Worst-case analysis for new online bin packing problems
Bibliographic record
Abstract
We consider two new online bin packing problems, the online Variable Cost and Size Bin Packing Problem (o-VCSBPP) and the online Generalized Bin Packing Problem (o-GBPP). We take two well-known bin packing algorithms to address them, the First Fit (FF) and the Best Fit (BF). We show that both algorithms have an asymptotic worst-case ratio bound equal to 2 for the o-VCSBPP and this bound is tight. When there are enough bins of a particular type to load all items, FF and BF also have an absolute worst-case ratio bound equal to 2 for the o-VCSBPP, and this bound is also tight. In addition, we prove that no worst-case ratio bound of FF and BF can be computed for the o-GBPP. Therefore, we consider a natural evolution of these algorithms, the First Fit with Rejection and the Best Fit with Rejection, able to reject inconvenient bins at the end of the process. Similarly, we prove that no worst-case ratio of these algorithms can be computed for the o-GBPP. Finally, we give sucient conditions under which algorithms do not admit any performance ratio, and conclude that the worst-case results obtained for the o-VCSBPP and the o-GBPP also hold for the oine variant of these two problems
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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 | 0.011 | 0.043 |
| Meta-epidemiology (narrow) | 0.004 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.002 |
| Bibliometrics | 0.003 | 0.005 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.006 | 0.012 |
| Open science | 0.009 | 0.004 |
| Research integrity | 0.003 | 0.006 |
| Insufficient payload (model declined to judge) | 0.007 | 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".