A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths
Bibliographic record
Abstract
The following multiprocessor scheduling problem was motivated by scheduling maintenance periods for aircraft. Each maintenance period is a job, and the maintenance facilities are machines. In this context, there are very few different types of maintenances performed, so it is natural to consider the problem with only a small, fixed number C of different types of jobs. Each job type has a processing time, and each machine is available for the same length of time. A machine can handle at most one job at a time, all jobs are released at time zero, there are no due dates or precedence constraints, and preemption is not allowed. The question is whether it is possible to finish all jobs. We call this problem the Multiprocessor Scheduling Problem with C job lengths (MSPC). Scheduling problems such as MSPC where we can partition the jobs into relatively few types such that all jobs of a certain type are identical are often called high-multiplicity problems. High-multiplicity problems are interesting because their input is very compact: The input to MSPC consists of only 2C + 2 numbers. For the case C = 2 we present a polynomial-time algorithm. We show that this algorithm guarantees a schedule that uses the minimum possible number of different one-machine schedules, namely three. Further, we extend this algorithm to the case of machine-dependent deadlines (uniform parallel machines), to a multi-parametric case (that contains the case of unrelated parallel machines), and to some related covering problems. Finally, we give some counterexamples showing why our results do not extend to the case C > 2.
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.003 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| Insufficient payload (model declined to judge) | 0.015 | 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".