Abstract
A path P between two points s and t in a polygonal subdivision T with obstacles and weighted regions defines a class of paths that can be deformed to P without passing over any obstacle. We present the first algorithm that, given P and a relative error tolerance epsilon is an element of (0, 1), computes a path from this class with cost at most 1 + epsilon times the optimum. The running time is O(h(3)/epsilon(2)kn polylog(k, n, 1/epsilon)), where k is the number of segments in P and h and n are the numbers of obstacles and vertices in T, respectively. The constant in the running time of our algorithm depends on some geometric parameters and the ratio of the maximum region weight to the minimum region weight.