Tiling with Squares and Packing Dominos in Polynomial Time

We consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. We give two polynomial time algorithms, one for deciding if $P$ can be tiled with $2\times 2$ squares (that is, deciding if $P$ is the union of a set of non-overlapping copies of the $2\times 2$ square) and one for packing $P$ with a maximum number of non-overlapping and axis-parallel $2\times 1$ dominos, allowing rotations of $90^\circ$. As packing is more general than tiling, the latter algorithm can also be used to decide if $P$ can be tiled by $2\times 1$ dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of $2\times 2$ squares is known to be NP-Hard [J.~Algorithms 1990]. For our three problems there are known pseudo-polynomial time algorithms, that is, algorithms with running times polynomial in the \emph{area} of $P$. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial time algorithms for the problems. Concretely, we give a simple $O(n\log n)$ algorithm for tiling with squares, and a more involved $O(n^4)$ algorithm for packing and tiling with dominos.