Assembly planning, which is a fundamental problem in robotics and automation, aims to design a sequence of motions that will bring the separate constituent parts of a product into their final placement in the product. It is convenient to study assembly planning in reverse order, where the following key problem, assembly partitioning, arises: Given a set of parts in their final placement in a product, partition them into two sets, each regarded as a rigid body, which we call a subassembly, such that these two subassemblies can be moved sufficiently far away from each other, without colliding with one another. The basic assembly planning problem is further complicated by practical consideration such as how to hold the parts in a subassembly together. Therefore, a desired property of a valid assembly partition is that each of the two subassemblies will be connected. We show that even an utterly simple case of the connected-assembly-partitioning problem is hard: Given a connected set $A$ of unit squares in the plane, each forming a distinct cell of the uniform integer grid, find a subset $S\subset A$ such that $S$ can be rigidly translated to infinity along a prescribed direction without colliding with $A\setminus S$, and both subassemblies $S$ and $A\setminus S$ are each connected. We show that this problem is NP-Complete, and by that settle an open problem posed by Wilson et al. (1995) a quarter of a century ago. We complement the hardness result with two positive results for the aforementioned problem variant of grid squares. First, we show that it is fixed parameter tractable and give an $O(2^k n^2)$-time algorithm, where $n=|A|$ and $k=|S|$. Second, we describe a special case of this variant where a connected partition can always be found in linear time. Each of the positive results sheds further light on the special geometric structure of the problem at hand.

中文翻译：

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论双手平面装配分割

装配规划是机器人和自动化领域的一个基本问题，旨在设计一系列运动，将产品的独立组成部分带到产品中的最终位置。以相反的顺序研究装配规划很方便，其中出现了以下关键问题，即装配分区：给定一组零件在产品中的最终放置，将它们分成两组，每组都视为一个刚体，我们调用一个子装配体，这样这两个子装配体可以相互远离足够远，而不会相互碰撞。基本装配规划问题由于实际考虑而进一步复杂化，例如如何将子装配中的零件固定在一起。所以，有效装配分区的理想属性是两个子装配中的每一个都将被连接。我们表明，即使是连接组装分区问题的一个完全简单的情况也很难：给定平面中单位正方形的连接集合 $A$，每个正方形形成统一整数网格的不同单元格，找到一个子集 $S\子集 A$ 使得 $S$ 可以沿指定方向严格转换为无穷大而不会与 $A\setminus S$ 碰撞，并且两个子集 $S$ 和 $A\setminus S$ 都是相连的。我们证明这个问题是 NP-Complete，并由此解决了 Wilson 等人提出的一个开放问题。(1995) 四分之一个世纪前。对于上述网格方块的问题变体，我们用两个积极的结果补充了硬度结果。第一的，我们证明它是固定参数易处理的，并给出了 $O(2^kn^2)$-time 算法，其中 $n=|A|$ 和 $k=|S|$。其次，我们描述了这种变体的一个特殊情况，其中始终可以在线性时间内找到连接的分区。每个积极的结果都进一步阐明了手头问题的特殊几何结构。