[Paper Review] Pruning Isomorphic Structural Sub-problems in Configuration

Stéphane Grandcolas, Laurent Hénocque+1 more|ArXiv.org|Jun 27, 2003

Model-Driven Software Engineering Techniques16 references3 citations

TL;DR

This paper proposes a formalism-independent method to eliminate isomorphic structural sub-problems in configuration by testing the canonicity of T-tree representations using a low-cost, amortizable algorithm. It enables configurators to prune redundant search spaces without modifying the problem model, reducing the number of non-canonical configurations by orders of magnitude in practice.

ABSTRACT

Configuring consists in simulating the realization of a complex product from a catalog of component parts, using known relations between types, and picking values for object attributes. This highly combinatorial problem in the field of constraint programming has been addressed with a variety of approaches since the foundation system R1(McDermott82). An inherent difficulty in solving configuration problems is the existence of many isomorphisms among interpretations. We describe a formalism independent approach to improve the detection of isomorphisms by configurators, which does not require to adapt the problem model. To achieve this, we exploit the properties of a characteristic subset of configuration problems, called the structural sub-problem, which canonical solutions can be produced or tested at a limited cost. In this paper we present an algorithm for testing the canonicity of configurations, that can be added as a symmetry breaking constraint to any configurator. The cost and efficiency of this canonicity test are given.

Motivation & Objective

• To address the challenge of structural isomorphisms in configuration problems, which lead to redundant search space in constraint programming.
• To develop a method that detects and eliminates isomorphisms arising from identical substructures without requiring model reconfiguration.
• To provide a general-purpose symmetry-breaking mechanism compatible with any existing configurator, independent of the underlying formalism.
• To formally define and compute the number of canonical versus non-canonical configurations in a simplified T-tree model.
• To demonstrate that canonical representation drastically reduces the number of equivalent solutions, improving configurator efficiency.

Proposed method

• The paper introduces the concept of a structural sub-problem, modeled as a T-tree, where nodes represent object types and edges represent composition relations.
• It defines a partial order ≤ on T-trees based on structural equivalence, leading to a canonical representative for each isomorphism class.
• A canonicity test is formulated using a recursive algorithm that checks whether a T-tree is lexicographically smallest under the ≤ relation.
• The test leverages multiset combinations of subtrees, using binomial coefficients to count and compare subtree arrangements efficiently.
• The algorithm is integrated as a symmetry-breaking constraint into any configurator, enabling pruning of isomorphic sub-trees during search.
• The method is formalized using recursive formulas: $ N_{p,k} = \frac{(N_{p-1,k})^{k+1} - 1}{N_{p-1,k} - 1} $ for total configurations and $ M_{p,k} = \binom{M_{p-1,k} + k}{k} $ for canonical ones.

Experimental results

Research questions

• RQ1 How can isomorphisms arising from structural equivalence in configuration problems be detected and eliminated without modifying the model?
• RQ2 What is the computational cost of testing whether a T-tree is canonical, and can this be done efficiently?
• RQ3 How much reduction in search space is achievable by filtering only canonical configurations?
• RQ4 What is the asymptotic ratio between the number of non-canonical and canonical configurations in a T-tree model?
• RQ5 Can a general-purpose, formalism-independent symmetry-breaking technique be designed for configuration tools?

Key findings

• The number of non-canonical k-connected T-trees of depth p, $ N_{p,k} $, grows as $ \Theta(k^{k^{p-1}}) $, indicating exponential explosion in redundant solutions.
• The number of canonical k-connected T-trees, $ M_{p,k} $, grows as $ \Theta(\frac{4^{k^{p-1}}}{k^{k^{p-2}}}) $, which is significantly smaller than $ N_{p,k} $ for large p and k.
• For $ p=3, k=4 $, the ratio $ N_{p,k}/M_{p,k} $ exceeds $ 3.6 \times 10^{11} / 1.13 \times 10^7 $, showing massive redundancy reduction via canonical pruning.
• Even for small values like $ p=2, k=2 $, the ratio $ N_{2,2}/M_{2,2} = 13/10 $, demonstrating early and substantial benefit.
• The canonicity test is computationally efficient and amortizable, making it suitable for integration into real-world configurators as a symmetry-breaking constraint.
• The method generalizes and outperforms prior approaches like object counting or interchangeability of unused components, as it handles isomorphisms at the structural level.