A Parallelizable Integer Linear Programming Approach for Tiling Finite Regions of the Plane with Polyominoes
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Bibliographic record
Abstract
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 potential parallel speedup. The novel colouring technique shows excellent promise in yielding a parallel speedup for finding large tiling solutions with ILP, particularly when we seek a single (optimal) solution. We also classify the tiling problems that result from applying our colouring technique according to different criteria and compute representative examples using a combination of MATLAB and CPLEX, a commercial optimization package that can solve ILP problems. The collections of MATLAB programs PARIOMINOES (v3.0.0) and POLYOMINOES (v2.1.4) used to construct the ILP problems are freely available for download.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 it