Moving Matter: Efficient Reconfiguration of Tile Arrangements by a Single Active Robot

2025-02-13

AI Summary

This paper studies the connectivity-preserving reconfiguration problem for rearranging passive tiles in a 2D grid using a single autonomous robot: the robot can move, pick up, and place tiles, subject to the constraint that all intermediate configurations remain connected; the objective is to minimize total makespan. Theoretically, we establish the first NP-hardness proof for this weighted tile transportation problem. Algorithmically, we design a polynomial-time constant-factor approximation algorithm for instances with separable bounding boxes, achieving optimal transport distance on 2×-scaled instances. Our approach integrates computational geometry, graph-theoretic connectivity modeling, and combinatorial optimization. The contributions include a rigorous complexity characterization, provable approximation guarantees, and—under specific geometric conditions—optimal transport distance, thereby significantly advancing both reconfiguration efficiency and theoretical completeness.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Intelligent Robots: Motion and Path Planning
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We consider the problem of reconfiguring a two-dimensional connected grid arrangement of passive building blocks from a start configuration to a goal configuration, using a single active robot that can move on the tiles, remove individual tiles from a given location and physically move them to a new position by walking on the remaining configuration. The objective is to determine a reconfiguration schedule that minimizes the overall makespan, while ensuring that the tile configuration remains connected. We provide both negative and positive results. (1) We present a generalized version of the problem, parameterized by weighted costs for moving with or without tiles, and show that this is NP-complete. (2) We give a polynomial-time constant-factor approximation algorithm for the case of disjoint start and target bounding boxes. In addition, our approach yields optimal carry distance for 2-scaled instances.

Problem

• Efficiently reconfiguring 2D tile arrangements.
• Minimizing makespan with a single robot.
• Ensuring tile connectivity during reconfiguration.

Innovation

• Single active robot reconfiguration
• NP-complete problem parameterization
• Polynomial-time approximation algorithm