Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

2026-01-24

AI Summary

This study investigates the computational complexity of stochastic scheduling on identical parallel machines to minimize the expected total weighted completion time. Despite the abundance of approximation algorithms, a rigorous theoretical foundation has long been lacking, with constant-factor approximations known only under strong distributional assumptions. This work establishes, for the first time, that the problem is #P-hard even in the restricted setting of unit weights and two-point processing time distributions, without relying on the hardness of the deterministic counterpart. Specifically, it proves that both deciding whether a scheduling policy exists whose expected cost meets a given threshold and computing the expected objective value of the classical (W)SEPT greedy policy are #P-hard. The result, obtained via a #P-completeness reduction, fills a longstanding gap in the complexity theory of stochastic scheduling for min-sum objectives.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Planning, Routing, and Scheduling: Scheduling under Uncertainty
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

Abstract

This paper considers the scheduling of stochastic jobs on parallel identical machines to minimize the expected total weighted completion time. While this is a classical problem with a significant body of research on approximation algorithms over the past two decades, constant-factor performance guarantees are currently known only under very restrictive assumptions on the input distributions, even when all job weights are identical. This algorithmic difficulty is striking given the lack of corresponding complexity results: to date, it is conceivable that the problem could be solved optimally in polynomial time. We address this gap with hardness results that demonstrate the problem's inherent intractability. For the special case of discrete two-point processing time distributions and unit weights, we prove that deciding whether there exists a scheduling policy with expected cost at most a given threshold is #P-hard. Furthermore, we show that evaluating the expected objective value of the standard (W)SEPT greedy policy is itself #P-hard. These represent the first hardness results for scheduling independent stochastic jobs and min-sum objective that do not merely rely on the intractability of the underlying deterministic counterparts.

Problem

• stochastic scheduling
• parallel machines
• completion time
• computational hardness
• #P-hard

Innovation

• stochastic scheduling
• #P-hardness
• parallel machines
• expected completion time
• WSEPT policy