Optimal polygon placement by translation1

Let M be an m-sided simple polygon and N be an n-sided polygon with holes. In this paper we consider the problem of computing the feasible region, i.e., the set of all placements by translation of M so that M lies inside N without intersecting any hole. First we propose an 0(mn2) time algorithm for computing the feasible region for the case when M is a monotone polygon. Then we consider the general case when M is a simple polygon and propose an 0(m2n2) time algorithm for computing the feasible region. Both algorithms are optimal upto a constant factor.