Tiling with Squares and Packing Dominos in Polynomial Time

TL;DR

This paper presents the first polynomial time algorithms for tiling a polyomino with squares and packing it with dominos, solving classical problems relevant to VLSI design efficiently.

Contribution

It introduces novel polynomial time algorithms for tiling with squares and packing with dominos, improving over previous pseudo-polynomial solutions.

Findings

Polynomial time algorithm for tiling with fixed-size squares.

Polynomial time algorithm for packing with 2x1 dominos.

Applications in VLSI design and related problems.

Abstract

A polyomino is a polygonal region with axis parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container $P$. We give two polynomial time algorithms, one for deciding if $P$ can be tiled with $k\times k$ squares for any fixed $k$ which can be part of the input (that is, deciding if $P$ is the union of a set of non-overlapping $k\times k$ squares) and one for packing $P$ with a maximum number of non-overlapping and axis-parallel $2\times 1$ dominos, allowing rotations by $90^\circ$. 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 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 $O(n^3\,\text{polylog}\, n)$ algorithm for packing and tiling with dominos, where $n$ is the number of corners of $P$.