We consider a variant of the NP-hard problem of assigning jobs to machines to minimize the completion time of the last job. Usually, precedence constraints are given by a partial order on the set of jobs, and each job requires all its predecessors to be completed before it can start. In this paper, we consider a different type of precedence relation that has not been discussed as extensively and is called OR-precedence. In order for a job to start, we require that at least one of its predecessors is completed—in contrast to all its predecessors. Additionally, we assume that each job has a release date before which it must not start. We prove that a simple List Scheduling algorithm due to Graham (Bell Syst Tech J 45(9):1563–1581, 1966) has an approximation guarantee of 2 and show that obtaining an approximation factor of \(4/3 - \varepsilon \) is NP-hard. Further, we present a polynomial-time algorithm that solves the problem to optimality if preemptions are allowed. The latter result is in contrast to classical precedence constraints where the preemptive variant is already NP-hard. Our algorithm generalizes previous results for unit processing time jobs subject to OR-precedence constraints, but without release dates. The running time of our algorithm is \(O(n^2)\) for arbitrary processing times and it can be reduced to O(n) for unit processing times, where n is the number of jobs. The performance guarantees presented here match the best-known ones for special cases where classical precedence constraints and OR-precedence constraints coincide.

我们考虑将作业分配给机器的NP难题的一种变体，以最大程度地减少上一个作业的完成时间。通常，优先约束由作业集的偏序给出，并且每个作业都需要完成其所有前驱才能开始。在本文中，我们考虑了一种尚未广泛讨论的不同类型的优先关系，称为 OR 优先级。为了开始一项工作，我们要求至少完成一个它的前辈——而不是所有的它的前辈。此外，我们假设每个工作都有一个发布日期，在该日期之前它不能开始。我们证明了由Graham（Bell Syst Tech J 45（9）：1563–1581，1966）提出的一种简单的列表调度算法具有大约2的近似保证，并证明获得了近似因子\（4/3-\ varepsilon \ )是 NP 难的。此外，我们提出了多项式时间算法，该算法可以在允许抢占的情况下将问题解决为最优。后一个结果与经典优先约束形成对比，其中抢占变体已经是 NP-hard。我们的算法概括了受 OR 优先约束但没有发布日期的单位处理时间作业的先前结果。我们算法的运行时间是\(O(n^2)\)对于任意处理时间，单位处理时间可以减少到O ( n )，其中n是作业数。对于经典优先约束和 OR 优先约束重合的特殊情况，此处提供的性能保证与最著名的保证相匹配。