Approximate shortest descent path on a terrain

A path from a point s to a point t on the surface of a polyhedral terrain is said to be descent if for every pair of points p = (x(p), y(p), z(p)) and q = (x(q), y(q), z(q)) on the path, if dist(s,p) < dist(s,q) then z(p) > z(q), where dist(s,p) denotes the distance of p from s along the aforesaid path. Although an efficient algorithm to decide if there is a descending path between two points is known for more than a decade, no efficient algorithm is yet known to find a shortest descending path from s to t in a polyhedral terrain. In this paper we propose an (1 + ε-approximation algorithm running in polynomial time for the same.