Tilt Automata: Gathering Particles With Uniform External Control

2026-03-03

AI Summary

This study addresses the problem of gathering dispersed particles to a single location in a hole-free, multiply-connected grid environment under a uniform external control signal, with applications such as targeted drug delivery in mind. All particles move synchronously in one of four cardinal directions until blocked by obstacles or boundaries. The goal is to design a sequence of movement directions that achieves complete aggregation. By establishing a connection between the tilt model and synchronous automata theory, we present a polynomial-time algorithm for the fully filled case, prove the hardness of approximating the shortest gathering sequence, and demonstrate that deciding gatherability in partially filled configurations is NP-hard, thereby advancing the parameterized complexity understanding of this problem.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Multiagent Systems: Distributed Problem Solving
• Intelligent Robots: Motion and Path Planning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• tilt automata
• particle gathering
• external control
• polyomino
• motion planning

Innovation

• tilt automata
• synchronizing automata
• particle gathering
• polyomino
• computational hardness