Complexity of Tiling a Polygon with Trominoes or Bars

Algorithms

The general problem of tiling finite regions of the plane with polyominoes is NP-complete, and so the associated computational geometry problem rapidly becomes intractable for large instances. Thus, the need to reduce algorithm complexity for tiling is important and continues as a fruitful area of research. Traditional approaches to tiling with polyominoes use backtracking, which is a refinement of the ‘brute-force’ solution procedure for exhaustively finding all solutions to a combinatorial search problem. In this work, we combine checkerboard colouring techniques with a recently introduced integer linear programming (ILP) technique for tiling with polyominoes. The colouring arguments often split large tiling problems into smaller subproblems, each represented as a separate ILP problem. Problems that are amenable to this approach are embarrassingly parallel, and our work provides proof of concept of a parallelizable algorithm. The main goal is to analyze when this approach yields a...

Algorithms, 2022

Checkerboard colouring arguments for proving that a given collection of polyominoes cannot tile a finite target region of the plane are well-known and typically applied on a case-by-case basis. In this article, we give a systematic mathematical treatment of such colouring arguments, based on the concept of a parity violation, which arises from the mismatch between the colouring of the tiles and the colouring of the target region. Identifying parity violations is a combinatorial problem related to the subset sum problem. We convert the combinatorial problem into linear Diophantine equations and give necessary and sufficient conditions for a parity violation. The linear Diophantine equation approach leads to an algorithm implemented in MATLAB for finding all possible parity violations of large tiling problems, and is the main contribution of this article. Numerical examples illustrate the effectiveness of our algorithm. The collection of MATLAB programs, POLYOMINO_PARITY (v2.0.0) is f...

Information and Control, 1984

Decomposing a polygon into simple shapes is a basic problem in computational geometry, with applications in pattern recognition and integrated circuit manufacture. Here we examine the special case of covering a rectilinear polygon (or polyomino) with the minimum number of rectangles, with overlapping allowed. The problem is NP-hard. However, we give here an O(v z) algorithm for constructing a minimum rectangle cover, when the polygon is vertically convex. (Here v is the number of vertices.) The problem is first reduced to a 1-dimensional interval "basis" problem. In showing our algorithm produces an optimal cover we give a new proof of a minimum basis-maximum independent set duality theorem first proved by E.

Operations Research Letters, 2011

We initiate a parameterized complexity study of the NP-hard problem to tile a positive integer matrix with rectangles, keeping the number of tiles and their maximum weight small. We show that the problem remains NP-hard even for binary matrices only using 1×1 and 2×2-squares as tiles and provide insight into the influence of naturally occurring parameters on the problem's complexity.

Journal of Combinatorial Theory, Series A, 1990

Discrete and Computational Geometry, 1992

Under what conditions can a simple polygon be tiled by parallelograms? In this paper we give matching necessary and sufficient conditions on the polygon to be tilable and characterize the set of possible tilings. We also provide an etficient algorithm for constructing a tiling.

Annals of the New York Academy of Sciences, 1985

Theoretical Computer Science, 2020

We show that the problem of deciding whether a collection of polyominoes, each fitting in a 2 × O(log n) rectangle, can be packed into a 3 × n box does not admit a 2 o(n/ log n)-time algorithm, unless the Exponential Time Hypothesis fails. We also give an algorithm that attains this lower bound, solving any instance of polyomino packing with total area n in 2 O(n/ log n) time. This establishes a tight bound on the complexity of Polyomino Packing, even in a very restricted case. In contrast, for a 2 × n box, we show that the problem can be solved in strongly subexponential time. 2012 ACM Subject Classification Theory of computation → Problems, reductions and completeness, Mathematics of computing → Combinatorial algorithms Keywords and phrases polyomino packing, exact complexity, exponential time hypothesis

2017

The Tokyo 2020 Olympic and Paralympic games emblems, called ‘harmonized chequered emblems,’ are designed with three kinds of rectangles. We enumerate all such tilings in a dodecagon with a hole.

Proceedings of the thirty-third annual ACM symposium on Theory of computing - STOC '01, 2001

We study the minimal complexity of tilings of a plane with a given tile set. We note that every tile set admits either no tiling or some tiling with O(n) Kolmogorov complexity of its (n × n)-squares. We construct tile sets for which this bound is tight: all (n × n)-squares in all tilings have complexity at least n. This adds a quantitative angle to classical results on non-recursivity of tilings -that we also develop in terms of Turing degrees of unsolvability.