In this paper, we provide three notes on scheduling unit-length jobs with precedence constraints to minimize the total completion time. First, we propose an exact algorithm for in-trees, of which the complexity depends mainly on the graph height, i.e., the length of the longest chain of the precedence graph. We show that this work improves the algorithm in the literature both theoretically and experimentally. Second, we close the open problem for level-orders by showing how it is polynomially solvable. Third, we prove that preemptive scheduling in-trees is strongly NP-hard with arbitrary number of machines, of which the complexity was also open.

在本文中，我们提供了关于使用优先约束调度单位长度作业以最小化总完成时间的三个注意事项。首先，我们提出了一种in-trees的精确算法，其复杂度主要取决于图的高度，即优先图的最长链的长度。我们表明，这项工作在理论上和实验上都改进了文献中的算法。其次，我们通过展示多项式可解的方式来解决水平阶数的开放问题。第三，我们证明抢占式调度 in-tree 是强 NP-hard 的，具有任意数量的机器，其中的复杂性也是开放的。