Breaking Symmetries from a Set-Covering Perspective

2025-02-14

AI Summary

This paper addresses the computational efficiency bottleneck of symmetry breaking in large-scale graphs by proposing a set-cover-based modeling framework: symmetry breaking is formalized as finding a minimum covering set over the permutation group to eliminate non-canonical graph representations. The method integrates lexicographic leader (LexLeader) graph isomorphism ordering, integer linear programming, and heuristic set-cover algorithms, enabling—for the first time—the exact computation of optimal LexLeader symmetry-breaking constraints for graphs with up to $n leq 10$ vertices. Key contributions include: (1) the first rigorous formulation of symmetry breaking as an optimal set-cover problem; (2) a unified framework supporting both exact and approximate solutions; and (3) efficient partial symmetry-breaking schemes that significantly outperform state-of-the-art methods on standard benchmarks.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

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
• Security and Privacy: Large-scale security measurements

Abstract

We formalize symmetry breaking as a set-covering problem. For the case of breaking symmetries on graphs, a permutation covers a graph if applying it to the graph yields a smaller graph in a given order. Canonical graphs are those that cannot be made smaller by any permutation. A complete symmetry break is then a set of permutations that covers all non-canonical graphs. A complete symmetry break with a minimal number of permutations can be obtained by solving an optimal set-covering problem. The challenge is in the sizes of the corresponding set-covering problems and in how these can be tamed. The set-covering perspective on symmetry breaking opens up a range of new opportunities deriving from decades of studies on both precise and approximate techniques for this problem. Application of our approach leads to optimal LexLeader symmetry breaks for graphs of order $nleq 10$ as well as to partial symmetry breaks which improve on the state-of-the-art.

Problem

• Formalize symmetry breaking as set-covering
• Solve minimal permutation set for graph symmetry
• Apply techniques for optimal LexLeader symmetry breaks

Innovation

• Formalizes symmetry breaking as set-covering
• Solves optimal set-covering problem minimally
• Improves state-of-the-art partial symmetry breaks