A generalized colouring method for a parallelizable integer linear programming approach to polyomino tiling
Bibliographic record
Abstract
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.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".