Abstract
A polyomino is a polygonal region with axisparallel 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 polynomialtime algorithms for deciding if P can be tiled with k× 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 nonoverlapping k× k squares) and for packing P with a maximum number of nonoverlapping and axisparallel 2× 1 dominos, allowing rotations by 90^∘. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2× 1 dominos.
These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2× 2 squares is known to be NPhard [J. Algorithms 1990]. For our three problems there are known pseudopolynomialtime algorithms, that is, algorithms with running times polynomial in the area or perimeter 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 polynomialtime algorithms for the problems. Concretely, we give a simple O(nlog n)time algorithm for tiling with squares, where n is the number of corners of P. We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O(n³) vertices. This leads to algorithms with running times O(n³(log³ n)/(log²log n)) and O(n³(log² n)/(log log n)), respectively.
BibTeX  Entry
@InProceedings{aamand_et_al:LIPIcs.SoCG.2022.1,
author = {Aamand, Anders and Abrahamsen, Mikkel and Ahle, Thomas and Rasmussen, Peter M. R.},
title = {{Tiling with Squares and Packing Dominos in Polynomial Time}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {1:11:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772273},
ISSN = {18688969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16009},
URN = {urn:nbn:de:0030drops160096},
doi = {10.4230/LIPIcs.SoCG.2022.1},
annote = {Keywords: packing, tiling, polyominos}
}
Keywords: 

packing, tiling, polyominos 
Collection: 

38th International Symposium on Computational Geometry (SoCG 2022) 
Issue Date: 

2022 
Date of publication: 

01.06.2022 