On the empirical time complexity of random 3-SAT at the phase transition
Bibliographic record
Abstract
The time complexity of problems and algorithms, i.e., the scaling of the time required for solving a problem instance as a function of instance size, is of key interest in theoretical computer science and practical applications. In this context, propositional satisfiability (SAT) is one of the most intensely studied problems, and it is generally believed that solving SAT requires exponential time in the worst case. For more than two decades, random 3-SAT at the solubility phase transition has played a pivotal role in the theoretical and empirical investigation of SAT solving, and to this day, it is arguably the most prominent model for difficult SAT instances. Here, we study the empirical scaling of the running time of several prominent, high-performance SAT solvers on random 3-SAT instances from the phase transition region. After introducing a refined model for the location of the phase transition point, we show that the median running time of three incomplete, SLS-based solvers - WalkSAT/SKC, BalancedZ and probSAT - scales polynomially with instance size. An analogous analysis of three complete, DPLL-based solvers - kcnfs, march_hi and march_br - clearly indicates exponential scaling of median running time. Moreover, exponential scaling is witnessed for these DPLL-based solvers when solving only satisfiable and only unsatisfiable instances, and the respective scaling models for each solver differ mostly by a constant factor.
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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 | 0.069 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.003 | 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".