Tight Approximation Results for General Covering Integer Programs
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Bibliographic record
Abstract
In this paper we study approximation algorithms for solving a general covering integer program. An-vector of nonnegative integers is sought, which minimizes � subject to � � � � � � � The entries of � � � � are nonnegative. Let be the number of rows of � � Covering problems have been heavily studied in combinatorial optimization. We focus on the effect of the multiplicity constraints, � �� on approximability. Two longstanding open questions remain for this general formulation with upper bounds on the variables. (i) The integrality gap of the standard LP relaxation is arbitrarily large. Existing approximation algorithms that achieve the well-known �-approximation with respect to the LP value do so at the expense of violating the upper bounds on the variables by the same � multiplicative factor. What is the smallest possible violation of the upper bounds that still achieves cost within � of the standard LP optimum? (ii) The best known approximation ratio for the problem has been � � � � �� � since 1982. This bound can be as bad as polynomial in the input size. Is an �-approximation, like the one known for the special case of Set Cover, possible? We settle these two open questions. To answer the first question we give an algorithm based on the relatively simple new idea of randomly rounding variables to smaller-thaninteger units. To settle the second question we give a reduction from approximating the problem while respecting multiplicity constraints to approximating the problem with a bounded violation of the multiplicity constraints. 1 Research partially supported by NSERC Grant 227809-00 and a CFI
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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 | 0.000 | 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 | 0.001 |
| 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 it