This paper mainly focuses on a resource leveling variant of a two-processor scheduling problem. The latter problem is to schedule a set of dependent UET jobs on two identical processors with minimum makespan. It is known to be polynomial-time solvable. In the variant we consider, the resource constraint on processors is relaxed and the objective is no longer to minimize makespan. Instead, a deadline is imposed on the makespan and the objective is to minimize the total resource use exceeding a threshold resource level of two. This resource leveling criterion is known as the total overload cost. Sophisticated matching arguments allow us to provide a polynomial algorithm computing the optimal solution as a function of the makespan deadline. It extends a solving method from the literature for the two-processor scheduling problem. Moreover, the complexity of related resource leveling problems sharing the same objective is studied. These results lead to polynomial or pseudo-polynomial algorithms or NP-hardness proofs, allowing for an interesting comparison with classical machine scheduling problems.

翻译：本文主要研究双处理器调度问题的一个资源均衡化变体。原问题是在两个相同处理器上调度一组具有依赖关系的UET作业，以最小化完工时间，该问题已知存在多项式时间解法。在我们考虑的变体中，处理器的资源约束被放宽，目标不再是最小化完工时间，而是对完工时间设置截止期限，目标转为最小化超过阈值资源水平（取值为二）的总资源使用量。该资源均衡化准则被称为总过载成本。通过精巧的匹配论证，我们提出了一个多项式算法，能够将最优解计算为完工时间截止期限的函数。该算法扩展了文献中针对双处理器调度问题的求解方法。此外，本文还研究了具有相同目标函数的其他相关资源均衡化问题的复杂性。这些结果推导出多项式或伪多项式算法，或NP难性证明，从而可与经典机器调度问题形成有意义的对比。