This paper focuses on the link scheduling problem in networks where signal delays between nodes are multiples of a time interval. To model such networks, a directed hypergraph is employed, along with an integer matrix that specifies the delays. The link scheduling problem is closely connected to the independent sets of the periodic hypergraph induced by the network model. However, due to the infinite number of vertices, it is impractical to enumerate the independent sets of the periodic hypergraph using generic graph algorithms. To tackle this challenge, a graphical approach is proposed in this paper. The link scheduling rate region is characterized using a finite directed graph called a scheduling graph, which is derived from the network model. A collision-free schedule of the network corresponds to a path in the scheduling graph, and the rate region is determined by the convex hull of the rate vectors associated with the cycles in the scheduling graph. Although existing cycle enumeration algorithms can be employed to calculate the rate region, their computational complexity becomes prohibitively high as the size of the scheduling graph grows exponentially with the number of network links. To address this issue, the dominance property of a special scheduling graph called the step-T scheduling graph is investigated. This property allows the utilization of specific subgraphs of the step-T scheduling graph to characterize the scheduling rate region, achieving a reduction in both the number of cycles and their lengths. For common problems such as calculating the rate region and maximizing a weighted sum of the scheduling rates, algorithms leveraging the dominance property are developed. These algorithms can be more efficient than using generic graph algorithms directly on the scheduling graphs.

翻译：本文研究节点间信号传播时延为时间间隔整数倍的网络中的链路调度问题。为建模此类网络，本文采用有向超图并引入整数矩阵描述时延。链路调度问题与网络模型诱导的周期超图的独立集密切相关。然而，由于周期超图含无限个顶点，直接使用通用图算法枚举其独立集并不可行。针对这一挑战，本文提出一种图论方法：通过从网络模型导出的有限有向图（称为调度图）来刻画链路调度速率区域。网络中的无冲突调度对应调度图中的一条路径，而速率区域由调度图中各环对应速率向量的凸包决定。尽管现有环枚举算法可用于计算速率区域，但随着调度图规模随网络链路数呈指数增长，其计算复杂度将急剧上升。为此，本文研究了阶梯T调度图这类特殊调度图的支配性质。该性质允许利用阶梯T调度图的特定子图刻画调度速率区域，从而减少环的数量与长度。针对速率区域计算及调度速率加权和最大化等常见问题，本文开发了基于支配性质的算法。这些算法相比直接在调度图上使用通用图算法具有更高效率。