Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

Benjamin Moseley; Kirk Pruhs; Marc Uetz; Rudy Zhou

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arXiv · 2601.17425

Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

Benjamin Moseley·Kirk Pruhs·Marc Uetz·Rudy Zhou

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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 list scheduling policy is itself #P-hard. These are 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.

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BibTeX RIS

Benjamin Moseley, Kirk Pruhs, Marc Uetz, Rudy Zhou. 2026-01-24. Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard. https://arxiv.org/abs/2601.17425

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