The Approximability and Integrality Gap of Interval Stabbing and Independence Problems
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Bibliographic record
Abstract
Motivated by problems such as rectangle stabbing in the plane, we study the minimum hitting set and maximum independent set problems for families of d-intervals and d-union-intervals. We obtain the following: (1) constructions yielding asymptotically tight lower bounds on the integrality gaps of the associated natural linear programming relaxations; (2) an LP-relative dapproximation for the hitting set problem ond-intervals; (3) a proof that the approximation ratios for independent set on families of 2-intervals and 2-union-intervals can be improved to match tight duality gap lower bounds obtained via topological arguments, if one has access to an oracle for a PPAD-complete problem related to nding Borsuk-Ulam xed-points.
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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 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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