We consider scheduling problems for unit jobs with release times, where the number or size of the gaps in the schedule is taken into consideration, either in the objective function or as a constraint. Except for several papers on minimum-energy scheduling, there is no work in the scheduling literature that uses performance metrics depending on the gap structure of a schedule. One of our objectives is to initiate the study of such scheduling problems. We focus on the model with unit-length jobs. First we examine scheduling problems with deadlines, where we consider two variants of minimum-gap scheduling: maximizing throughput with a budget for the number of gaps and minimizing the number of gaps with a throughput requirement. We then turn to other objective functions. For example, in some scenarios gaps in a schedule may be actually desirable, leading to the problem of maximizing the number of gaps. A related problem involves minimizing the maximum gap size. The second part of the paper examines the model without deadlines, where we focus on the tradeoff between the number of gaps and the total or maximum flow time. For all these problems we provide polynomial time algorithms, with running times ranging from \(O(n\log n)\) for some problems to \(O(n^7)\) for other. The solutions involve a spectrum of algorithmic techniques, including different dynamic programming formulations, speed-up techniques based on searching Monge arrays, searching \(X+Y\) matrices, or implicit binary search. Throughout the paper, we also draw a connection between gap scheduling problems and their continuous analogues, namely hitting set problems for intervals of real numbers. As it turns out, for some problems the continuous variants provide insights leading to efficient algorithms for the corresponding discrete versions, while for other problems completely new techniques are needed to solve the discrete version.

我们考虑具有发布时间的单元作业的调度问题，其中在目标函数中或作为约束考虑了调度中间隙的数量或大小。除了几篇关于最小能量调度的论文外，调度文献中没有使用依赖于调度间隙结构的性能指标的工作。我们的目标之一是开始研究此类调度问题。我们专注于具有单位长度作业的模型。首先，我们检查具有截止日期的调度问题，其中我们考虑最小间隙调度的两种变体：利用间隙数量的预算最大化吞吐量和最小化具有吞吐量要求的间隙数量。然后我们转向其他目标函数。例如，在某些情况下，时间表中的差距实际上可能是可取的，导致最大化间隙数量的问题。一个相关的问题涉及最小化最大间隙尺寸。论文的第二部分检查了没有最后期限的模型，我们重点讨论了间隙数量与总或最大流动时间之间的权衡。对于所有这些问题，我们提供多项式时间算法，运行时间从\(O(n\log n)\)对于某些问题到\(O(n^7)\)对于其他问题。这些解决方案涉及一系列算法技术，包括不同的动态规划公式、基于搜索 Monge 数组的加速技术、搜索\(X+Y\)矩阵或隐式二分搜索。在整篇论文中，我们还在间隙调度问题和它们的连续类似物之间建立了联系，即针对实数区间的集合问题。事实证明，对于某些问题，连续变体提供了导致相应离散版本的有效算法的见解，而对于其他问题，需要全新的技术来解决离散版本。