Minimizing Completion Times of Stochastic Jobs on Parallel Machines is Hard

TL;DR

This paper proves that scheduling stochastic jobs on parallel machines to minimize expected total weighted completion time is computationally intractable, even in simplified cases, highlighting fundamental complexity challenges in this classical problem.

Contribution

It provides the first hardness results for scheduling independent stochastic jobs with a min-sum objective, demonstrating the problem's inherent computational difficulty.

Findings

Deciding scheduling policies with expected cost thresholds is #P-hard.

Evaluating the expected value of the (W)SEPT greedy policy is #P-hard.

The problem remains hard even with discrete two-point distributions and unit weights.

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.