Bibliographic record
Abstract
A decomposition of a polygon P is a set of polygons whose geometric union is exactly P . We consider the problem of decomposing a polygon, which may contain holes, using subpolygons that have a bounded diameter. We show that this problem is NP-complete via a reduction from P lanar 3, 4SAT. Polygon decomposition problems arise in applications where objects represented by polygons need to be subdivided for the sake of tractability. Many variations of decomposition problems have recieved attention in the literature. The reader is directed towards [5] for a synopsis of recent polygon decomposition results. Of particular interest are those results concerning the decomposition of non-simple polygons. The problem of minimally decomposing a polygon that may contain holes has proven to be difficult, and is typically NP-hard. Bounding box heuristics are commonly used in object intersection algorithms. It has been shown that these algorithms have better performance guarantees when the bounding boxes have similar sizes [6]. This result motivates Damian-Iordache [3] to explore the idea of restricting the diameter of the components in the decomposition of a polygon. Damian-Iordache is able to develop a polynomial time algorithm for partitioning a simple polygon into the minimum number of components that have a maximum diameter of α. Here α is a fixed real number that is part of the input to the partioning algorithm. The problem of decomposing a polygon, which may have holes, with the minimum number of diameter bounded components is conjectured to be NP-hard [3]. We confirm this conjecture by reducing P lanar 3, 4SAT to the corresponding covering and partitioning decision problems.
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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 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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".