Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph is represented by assigning a set of integer time-labels to every edge $e$ of $G$, indicating the discrete time steps at which $e$ is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) $(e,t)$ such that no vertex is matched more than once within any time window of $\Delta$ consecutive time slots, where $\Delta \in \mathbb{N}$ is given. The requirement that a vertex cannot be matched twice in any $\Delta$-window models some necessary "recovery" period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms.

中文翻译：

---

计算时态图中的最大匹配

时间图是其拓扑结构随时间发生离散变化的图。给定一个静态底层图 $G$，时间图通过为 $G$ 的每条边 $e$ 分配一组整数时间标签来表示，指示 $e$ 处于活动状态的离散时间步长。我们介绍和研究经典图问题最大匹配的自然时间扩展的复杂性，同时考虑到时间图的动态特性。在我们的问题，最大时间匹配中，我们正在寻找最大可能数量的时间标记边（简称时间边）$(e,t)$，这样在 $\ 的任何时间窗口内没有顶点匹配一次以上Delta$ 连续时隙，其中给出了 $\Delta \in \mathbb{N}$。一个顶点不能在任何 $\Delta$-window 模型中被匹配两次的要求是一些必要的“