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

TL;DR

This paper proves that reconfiguring arrangements of identical unit squares or disks within polygons is PSPACE-hard, even in simple settings, advancing understanding of computational complexity in geometric reconfiguration problems.

Contribution

It establishes PSPACE-hardness for reconfiguring unlabeled unit squares in simple polygons, extending previous results to more restricted settings and introducing new complexity proofs.

Findings

Reconfiguration of unit squares in simple polygons is PSPACE-hard.

Reconfiguration of unit disks in polygons with holes is PSPACE-hard.

Complexity results extend previous NP-hardness findings to PSPACE-hardness.

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].