We consider the problem of counting the number of vertices reachable from each vertex in a digraph G, which is equal to computing all the out-degrees of the transitive closure of G. The current (theoretically) fastest algorithms run in quadratic time; however, Borassi has shown that this problem is not solvable in truly subquadratic time unless the Strong Exponential Time Hypothesis fails [Borassi, 2016 [13]]. In this paper, we present an $\mathcal{O}(f^{3}n)$-time exact algorithm, where n is the number of vertices in G and f is the feedback edge number of G. Our algorithm thus runs in truly subquadratic time for digraphs of $f=\mathcal{O}(n^{\frac{1}{3}-ϵ})$ for any $ϵ>0$, i.e., the number of edges is n plus $\mathcal{O}(n^{\frac{1}{3}-ϵ})$, and is fully polynomial fixed parameter tractable, the notion of which was first introduced by Fomin et al. (2018) [22]. We also show that the same result holds for vertex-weighted digraphs, where the task is to compute the total weights of vertices reachable from each vertex.

我们考虑对有向图G中每个顶点可到达的顶点进行计数的问题，该问题等于计算G的传递闭包的所有向外度。当前（理论上）最快的算法在二次时间内运行；然而，Borassi指出，除非强指数时间假说失败，否则这个问题在真正的次二次时代是无法解决的[Borassi，2016 [13]]。在本文中，我们提出了一个$\mathcal{Ø}（F^3ñ）$-time精确算法，其中Ñ是在顶点的数目ģ和˚F是反馈边沿数ģ。因此，对于$F=\mathcal{Ø}（{ñ}^{\frac{\mathrm{1个}}{3}-ϵ}）$ 对于任何 $ϵ>0$，即边数为n加$\mathcal{Ø}（{ñ}^{\frac{\mathrm{1个}}{3}-ϵ}）$，并且是完全多项式固定参数易处理的，其概念最早是由Fomin等人引入的。（2018）[22]。我们还显示出相同的结果适用于顶点加权有向图，其中的任务是计算从每个顶点可到达的顶点的总权重。