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

TL;DR

The paper addresses the complexity of reconfiguring simple geometric objects—unlabeled or labeled unit squares and unit disks—inside polygons. It leverages reductions from Nondeterministic Constrained Logic (NCL) and Monotone-Planar-3-SAT to encode logical constraints into movement constraints, using gadgets, reference centers, and a nine-position discretization to control local motion. The main contributions are PSPACE-hardness results for reconfiguring unit squares in a simple polygon and for unit disks in a polygon with holes (extending prior hole-based hardness to simpler settings), along with a framework that connects SAT reconfiguration to geometric motion via a hypercube-like solution space $G(\phi)$. These findings deepen the understanding of motion-planning complexity for deceptively simple shapes and demonstrate that even highly structured, low-dimensional reconfiguration tasks can be PSPACE-hard, with implications for algorithm design in robotics and computational geometry.

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].

Figures (18)

• Figure 1: Can the left configuration of unit squares in a simple polygon be reconfigured to the right configuration of unit squares?
• Figure 2: An instance of NCL. Can the left orientation be transformed to the right one by edge reversals such that all intermediate orientations are feasible?
• Figure 3: Mapping an NCL-Instance to an embedding of a planar 3-Sat formula. Variable nodes are drawn as squares, clause nodes are drawn as disks.
• Figure 4: Graph $G$ of an NCL instance and corresponding embedded 3-Sat formula $\phi$. (a) NCL instance $G$. (b) We place a variable node in the middle of each edge of $G$. (c) We replace each vertex of $G$ by one or two clause nodes, and connect them to their incident variable nodes appropriately.
• Figure 8: A grid polygon that has reference centers for $b_0=1, b_1=6, b_2=5$ and an $8\times 8$ square intersecting four reference centers; the area of intersection is 16.

Theorems & Definitions (20)

• Theorem 1
• Theorem 2
• Corollary 2
• Theorem 3
• Corollary 3
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7