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(ε2n2logεn) a path of total cost
at most (1+ε) times the cost of the optimal path.Comment: A preliminary version of this work appear in the Proceedings of the
27th Annual ACM-SIAM Symposium on Discrete Algorithm