Scheduling on a Stochastic Number of Machines

TL;DR

This work studies scheduling on parallel identical machines when the exact number of machines is drawn from a known distribution and revealed only after jobs are partitioned into bags. It delivers two PTASes: (i) a PTAS for minimizing the expected makespan by combining bag-size guessing with bin-packing and makespan PTAS techniques, and (ii) a PTAS for maximizing the expected minimum load via a novel dynamic program over rounded processing times that aggregates interval-based subproblems. A key technical contribution is bounding bag sizes and using sand bags to manage small bags, enabling a polynomial-time PTAS for the stochastic setting where the number of bags and machines is part of the input. The results match the best-known deterministic approximations in the stochastic setting and offer a foundation for further exploration of stochastic machine environments in scheduling problems, including potential extensions to related machine speeds or more complex objectives.

Abstract

We consider a new scheduling problem on parallel identical machines in which the number of machines is initially not known, but it follows a given probability distribution. Only after all jobs are assigned to a given number of bags, the actual number of machines is revealed. Subsequently, the jobs need to be assigned to the machines without splitting the bags. This is the stochastic version of a related problem introduced by Stein and Zhong [SODA 2018, TALG 2020] and it is, for example, motivated by bundling jobs that need to be scheduled by data centers. We present two PTASs for the stochastic setting, computing job-to-bag assignments that (i) minimize the expected maximum machine load and (ii) maximize the expected minimum machine load (like in the Santa Claus problem), respectively. The former result follows by careful enumeration combined with known PTASs. For the latter result, we introduce an intricate dynamic program that we apply to a suitably rounded instance.

Figures (1)

• Figure 1: Visualization of relation between jobs in $J_k$ and bags in $\mathcal{B}^*_{k}$, $\mathcal{B}^*_{k+1}$ and $\bigcup_{k' = k+2}^{K} \mathcal{B}^*_{k'}$.

Theorems & Definitions (50)

• Theorem 1
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• Lemma 1
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• Lemma 7