Scheduling by Edge Reversal provides an order of operation for nodes in a graph, but maximizing or minimizing the resulting concurrency is hard. In this paper, we discuss a series of real‐world applications for this technique and propose algorithms for both problems. For maximum concurrency, we prove its general inapproximability and introduce approximation algorithms for classes of graphs. For minimum concurrency, we use hardness and inapproximability results to establish its relation to longest cycles, while also introducing a novel application for assembling musical phrases.

中文翻译：

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通过边缘反转优化调度中的并发

通过边缘反转进行调度可以为图中的节点提供操作顺序，但是要最大化或最小化最终的并发性是很困难的。在本文中，我们讨论了该技术的一系列实际应用，并针对这两个问题提出了算法。为了获得最大的并发性，我们证明了它的一般不近似性，并为图类引入了近似算法。对于最小的并发，我们使用硬度和不可逼近度的结果来建立其与最长循环的关系，同时还引入了一种新颖的应用来组装乐句。