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

We study a path-planning problem amid a set O of obstacles in R², in which we wish to compute a short path between two points while also maintaining a high clearance from O; the clearance of a point is its distance from a nearest obstacle in 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 ε ∈ (0, 1]. Our algorithm computes in time O(n²/ε² log n/ε) a path of total cost at most (1 + ε) times the cost of the optimal path.