A dichotomy theorem for constraint satisfaction problems on a 3-element set
Bibliographic record
Abstract
The Constraint Satisfaction Problem (CSP) provides a common framework for many combinatorial problems. The general CSP is known to be NP-complete; however, certain restrictions on a possible form of constraints may affect the complexity and lead to tractable problem classes. There is, therefore, a fundamental research direction, aiming to separate those subclasses of the CSP that are tractable and those which remain NP-complete.Schaefer gave an exhaustive solution of this problem for the CSP on a 2-element domain. In this article, we generalise this result to a classification of the complexity of the CSP on a 3-element domain. The main result states that every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al. [2005] and Bulatov and Jeavons [2001b]. We also characterize those subproblems for which standard constraint propagation techniques provide a decision procedure. Finally, we exhibit a polynomial time algorithm which, for a given set of allowed constraints, outputs if this set gives rise to a tractable problem class. To obtain the main result and the algorithm, we extensively use the algebraic technique for the CSP developed in Jeavons [1998b], Bulatov et al.[2005], and Bulatov and Jeavons [2001b].
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 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 | 0.004 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.002 | 0.006 |
| Scholarly communication | 0.006 | 0.013 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.011 |
| Insufficient payload (model declined to judge) | 0.013 | 0.002 |
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".