Let P and Q be two simple polygons in the plane of total complexity n, each of which can be decomposed into at most k convex parts. We present an (1 − ε)-approximation algorithm, for finding the translation of Q, which maximizes its area of overlap with P. Our algorithm runs in O(cn) time, where c is a constant that depends only on k and ε. This suggest that for polygons that are “close” to being convex, the problem can be solved (approximately), in near linear time.