Rearrangement on Lattices with Pick-n-Swaps: Optimality Structures and Efficient Algorithms

TL;DR

This paper investigates optimal rearrangement of items on lattices using a robotic pick-n-swap model, analyzing the problem's structure, developing efficient algorithms for 1D lattices, and addressing computational complexity in higher dimensions.

Contribution

It introduces a novel pick-n-swap rearrangement model, analyzes its optimality structure, and develops polynomial-time algorithms for 1D lattices, while providing asymptotic solutions for higher dimensions.

Findings

Optimal algorithms for 1D lattices under various labeling conditions.

NP-hardness of rearrangement in 2D and higher dimensions.

Asymptotically optimal algorithms for 2D lattices based on cycle analysis.

Abstract

We study a class of rearrangement problems under a novel pick-n-swap prehensile manipulation model, in which a robotic manipulator, capable of carrying an item and making item swaps, is tasked to sort items stored in lattices of variable dimensions in a time-optimal manner. We systematically analyze the intrinsic optimality structure, which is fairly rich and intriguing, under different levels of item distinguishability (fully labeled, where each item has a unique label, or partially labeled, where multiple items may be of the same type) and different lattice dimensions. Focusing on the most practical setting of one and two dimensions, we develop low polynomial time cycle-following-based algorithms that optimally perform rearrangements on 1D lattices under both fully- and partially-labeled settings. On the other hand, we show that rearrangement on 2D and higher-dimensional lattices become computationally intractable to optimally solve. Despite their NP-hardness, we prove that efficient cycle-following-based algorithms remain optimal in the asymptotic sense for 2D fully- and partially-labeled settings, in expectation, using the interesting fact that random permutations induce only a small number of cycles. We further improve these algorithms to provide $1.x$-optimality when the number of items is small. Simulation studies corroborate the effectiveness of our algorithms. The implementation of the algorithms from the paper can be found at github.com/arc-l/lattice-rearrangement.