Parameterized Reductions
Bibliographic record
Abstract
In this chapter we introduce the notion of parameterized reductions. We explain how this technique can be used to transform an input for a parameterized problem $K$-$A$ into an input or parameterized problem $K$-$B$, mapping yes-instances for $K$-$A$ to yes-instances for $K$-$B$ and vice versa. If this transformation can be done in fixed-parameter tractable time, this implies that if $K$-$B$ is fixed-parameter-tractable, then so is $K$-$A$; conversely, if $K$-$A$ is not fixed-parameter tractable, then neither is $K$-$B$. Like the polynomial-time reductions introduced in Chapter 3, parameterized reductions are a powerful technique for relating problems to each other. We will demonstrate parameterized analogues of each of the reduction strategies described in Chapter 3. We also include several exercises for practicing this technique.
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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 | 0.001 | 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".