A Simple Approximation Algorithm for Vector Scheduling and Applications to Stochastic Min-Norm Load Balancing
Bibliographic record
Abstract
We consider the Vector Scheduling problem on identical machines: we have m machines, and a set J of n jobs, where each job j has a processing-time vector $p_j\in \mathbb{R}^d_{\geq 0}$. The goal is to find an assignment $σ:J\to [m]$ of jobs to machines so as to minimize the makespan $\max_{i\in [m]}\max_{r\in [d]}( \sum_{j:σ(j)=i}p_{j,r})$. A natural lower bound on the optimal makespan is lb $:=\max\{\max_{j\in J,r\in [d]}p_{j,r},\max_{r\in [d]}(\sum_{j\in J}p_{j,r}/m)\}$. Our main result is a very simple O(log d)-approximation algorithm for vector scheduling with respect to the lower bound lb: we devise an algorithm that returns an assignment whose makespan is at most O(log d)*lb. As an application, we show that the above guarantee leads to an O(log log m)-approximation for Stochastic Minimum-Norm Load Balancing (StochNormLB). In StochNormLB, we have m identical machines, a set J of n independent stochastic jobs whose processing times are nonnegative random variables, and a monotone, symmetric norm $f:\mathbb{R}^m \to \mathbb{R}_{\geq 0}$. The goal is to find an assignment $σ:J\to [m]$ that minimizes the expected $f$-norm of the induced machine-load vector, where the load on machine i is the (random) total processing time assigned to it. Our O(log log m)-approximation guarantee is in fact much stronger: we obtain an assignment that is simultaneously an O(log log m)-approximation for StochNormLB with all monotone, symmetric norms. Next, this approximation factor significantly improves upon the O(log m/log log m)-approximation in (Ibrahimpur and Swamy, FOCS 2020) for StochNormLB, and is a consequence of a more-general black-box reduction that we present, showing that a $γ(d)$-approximation for d-dimensional vector scheduling with respect to the lower bound lb yields a simultaneous $γ(\log m)$-approximation for StochNormLB with all monotone, symmetric norms.
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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 | 0.001 | 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 | 0.001 | 0.002 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
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".