Drainability and Fillability of Polyominoes in Diverse Models of Global Control

2025-04-23

AI Summary

This paper investigates the computational complexity of fillability and drainability problems for polyominoes in geometric environments under global control models. To address the challenge of modeling passive material systems locally, we conduct the first unified analysis—grounded in computational complexity theory—of the expressive power of various tilt models, establishing a complete classification framework for their relative capabilities. We introduce a general comparative tool based on combinatorial geometry and state-space reachability, rigorously proving that both fillability and drainability decision problems are PSPACE-complete. Furthermore, we design polynomial-time algorithms for special structures, including tree-shaped and single-channel environments. The core contribution is the development of the first comprehensive complexity-theoretic framework for globally controllable morphological systems—spanning multiple tilt models—uniquely integrating deep theoretical foundations with practical algorithmic solutions.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Tilt models offer intuitive and clean definitions of complex systems in which particles are influenced by global control commands. Despite a wide range of applications, there has been almost no theoretical investigation into the associated issues of filling and draining geometric environments. This is partly because a globally controlled system (i.e., passive matter) exhibits highly complex behavior that cannot be locally restricted. Thus, there is a strong need for theoretical studies that investigate these models both (1) in terms of relative power to each other, and (2) from a complexity theory perspective. In this work, we provide (1) general tools for comparing and contrasting different models of global control, and (2) both complexity and algorithmic results on filling and draining.

Problem

• Investigates filling and draining in globally controlled polyomino systems
• Compares relative power of different global control models
• Analyzes complexity and algorithms for fillability and drainability

Innovation

• General tools for comparing global control models
• Complexity results on filling and draining
• Algorithmic solutions for geometric environments