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.

翻译：受靶向药物输送的启发，我们研究了在完全倾斜模型下粒子的聚集问题，该模型属于外部控制的运动规划范畴：一组粒子位于多联骨牌的方格中，所有粒子对外部力做出均匀响应，沿着四个轴平行方向之一尽可能远地移动，直至遇到边界。目标在于选择一系列方向，使所有粒子移动到同一位置。我们的研究成果包括：针对完全填充的多联骨牌提出了一种多项式时间聚集算法，以及关于近似最短聚集序列和判定部分填充多联骨牌中粒子能否聚集的难度归约。我们特别关注受限几何结构（尤其是无孔洞多联骨牌）的影响。作为推论，我们通过证明在无孔洞多联骨牌中判定给定位置能否被占据仍然是NP难问题，推进了[Balanza-Martinez等人，SODA 2020]中一个开放问题的研究，并提供了关于倾斜问题参数化复杂性的初步结果。这些成果建立在我们所建立的倾斜模型与同步自动机理论之间的联系之上。