Approximating shortest paths on a nonconvex polyhedron

We present an approximation algorithm that, given the boundary P of a simple, nonconvex polyhedron in R³, and two points s and t on P, constructs a path on P between s and t whose length is at most 7(1+ε)d_P(s,t), where d_P(s,t) is the length of the shortest path between s and t on P, and ε>0 is an arbitrarily small positive constant. The algorithm runs in O(n^5/3 log^5/3 n) time, where n is the number of vertices in P. We also present a slightly faster algorithm that runs in O(n^8/5 log^8/5 n) time and returns a path whose length is at most 15(1+ε)d_P(s,t)