Bibliographic record
Abstract
The 2-page upward book embedding (2UBE) problem, recently proven NP-complete, is a fundamental problem in graph theory with various applications. Given its computational intractability, SAT-based approaches offer a promising direction for efficiently determining embeddability. In this paper, we present two practical SAT encodings, SAT-1 (largely adopted from the literature) and SAT-2 (specifically tailored for 2UBE). We also introduce a Constraint Programming (CP) formulation as an alternative approach. Our empirical evaluation on benchmark datasets demonstrates that SAT solvers significantly outperform CP in solving 2UBE instances. Additionally, we analyze the scalability of these methods on large grid graphs, revealing that SAT-2 achieves up to a 40% speedup over SAT-1. Using SAT-2, we discovered a phase transition in 2UBE, which occurs when the edge-to-node ratio (m/n) ≈ 1.5. In general, we have established SAT-based approaches as both a practical and scalable solution for book embedding problems and a tool for evaluating their structural properties.
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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 | 0.005 | 0.058 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.005 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.006 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.016 | 0.004 |
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".