An Efficient Algorithm for Computing High-Quality Paths amid Polygonal Obstacles

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 $\epsilon \in (0,1]$. Our algorithm computes in time $O(\frac{n^2}{{\epsilon }^2}\log \frac{n}{\epsilon })$ a path of total cost at most $(1+\epsilon )$ times the cost of the optimal path.