Computing simple paths among obstacles

Given a set X of points in the plane, two distinguished points s; t 2 X, and a set of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in. We present two results: (1) We show that nding such simple paths among arbitrary obstacles is NP-complete. (2) We give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P. Our algorithm runs in time O(m 2 n 2), where m is the number of vertices of P and n is the number of points in X.