Lp-Based Approximation Algorithms For Scheduling And Inventory Management Problems
Bibliographic record
Abstract
There are two fundamental approaches for using linear programming in designing approximation algorithms: LP-rounding and the primal-dual method. In this thesis, we develop LP-based approximation algorithms for several scheduling and inventory management problems, using both LP-rounding and the primal-dual method. In machine scheduling, we consider a general class of single-machine scheduling problem of minimizing the total cost summing over all jobs, and the only requirement on the cost function of each job is that it is non-negative and non-decreasing. Using the primal-dual method, we give a simple algorithm for this problem that is guaranteed to return a solution that costs at most twice the optimal. To obtain this result, we add an exponential number of valid inequalities to strengthen the natural LP-relaxation, then design a primal-dual method that works on this exponential-sized LP. We then show how to modify our algorithm for scheduling problems with machine breakdown. In inventory management, we consider several generalization of the classical Joint Replenishment Problem (JRP): the tree JRP and the cardinality JRP. Using the LP-rounding technique, we give novel algorithms for the tree JRP and cardinality JRP that are guaranteed to generate a solution with cost within a factor of three and five, respectively, of the optimal.
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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 | 0.004 | 0.008 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.003 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.005 |
| Insufficient payload (model declined to judge) | 0.005 | 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".