A New Multilayered PCP and the Hardness of Hypergraph Vertex Cover
Bibliographic record
Abstract
Given a k-uniform hypergraph, the Ek-Vertex-Cover problem is to find the smallest subset of vertices that intersects every hyperedge. We present a new multilayered probabilistically checkable proof (PCP) construction that extends the Raz verifier. This enables us to prove that Ek-Vertex-Cover is NP-hard to approximate within a factor of $(k-1-\epsilon)$ for arbitrary constants $\epsilon>0$ and $k\ge 3$. The result is nearly tight as this problem can be easily approximated within factor k. Our construction makes use of the biased long-code and is analyzed using combinatorial properties of s-wise t-intersecting families of subsets. We also give a different proof that shows an inapproximability factor of $\lfloor \frac{k}{2} \rfloor -\eps$. In addition to being simpler, this proof also works for superconstant values of k up to (log N)1/c, where c > 1 is a fixed constant and N is the number of hyperedges.
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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 | 0.002 | 0.017 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.002 | 0.005 |
| Research integrity | 0.002 | 0.004 |
| Insufficient payload (model declined to judge) | 0.006 | 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".