Reconfiguration of unit squares and disks: PSPACE-hardness in simple settings

2024-12-30

AI Summary

This paper investigates the feasibility of label-free reconfiguration for unit squares and unit disks within simple or hole-containing polygons. Using a reduction-based approach, we establish a new lemma proving PSPACE-completeness of reconfiguration for monotone planar 3SAT, combined with computational-geometric constructions and logical encoding techniques. Our contributions are: (1) Reconfiguration of unit squares in *simple* polygons is PSPACE-hard—significantly lowering the prior geometric requirement that necessitated polygonal holes; (2) Reconfiguration of unit disks in *hole-containing* polygons is also PSPACE-hard—the first such hardness result for a single disk radius. These results extend the geometric boundaries of rigidity-aware reconfiguration complexity theory and provide foundational complexity characterizations for motion planning and automated assembly applications.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study two well-known reconfiguration problems. Given a start and a target configuration of geometric objects in a polygon, we wonder whether we can move the objects from the start configuration to the target configuration while avoiding collisions between the objects and staying within the polygon. Problems of this type have been considered since the early 80s by roboticists and computational geometers. In this paper, we study some of the simplest possible variants where the objects are unlabeled unit squares or unit disks. In unlabeled reconfiguration, the objects are identical, so that any object is allowed to end at any of the targets positions. We show that it is PSPACE-hard to decide whether there exists a reconfiguration of unit squares even in a simple polygon. Previously, it was only known to be PSPACE-hard in a polygon with holes [Solovey and Halperin, Int. J. Robotics Res. 2016]. Our proof is based on a result of independent interest, namely that reconfiguration between two satisfying assignments of a formula of Monotone-Planar-3SAT is also PSPACE-complete. The reduction from reconfiguration of Monotone-Planar-3SAT to reconfiguration of unit squares extends techniques recently developed to show NP-hardness of packing unit squares in a simple polygon [Abrahamsen and Stade, FOCS 2024]. We also show PSPACE-hardness of reconfiguration of unit disks in a polygon with holes. Previously, it was only known that reconfiguration of disks of two different sizes was PSPACE-hard [Brocken, van der Heijden, Kostitsyna, Lo-Wong and Surtel, FUN 2021].

Problem

• PSPACE-hard problems
• Collision-free motion
• Polygonal environments

Innovation

• PSPACE-hard
• Simple Polygons
• Disk Motion Planning