Assembly planning is a fundamental problem in robotics and automation, which involves designing a sequence of motions to bring the separate constituent parts of a product into their final placement in the product. Assembly planning is naturally cast as a disassembly problem, giving rise to the assembly partitioning problem: Given a set $A$ of parts, find a subset $S\subset A$, referred to as a subassembly, such that $S$ can be rigidly translated to infinity along a prescribed direction without colliding with $A\setminus S$. While assembly partitioning is efficiently solvable, it is further desirable for the parts of a subassembly to be easily held together. This motivates the problem that we study, called connected-assembly-partitioning, which additionally requires each of the two subassemblies, $S$ and $A\setminus S$, to be connected. We show that this problem is NP-complete, settling an open question posed by Wilson et al. (1995) a quarter of a century ago, even when $A$ consists of unit-grid squares (i.e., $A$ is polyomino-shaped). Towards this result, we prove the NP-hardness of a new Planar 3-SAT variant having an adjacency requirement for variables appearing in the same clause, which may be of independent interest. On the positive side, we give an $O(2^k n^2)$-time fixed-parameter tractable algorithm (requiring low degree polynomial-time pre-processing) for an assembly $A$ consisting of polygons in the plane, where $n=|A|$ and $k=|S|$. We also describe a special case of unit-grid square assemblies, where a connected partition can always be found in $O(n)$-time.

翻译：装配规划是机器人与自动化领域的基础问题，涉及设计一系列运动序列以将产品的独立组成部件按最终位置组装。装配规划自然被视为拆卸问题，由此引出了装配分割问题：给定部件集合 $A$，寻找子集 $S\subset A$（称为子装配体），使得 $S$ 可沿指定方向刚性平移至无穷远而不与 $A\setminus S$ 碰撞。尽管装配分割问题可高效求解，但进一步要求子装配体的部件易于固定在一起。这促使我们研究称为“连通装配分割”的问题，该问题额外要求两个子装配体 $S$ 和 $A\setminus S$ 均保持连通。我们证明该问题是NP完全的，解决了Wilson等人（1995）四分之一世纪前提出的开放问题——即便当 $A$ 由单位网格正方形构成（即 $A$ 为多联骨牌形状）时结论仍成立。为得到该结果，我们证明了新型平面3-SAT变体的NP难性，该变体对同一子句中变量存在邻接约束，此结果可能具有独立研究价值。在正面结果方面，针对由平面多边形构成的装配体 $A$（其中 $n=|A|$，$k=|S|$），我们提出了时间复杂度为 $O(2^k n^2)$ 的固定参数可处理算法（需低次多项式时间预处理）。我们还描述了单位网格正方形装配体的特例，其中连通分割可在 $O(n)$ 时间内找到。