Treffer: A generalized colouring method for a parallelizable integer linear programming approach to polyomino tiling.

This article presents a generalized checkerboard colouring method for solving large polyomino tiling problems using integer linear programming (ILP). Extending a previous two-colour approach, our method scales to three or more colours, improving runtimes in cases where the two-colour method offers no advantage. We rigorously derive ILP formulations for both the coloured and uncoloured cases and propose a proof-of-concept parallel implementation to improve scalability in these NP -complete tiling problems. Numerical experiments using MATLAB, CPLEX, and Gurobi demonstrate significant reductions in problem size and computation time. This approach has the potential to solve tiling problems substantially larger than those previously tractable using ILP-based methods. Although our colouring method generates many subproblems, it allows for the efficient exploration of a manageable subset to find feasible solutions. Our publicly available MATLAB package, CHROMINOES (v1.0.0), constructs ILP formulations, plots solutions, and is available for download on Zenodo.org. We conclude by discussing the implications of these results and outlining key challenges in developing a fully practical parallel implementation, including load balancing and managing the overhead associated with processing a large number of subproblems. [ABSTRACT FROM AUTHOR]