Combination can be hard: approximability of the unique coverage problem
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
We prove semi-logarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, we prove O(1/ logσ(e)n) inapproximability assuming that NP n BPTIME(2ne) for some e > 0. We also prove O(1/log1/3-en) inapproximability, for any e > 0, assuming that refuting random instances of 3SAT is hard on average; and prove O(1/log n) inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish matching upper bounds up to exponents, even for a more general (budgeted) setting, giving an Ω(1/log n)-approximation algorithm as well as an Ω(1/log B)-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, and radio broad-casting.
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.002 | 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 it