论文标题
最大0-1在时间图上的定时匹配
Maximum 0-1 Timed Matching on Temporal Graphs
论文作者
论文摘要
时间图是图形随时间变化的拓扑和/或其他属性的图形。它们已被用来模拟各个域中的时间信息的应用程序。由于拓扑变化，静态图上的问题在时间图中求解更具挑战性，并且许多最近的作品探索了时间图上的图形问题。在本文中，我们为时间图定义了一种称为{\ em 0-1时匹配}的匹配类型，并研究了针对不同类别的时间图找到{\ em mument mument 0-1 timed匹配}的问题。我们首先证明，当每个边缘与两个或多个时间间隔关联时，植根于颞树的问题是NP算法。然后，当每个边缘与一个时间间隔相关联时，我们建议在带有$ n $ nodes的扎根颞树上的$ O（n \ log n）$ time算法。然后，即使每个边缘与单个时间间隔关联，每个节点的度量也与二分的时间图相关，并且每个节点的度数也是由常数$ k \ geq 3 $界定的。接下来，我们研究了每个边缘与一个以上时间间隔相关联的时间图的问题的近似算法。首先表明，没有$ \ frac {1} {n^{1-ε}} $ - 对于任何$ε> 0 $的问题，即使在带有$ n $ nodes的扎根的暂时树上，对于任何$ε> 0 $，除非np = zpp。然后，我们提出了一个$ \ frac {5} {2 \ mathcal {n}^* + 3} $ - 因子图表的问题的因子近似算法，其中$ \ mathcal {n^*} $是$ \ mathcal {n^*} $是$ \ mathcal {n^*} $是时间范围的平均边数与时间图中的每个范围重叠。相同的算法也是用于界限时间图的恒定因素近似算法。
Temporal graphs are graphs where the topology and/or other properties of the graph change with time. They have been used to model applications with temporal information in various domains. Problems on static graphs become more challenging to solve in temporal graphs because of dynamically changing topology, and many recent works have explored graph problems on temporal graphs. In this paper, we define a type of matching called {\em 0-1 timed matching} for temporal graphs, and investigate the problem of finding a {\em maximum 0-1 timed matching} for different classes of temporal graphs. We first prove that the problem is NP-Complete for rooted temporal trees when each edge is associated with two or more time intervals. We then propose an $O(n \log n)$ time algorithm for the problem on a rooted temporal tree with $n$ nodes when each edge is associated with exactly one time interval. The problem is then shown to be NP-Complete also for bipartite temporal graphs even when each edge is associated with a single time interval and degree of each node is bounded by a constant $k \geq 3$. We next investigate approximation algorithms for the problem for temporal graphs where each edge is associated with more than one time intervals. It is first shown that there is no $\frac{1}{n^{1-ε}}$-factor approximation algorithm for the problem for any $ε> 0$ even on a rooted temporal tree with $n$ nodes unless NP = ZPP. We then present a $\frac{5}{2\mathcal{N}^* + 3}$-factor approximation algorithm for the problem for general temporal graphs where $\mathcal{N^*}$ is the average number of edges overlapping in time with each edge in the temporal graph. The same algorithm is also a constant-factor approximation algorithm for degree bounded temporal graphs.
