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 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 | 0.000 |
| 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".