Efficient Reconfiguration of Tile Arrangements by a Single Active Robot

TL;DR

This paper addresses the problem of reconfiguring a grid of passive tiles into a new arrangement using a single active robot, focusing on minimizing makespan while maintaining connectivity, and provides complexity results and approximation algorithms.

Contribution

It introduces a generalized NP-hard version with weighted costs and offers a polynomial-time approximation algorithm for specific cases.

Findings

Weighted movement costs make the problem NP-hard.

A polynomial-time constant-factor approximation algorithm is developed.

The algorithm achieves optimal carry distance for 2-scaled instances.

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 schedule that minimizes the overall makespan, while keeping the tile configuration connected. We provide both negative and positive results. (1) We generalize the problem by introducing weighted movement costs, which can vary depending on whether tiles are carried or not, and prove that this variant is NP-hard. (2) We give a polynomial-time constant-factor approximation algorithm for the case of disjoint start and target bounding boxes, which additionally yields optimal carry distance for 2-scaled instances.