The betweenness centrality of a vertex v is an important centrality measure that quantifies how many optimal paths between pairs of other vertices visit v. Computing betweenness centrality in a temporal graph, in which the edge set may change over discrete timesteps, requires us to count temporal paths that are optimal with respect to some criterion. For several natural notions of optimality, including foremost or fastest temporal paths, this counting problem reduces to #Temporal Path, the problem of counting all temporal paths between a fixed pair of vertices; like the problems of counting foremost and fastest temporal paths, #Temporal Path is #P-hard in general. Motivated by the many applications of this intractable problem, we initiate a systematic study of the prameterised and approximation complexity of #Temporal Path. We show that the problem presumably does not admit an FPT-algorithm for the feedback vertex number of the static underlying graph, and that it is hard to approximate in general. On the positive side, we proved several exact and approximate FPT-algorithms for special cases.

翻译：顶点v的介数中心性是一种重要的中心性度量，用于量化其他顶点对之间有多少条最优路径经过v。在时间图（边集可能随时间步离散变化）中计算介数中心性时，需要计数满足某些准则的最优时间路径。对于若干自然的最优性概念（例如最早或最快时间路径），该计数问题可归约为#Temporal Path问题，即计算固定顶点对之间所有时间路径的数量；与计算最早和最快时间路径问题类似，#Temporal Path在一般情况下是#P-难的。受此难解问题诸多应用需求的驱动，我们首次系统研究了#Temporal Path的参数化与近似复杂度。我们证明该问题对于静态底层图的反馈顶点集可能不存在FPT算法，且一般情况下难以近似。在积极方面，我们证明了若干特殊情形下的精确与近似FPT算法。