Restricted Adaptivity in Stochastic Scheduling

TL;DR

This paper introduces a new class of scheduling policies that interpolate between fixed and adaptive strategies, achieving significantly better performance than fixed policies with minimal adaptivity in stochastic parallel machine scheduling.

Contribution

The paper proposes $ au$-shift and $ au$-delay policies with controllable adaptivity, providing an $ ext{O}(\log \log m)$-approximation and matching lower bounds, improving upon fixed assignment policies.

Findings

New policies achieve exponential improvement over fixed policies.

A matching lower bound demonstrates the limits of non-adaptive policies.

The policies are effective with small degrees of adaptivity.

Abstract

We consider the stochastic scheduling problem of minimizing the expected makespan on $m$ parallel identical machines. While the (adaptive) list scheduling policy achieves an approximation ratio of $2$, any (non-adaptive) fixed assignment policy has performance guarantee $\Omega\left(\frac{\log m}{\log \log m}\right)$. Although the performance of the latter class of policies are worse, there are applications in which non-adaptive policies are desired. In this work, we introduce the two classes of $\delta$-delay and $\tau$-shift policies whose degree of adaptivity can be controlled by a parameter. We present a policy - belonging to both classes - which is an $\mathcal{O}(\log \log m)$-approximation for reasonably bounded parameters. In other words, an exponential improvement on the performance of any fixed assignment policy can be achieved when allowing a small degree of adaptivity. Moreover, we provide a matching lower bound for any $\delta$-delay and $\tau$-shift policy when both parameters, respectively, are in the order of the expected makespan of an optimal non-anticipatory policy.