The Meet-in-the-Middle Principle for Cutting and Packing Problems
Bibliographic record
Abstract
Cutting and packing (C&P) is a fundamental research area that models a large number of managerial and industrial optimization issues. A solution to a C&P problem basically consists of a set of one-dimensional or multidimensional items packed in/cut from one or more bins, by satisfying problem constraints and minimizing a given objective function. Normal patterns are a well-known C&P technique used to build solutions where each item is aligned to the bottom of the bin along each dimension. They are used in several C&P techniques because they can reduce the search space while preserving optimality, but their limit is that their number grows consistently when number of items and size of the bin increase. In this paper we propose a new set of patterns, called meet in the middle, that preserves optimality and leads to several interesting results. Their computation is achieved with the same time complexity as that of the normal patterns, but their number is never higher, and in practical applications it frequently shows reductions of about 50%. These new patterns are applied to improve some exact state-of-the-art C&P techniques, including arc-flow formulations, combinatorial branch-and-bound algorithms, and mixed-integer linear programs. The efficacy of the improved techniques is assessed by extensive computational tests on a number of relevant applications. The online appendix is available at https://doi.org/10.1287/ijoc.2018.0806 .
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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 | 0.003 | 0.009 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.003 |
| Bibliometrics | 0.002 | 0.004 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.004 | 0.007 |
| Open science | 0.003 | 0.004 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.010 | 0.004 |
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".