We consider a problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension n×m and ask whether or not a tiling of this region can be rearranged so that tiling of the n×m rectangle can be realized as a tiling of an n×m′
Nearby in the stack
rectangle and an
n×m"
rectangle,
m=m′+m"
. We call this a subtiling. We show that the associated decision problem is
NP
-complete when restricted to rectangular polyominoes. We also show that for certain finite libraries of polyominoes, if
m
is sufficiently large, a subtiling always exists and give bounds.