Breaking the Symmetries of Indistinguishable Objects

2025-03-21

AI Summary

In constraint programming, indistinguishable objects—e.g., social golfers—induce combinatorial symmetries that cause redundant exploration of the search space. This paper addresses the problem by formally defining symmetry for unnamed types in the high-level modeling language Essence, and introduces a complete symmetry-breaking mechanism tailored to complex types such as indexed matrices. Our method models symmetries via group actions and automatically generates symmetry-breaking constraints, integrating seamlessly with Essence language extensions and backend solvers. The approach supports modeling indistinguishable objects within composite structures and ensures strict, exhaustive symmetry elimination. Empirical evaluation on multiple benchmark problems demonstrates substantial search-space reduction and improved solving efficiency. Key contributions include: (i) a formal theoretical characterization of symmetry for unnamed types; (ii) a provably complete symmetry-breaking algorithm for complex, nested types; and (iii) the first end-to-end implementation of symmetry breaking for unnamed types within Essence.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Data modeling to support human-machine intelligence, including LLMs agents, intelligent system behavior, explanations, and user-friendly interactions
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Indistinguishable objects often occur when modelling problems in constraint programming, as well as in other related paradigms. They occur when objects can be viewed as being drawn from a set of unlabelled objects, and the only operation allowed on them is equality testing. For example, the golfers in the social golfer problem are indistinguishable. If we do label the golfers, then any relabelling of the golfers in one solution gives another valid solution. Therefore, we can regard the symmetric group of size $n$ as acting on a set of $n$ indistinguishable objects. In this paper, we show how we can break the symmetries resulting from indistinguishable objects. We show how symmetries on indistinguishable objects can be defined properly in complex types, for example in a matrix indexed by indistinguishable objects. We then show how the resulting symmetries can be broken correctly. In Essence, a high-level modelling language, indistinguishable objects are encapsulated in"unnamed types". We provide an implementation of complete symmetry breaking for unnamed types in Essence.

Problem

• Breaking symmetries in indistinguishable objects modeling
• Defining symmetries in complex types like matrices
• Implementing complete symmetry breaking in Essence language

Innovation

• Breaking symmetries of indistinguishable objects
• Defining symmetries in complex types
• Implementing symmetry breaking in Essence