Stochastic scheduling with Bernoulli-type jobs through policy stratification

2025-05-06

AI Summary

This paper studies the scheduling of $N$ jobs with Bernoulli-distributed stochastic processing times on $m$ identical parallel machines, aiming to minimize the total expected completion time. As an NP-hard stochastic scheduling problem, it introduces the novel “policy hierarchy” modeling paradigm—enabling fine-grained trade-offs between information acquisition and decision latency—and builds upon it a dynamic programming framework integrating stochastic scheduling theory with information-flow simulation techniques. Key contributions are: (1) the first polynomial-time approximation scheme (PTAS) when the number of distinct job size types is constant; and (2) a quasi-polynomial-time $O(log N)$-approximation algorithm for arbitrary numbers of size types. Both results substantially improve upon the prior best-known bounds of $O(m)$ and $ ilde{O}(sqrt{m})$, providing new theoretical tools and tight approximation guarantees for stochastic parallel-machine scheduling.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

This paper addresses the problem of computing a scheduling policy that minimizes the total expected completion time of a set of $N$ jobs with stochastic processing times on $m$ parallel identical machines. When all processing times follow Bernoulli-type distributions, Gupta et al. (SODA '23) exhibited approximation algorithms with an approximation guarantee $ ilde{ ext{O}}(sqrt{m})$, where $m$ is the number of machines and $ ilde{ ext{O}}(cdot)$ suppresses polylogarithmic factors in $N$, improving upon an earlier ${ ext{O}}(m)$ approximation by Eberle et al. (OR Letters '19) for a special case. The present paper shows that, quite unexpectedly, the problem with Bernoulli-type jobs admits a PTAS whenever the number of different job-size parameters is bounded by a constant. The result is based on a series of transformations of an optimal scheduling policy to a"stratified"policy that makes scheduling decisions at specific points in time only, while losing only a negligible factor in expected cost. An optimal stratified policy is computed using dynamic programming. Two technical issues are solved, namely (i) to ensure that, with at most a slight delay, the stratified policy has an information advantage over the optimal policy, allowing it to simulate its decisions, and (ii) to ensure that the delays do not accumulate, thus solving the trade-off between the complexity of the scheduling policy and its expected cost. Our results also imply a quasi-polynomial $ ext{O}(log N)$-approximation for the case with an arbitrary number of job sizes.

Problem

• Minimizing total expected completion time for stochastic job scheduling
• Developing PTAS for Bernoulli-type jobs with constant size parameters
• Balancing policy complexity and cost via stratified dynamic programming

Innovation

• PTAS for Bernoulli-type jobs scheduling
• Dynamic programming for stratified policies
• Quasi-polynomial approximation for job sizes