Approximation Algorithms for Stochastic Minimum-Norm Combinatorial Optimization
Bibliographic record
Abstract
Motivated by the need for, and growing interest in, modeling uncertainty in data, we introduce and study stochastic minimum-norm optimization. We have an underlying combinatorial optimization problem where the costs involved are random variables with given distributions; each feasible solution induces a random multidimensional cost vector, and given a certain objective function, the goal is to find a solution (that does not depend on the realizations of the costs) that minimizes the expected objective value. For instance, in stochastic load balancing, jobs with random processing times need to be assigned to machines, and the induced cost vector is the machine-load vector. The choice of objective is typically the maximum- or sum-of the entries of the cost vector, or in some cases some other lp norm of the cost vector. Recently, in the deterministic setting, Chakrabarty and Swamy [7] considered a much broader suite of objectives, wherein we seek to minimize the f-norm of the cost vector under a given arbitrary monotone, symmetric norm f. In stochastic minimum-norm optimization, we work with this broad class of objectives, and seek a solution that minimizes the expected f-norm of the induced cost vector. The class of monotone, symmetric norms is versatile and includes lp-norms, and Topl-norms (sum of l largest coordinates in absolute value), and enjoys various closure properties; in particular, it can be used to incorporate multiple norm budget constraints, fl(x) ≤ Bl, l = 1, ..., k. We give a general framework for devising algorithms for stochastic minimum-norm combinatorial optimization, using which we obtain approximation algorithms for the stochastic minimum-norm versions of the load balancing and spanning tree problems. We obtain the following concrete results. (a) An O(1)-approximation for stochastic minimum-norm load balancing on unrelated machines with: (i) arbitrary monotone symmetric norms and job sizes that are Bernoulli random variables; and (ii) Topl norms and arbitrary job-size distributions. (b) An O(log m/log log m)-approximation for the general stochastic minimum-norm load balancing problem, where m is the number of machines. (c) An O(1)-approximation for stochastic minimum-norm spanning tree with arbitrary monotone symmetric norms and arbitrary edge-weight distributions; this guarantee extends to the stochastic minimum-norm matroid basis problem. Two key technical contributions of this work are: (1) a structural result of independent interest connecting stochastic minimum-norm optimization to the simultaneous optimization of a (small) collection of expected Topl-norms; and (2) showing how to tackle expected Topl-norm minimization by leveraging techniques used to deal with minimizing the expected maximum, circumventing the difficulties posed by the non-separable nature of Toplnorms.
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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 | 0.016 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.002 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.003 | 0.003 |
| 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".