Tilt Automata: Gathering Particles With Uniform External Control

TL;DR

This paper studies the problem of gathering particles in a full tilt model using external forces, providing algorithms, hardness results, and complexity insights, especially for polyominoes without holes, with applications to particle control.

Contribution

It introduces a polynomial-time algorithm for gathering particles in filled polyominoes and establishes hardness and complexity results for partial fillings and restricted geometries.

Findings

Polynomial-time algorithm for gathering in filled polyominoes

NP-hardness of deciding gatherability in certain cases

Progress on open problem regarding position occupancy in hole-free polyominoes

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.