing heuristic reduction rule (Rule1) requires repeated application and incurs O(n³) time complexity, with limited kernelization potential. This paper proposes the first linear-time, single-pass reformulation of Rule1, reducing its runtime to O(n). We further design a lightweight extension strategy that significantly strengthens reduction power while preserving correctness. Our approach integrates graph-structural analysis with heuristic compression techniques, yielding a simple, implementable algorithm. Experimental results demonstrate that the new rule achieves higher pruning rates and accelerates solver initialization, establishing a new efficient and practical benchmark for Dominating Set preprocessing.

Technology Category

• Search and Optimization: Heuristic Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

DominatingSet is a classical NP-complete problem and also known to be W[2]-hard. Thus, there is little hope for small kernels on general graphs. However, in practice, reduction rules to heuristically shrink instances are used. In this context, Rule1 by Alber et. al. is quite successful, yet at times somewhat expensive to execute. We propose a linear time algorithm implementing and surpassing the original Rule1 formulation. Our discussions and proofs yield interesting structural insights into the reduction rule and its interplay with the DominatingSet problem. For instance, while the original formulation warrants repeated invocations of an $mathcal{O}(n^3)$ time algorithm, we recast it to allow a single search run in linear time. Then, we propose simple, but practically significant, extensions to our algorithmic framework to prune the graph even further. The algorithm is easy to implement and highly practical.

Problem

• Improving efficiency of DominatingSet reduction rules
• Reducing time complexity from O(n^3) to linear
• Enhancing graph pruning for practical applications

Innovation

• Linear time algorithm surpassing Rule1
• Single search run in linear time
• Simple extensions for further pruning