时间图的可达集的延迟优化
Optimizing Reachability Sets in Temporal Graphs by Delaying
摘要：
时间图是一个动态图，其中每个边都被分配了一组整数时间标签，指示边在哪个离散时间步长可用。在本文中，我们研究了时间标签的变化，对应于延迟的边缘的可用性，影响给定的源的可达集。关于可达集的问题是由时间图在网络流行病学中的许多应用引起的，这些应用的目的是最大限度地减少感染的传播，以及制造业供应网络中的调度问题，其目标相反，即最大限度地提高覆盖率和生产率。我们引入控制机制的可达性集是基于两个自然操作的延迟。第一个操作，称为合并，是全局的，并在整个网络中同时将连续的时间标签批量合并为单个时间标签。这相当于将所有事件推迟到特定时间。第二种方法是在图的每条边的时间标签上施加独立的延迟。我们提供了一个彻底的调查计算复杂性的不同目标相关的可达性集时，这些操作被使用。对于合并操作，即全球锁定效应，我们证明了NP-硬度的结果，几个最小化和最大化的可达性目标，即使是非常简单的图形结构。对于第二个操作，独立的延迟，我们证明了最小化问题是NP-困难时，允许的延迟的数量是有界的。我们补充这一点与多项式时间算法最小化的情况下，无界延迟的可达性集。
英文摘要：
A temporal graph is a dynamic graph where every edge is assigned a set of integer time labels that indicate at which discrete time step the edge is available. In this paper, we study how changes of the time labels, corresponding to delays on the availability of the edges, affect the reachability sets from given sources. The questions about reachability sets are motivated by numerous applications of temporal graphs in network epidemiology, which aim to minimise the spread of infection, and scheduling problems in supply networks in manufacturing with the opposite objectives of maximising coverage and productivity. We introduce control mechanisms for reachability sets that are based on two natural operations of delaying. The first operation, termed merging, is global and batches together consecutive time labels into a single time label in the whole network simultaneously. This corresponds to postponing all events until a particular time. The second, imposes independent delays on the time labels of every edge of the graph. We provide a thorough investigation of the computational complexity of different objectives related to reachability sets when these operations are used. For the merging operation, i.e. global lockdown effect, we prove NP-hardness results for several minimization and maximization reachability objectives, even for very simple graph structures. For the second operation, independent delays, we prove that the minimization problems are NP-hard when the number of allowed delays is bounded. We complement this with a polynomial-time algorithm for minimising the reachability set in case of unbounded delays.
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