We consider the problem of efficiently scheduling jobs with precedence constraints on a set of identical machines in the presence of a uniform communication delay. Such precedence-constrained jobs can be modeled as a directed acyclic graph, $G = (V, E)$. In this setting, if two precedence-constrained jobs $u$ and $v$, with $v$ dependent on $u$ ($u \prec v$), are scheduled on different machines, then $v$ must start at least $\rho$ time units after $u$ completes. The scheduling objective is to minimize makespan, i.e. the total time from when the first job starts to when the last job finishes. The focus of this paper is to provide an efficient approximation algorithm with near-linear running time. We build on the algorithm of Lepere and Rapine [STACS 2002] for this problem to give an $O\left(\frac{\ln \rho}{\ln \ln \rho} \right)$-approximation algorithm that runs in $\tilde{O}(|V| + |E|)$ time.

翻译：我们考虑的是,在统一的通信延迟的情况下,如何高效地安排工作,对一套相同的机器有优先限制,但有统一的通信延误,这种受优先限制的工作可模拟成定向的单程图,$G=(V,E)$。在这种环境下,如果两个受优先限制的工作是美元和美元,如果两个受优先限制的工作是美元和美元,而美元则取决于美元($$$),则在不同机器上安排工作,那么,美元必须在美元完成后至少开始支付每小时单位的美元。这种排程目标是尽量减少每小时,即从第一份工作开始到最后一份工作结束的总时间。本文的重点是提供一种具有近线运行时间的高效近似近算法。我们利用莱佩尔和拉宾[STACS 2002]的算法,为这一问题提供一个以美元计时值($tilde{V}+E)运行的折价算法。