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  "path": "/abs/2603.02796v1",
  "publishedAt": "2026-03-04T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Sándor P. Fekete",
    "Jonas Friemel",
    "Peter Kramer",
    "Jan-Marc Reinhardt",
    "Christian Rieck",
    "Christian Scheffer"
  ],
  "textContent": "**Authors:** Sándor P. Fekete, Jonas Friemel, Peter Kramer, Jan-Marc Reinhardt, Christian Rieck, Christian Scheffer\n\nMotivated 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.",
  "title": "Tilt Automata: Gathering Particles With Uniform External Control"
}