Congruent Partitions of Polygons

Given a polygon $P$ and integer $n>0$, choose $n$ pairwise interior-disjoint mutually congruent polygonal pieces contained in $P$ so as to minimize the uncovered area of $P$. Determine the optimum, decide when it is zero, and characterize the complexity for $n>2$, with congruence either allowing or forbidding reflection.