Unfolding Boxes with Local Constraints

2025-06-01

AI Summary

This work addresses the automated discovery and enumeration of polyominoes that serve as common unfoldings (nets) for multiple non-isomorphic rectangular boxes—i.e., a single polyomino foldable into two or more distinct box shapes. Conventional approaches based on SAT solvers, randomized algorithms, or decision diagrams suffer from poor scalability due to inefficient encoding of global constraints such as connectivity and acyclicity. To overcome this, we propose a novel modeling paradigm that replaces global constraints with localized geometric constraints, thereby significantly enhancing constraint propagation and scalability of SAT solving. Experiments demonstrate substantial improvements: the maximum searchable area for two-box common unfoldings increases from 88 to over 150; for three-box enumeration, the upper bound rises from 30 to 60. Moreover, we refute Xu et al.’s conjecture that 46, 54, and 58 are minimal feasible areas, establishing tighter lower bounds.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• 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
• Web Mining and Content Analysis: Models for Web evolution
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

We consider the problem of finding and enumerating polyominos that can be folded into multiple non-isomorphic boxes. While several computational approaches have been proposed, including SAT, randomized algorithms, and decision diagrams, none has been able to perform at scale. We argue that existing SAT encodings are hindered by the presence of global constraints (e.g., graph connectivity or acyclicity), which are generally hard to encode effectively and hard for solvers to reason about. In this work, we propose a new SAT-based approach that replaces these global constraints with simple local constraints that have substantially better propagation properties. Our approach dramatically improves the scalability of both computing and enumerating common box unfoldings: (i) while previous approaches could only find common unfoldings of two boxes up to area 88, ours easily scales beyond 150, and (ii) while previous approaches were only able to enumerate common unfoldings up to area 30, ours scales up to 60. This allows us to rule out 46, 54, and 58 as the smallest areas allowing a common unfolding of three boxes, thereby refuting a conjecture of Xu et al. (2017).

Problem

• Finding polyominos foldable into multiple non-isomorphic boxes
• Overcoming limitations of global constraints in SAT encodings
• Scaling computation and enumeration of common box unfoldings

Innovation

• Replaces global constraints with local ones
• Uses SAT-based approach for better scalability
• Enables enumeration of larger box unfoldings