An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles
This solves a specific path-planning problem for robotics or computational geometry, but it is incremental as it builds on existing approximation schemes.
The paper tackles the problem of computing a path that is both short and maintains high clearance from obstacles in 2D, presenting the first polynomial-time approximation scheme that finds a path with cost at most (1+ε) times optimal in O(n²/ε² log(n/ε)) time.
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.