Pankaj K. Agarwal

Pankaj K. Agarwal, Boris Aronov, Tzvika Geft et al.

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.

Pankaj K. Agarwal, Kyle Fox, Oren Salzman

We study a path-planning problem amid a set $\mathcal{O}$ of obstacles in $\mathbb{R}^2$, in which we wish to compute a short path between two points while also maintaining a high clearance from $\mathcal{O}$; the clearance of a point is its distance from a nearest obstacle in $\mathcal{O}$. Specifically, the problem asks for a path minimizing the reciprocal of the clearance integrated over the length of the path. We present the first polynomial-time approximation scheme for this problem. Let $n$ be the total number of obstacle vertices and let $\varepsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{\varepsilon ^2} \log \frac{n}{\varepsilon})$ a path of total cost at most $(1+\varepsilon)$ times the cost of the optimal path.