Tilt Automata: Gathering Particles With Uniform External Control
Abstract
Motivated by targeted drug delivery, we investigate the gathering of particles in the full tilt model of externally controlled motion planning: A set of particles is located at the tiles of a polyomino with all particles reacting uniformly to an external force by moving as far as possible in one of the four axis-parallel directions until they hit the boundary. The goal is to choose a sequence of directions that moves all particles to a common position. Our results include a polynomial-time algorithm for gathering in a completely filled polyomino as well as hardness reductions for approximating shortest gathering sequences and for determining whether the particles in a partially filled polyomino can be gathered. We pay special attention to the impact of restricted geometry, particularly polyominoes without holes. As corollaries, we make progress on an open question from [Balanza-Martinez et al., SODA 2020] by showing that deciding whether a given position can be occupied remains NP-hard in polyominoes without holes and provide initial results on the parameterized complexity of tilt problems. Our results build on a connection we establish between tilt models and the theory of synchronizing automata.
Cite
@article{arxiv.2603.02796,
title = {Tilt Automata: Gathering Particles With Uniform External Control},
author = {Sándor P. Fekete and Jonas Friemel and Peter Kramer and Jan-Marc Reinhardt and Christian Rieck and Christian Scheffer},
journal= {arXiv preprint arXiv:2603.02796},
year = {2026}
}
Comments
30 pages, 13 figures. Full version of an extended abstract to appear in the proceedings of the 42nd International Symposium on Computational Geometry (SoCG 2026)