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.

翻译：我们在一个新颖的Pick-n-swap先入为主的操纵模型下研究一系列重新安排问题,在这个模型中,一个机器人操纵器,能够携带一个项目并进行项目交换,任务是以最理想的时速方式对存储于变量维度层的物品进行分类。我们系统地分析内在的最佳性结构,这种结构相当丰富和令人着迷,在不同的项目可辨度水平下(完全贴上标签,每个项目都有独特的标签,或部分贴上标签,其中多个项目可能属于同一类型)和不同的衬衣尺寸。我们注重一个和两个维度的最实用设置,我们开发了基于低多边周期的基于功能的算法,在完全和部分贴上标签的设置下,对1D级的变量进行优化的重新排列。另一方面,我们显示2D级和较高维度的缩略图的重新排列变得难以计算。尽管它们具有NP-hart性质,但我们证明高效的循环跟踪纸质算法在一个和两个维度两个维度层面的最优化的设置中,我们在2D级的逻辑周期周期周期周期内,只能对2级的精度的精度进行精确性研究,这些逻辑的精度的精度的精度的精度的精度的精度只能提供。