Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Faster Symmetry Breaking Constraints for Abstract Structures

Özgür Akgün, Mun See Chang, Ian P. Gent, Christopher Jefferson

TL;DR

The paper tackles the challenge of symmetry breaking for abstract structures in high-level constraint modelling languages like Essence, where traditional lex-leader approaches lead to unwieldy constraints. It introduces two innovations—representation-dependent ordering and delayed symmetry application—to produce compact, efficient symmetry-breaking constraints and applies them to unnamed types via Conjure. Empirical results (including a BIBD case study) show substantial runtime and search-space reductions, often by orders of magnitude, compared with prior methods. The approach is incomplete but significantly simplifies implementation and scales better, with future work focusing on adaptive symmetry-breaking levels and counting solutions modulo symmetries.

Abstract

In constraint programming and related paradigms, a modeller specifies their problem in a modelling language for a solver to search and return its solution(s). Using high-level modelling languages such as Essence, a modeller may express their problems in terms of abstract structures. These are structures not natively supported by the solvers, and so they have to be transformed into or represented as other structures before solving. For example, nested sets are abstract structures, and they can be represented as matrices in constraint solvers. Many problems contain symmetries and one very common and highly successful technique used in constraint programming is to "break" symmetries, to avoid searching for symmetric solutions. This can speed up the solving process by many orders of magnitude. Most of these symmetry-breaking techniques involve placing some kind of ordering for the variables of the problem, and picking a particular member under the symmetries, usually the smallest. Unfortunately, applying this technique to abstract variables produces a very large number of complex constraints that perform poorly in practice. In this paper, we demonstrate a new incomplete method of breaking the symmetries of abstract structures by better exploiting their representations. We apply the method in breaking the symmetries arising from indistinguishable objects, a commonly occurring type of symmetry, and show that our method is faster than the previous methods proposed in (Akgün et al. 2025).

Faster Symmetry Breaking Constraints for Abstract Structures

TL;DR

The paper tackles the challenge of symmetry breaking for abstract structures in high-level constraint modelling languages like Essence, where traditional lex-leader approaches lead to unwieldy constraints. It introduces two innovations—representation-dependent ordering and delayed symmetry application—to produce compact, efficient symmetry-breaking constraints and applies them to unnamed types via Conjure. Empirical results (including a BIBD case study) show substantial runtime and search-space reductions, often by orders of magnitude, compared with prior methods. The approach is incomplete but significantly simplifies implementation and scales better, with future work focusing on adaptive symmetry-breaking levels and counting solutions modulo symmetries.

Abstract

In constraint programming and related paradigms, a modeller specifies their problem in a modelling language for a solver to search and return its solution(s). Using high-level modelling languages such as Essence, a modeller may express their problems in terms of abstract structures. These are structures not natively supported by the solvers, and so they have to be transformed into or represented as other structures before solving. For example, nested sets are abstract structures, and they can be represented as matrices in constraint solvers. Many problems contain symmetries and one very common and highly successful technique used in constraint programming is to "break" symmetries, to avoid searching for symmetric solutions. This can speed up the solving process by many orders of magnitude. Most of these symmetry-breaking techniques involve placing some kind of ordering for the variables of the problem, and picking a particular member under the symmetries, usually the smallest. Unfortunately, applying this technique to abstract variables produces a very large number of complex constraints that perform poorly in practice. In this paper, we demonstrate a new incomplete method of breaking the symmetries of abstract structures by better exploiting their representations. We apply the method in breaking the symmetries arising from indistinguishable objects, a commonly occurring type of symmetry, and show that our method is faster than the previous methods proposed in (Akgün et al. 2025).

Paper Structure

This paper contains 14 sections, 3 theorems, 1 equation, 2 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Representation-dependent ordering
  4. Delayed symmetry application
  5. Breaking the symmetries of unnamed types
  6. Method summary
  7. Experiments
  8. Case Study: BIBD
  9. Conclusion and future work
  10. \ref{['def: dot less']} is well-defined
  11. Proof of \ref{['lemma: quick works for occ and exp of sets']}
  12. Proof of \ref{['lemma: quick is ok for conjure rep']}
  13. Representations of sets and multisets
  14. Representations of other types

Key Result

Lemma 4.1

Recalling rem: representation, suppose that a variable $X$ of type $T$ is to be represented as a variable $X'$ of type $T'$ using an underlying bijection $\phi: \mathrm{Val}(T) \rightarrow \mathrm{Val}(T')$. Let $\mathrel{\preceq}_T$ and $\mathrel{\preceq}_{T'}$ be total orderings of $\mathrm{Dom}(X then for any $g \in G$, the constraint $X' \mathrel{\preceq}_{T'} (X')^g$ is implied by the constra

Theorems & Definitions (15)

  • Remark 2.1
  • Remark 2.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.1
  • Example 2.2
  • Example 3.1
  • Definition 3.1
  • Example 3.2
  • Example 4.1
  • ...and 5 more