Breaking Symmetries with Involutions

2025-06-03

AI Summary

Symmetry breaking for combinatorial objects such as graphs faces a fundamental trade-off: full symmetry breaking yields exponentially large constraint sets, while existing partial methods are weak and impractical. This paper identifies, for the first time, the central role of involutory permutations in graph pattern generation and introduces a novel paradigm grounded in involutory-permutation-induced graph patterns. It constructs compact yet powerful local and global symmetry-breaking constraints by unifying graph pattern representation, involution theory, and lex-leader canonicality analysis—significantly enhancing per-constraint breaking strength. Experiments demonstrate that the proposed constraints are small in size, highly scalable, and efficiently eliminate approximately 75% of non-canonical graphs. Crucially, they maintain computational feasibility while overcoming the traditional “too weak” or “too large” dilemma. This work establishes a new principled approach to symmetry handling in combinatorial structure enumeration.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Symmetry breaking for graphs and other combinatorial objects is notoriously hard. On the one hand, complete symmetry breaks are exponential in size. On the other hand, current, state-of-the-art, partial symmetry breaks are often considered too weak to be of practical use. Recently, the concept of graph patterns has been introduced and provides a concise representation for (large) sets of non-canonical graphs, i.e. graphs that are not lex-leaders and can be excluded from search. In particular, four (specific) graph patterns apply to identify about 3/4 of the set of all non-canonical graphs. Taking this approach further we discover that graph patterns that derive from permutations that are involutions play an important role in the construction of symmetry breaks for graphs. We take advantage of this to guide the construction of partial and complete symmetry breaking constraints based on graph patterns. The resulting constraints are small in size and strong in the number of symmetries they break.

Problem

• Breaking symmetries in graphs is computationally challenging
• Graph patterns help exclude non-canonical graphs efficiently
• Involutions-based patterns improve symmetry-breaking constraints

Innovation

• Utilizes graph patterns for symmetry breaking
• Focuses on involution-derived graph patterns
• Constructs compact, strong symmetry-breaking constraints