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.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
How this classification was reachedexpand
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".