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 probl m is not solvable in truly subquadratic time unless the Strong Exponential Time Hypothesis fails [Inf. Process. Lett., 116(10):628--630, 2016]. In this paper, we present an $\mathcal{O}(f^3n)$-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 $f=\mathcal{O}(n^{\frac{1}{3}-\epsilon})$ for any $\epsilon > 0$, i.e., the number of edges is $n$ plus $\mathcal{O}(n^{\frac{1}{3}-\epsilon})$, and is fully polynomial fixed parameter tractable, the notion of which was first introduced by Fomin, Lokshtanov, Pilipczuk, Saurabh, and Wrochna [ACM Trans. Algorithms, 14(3):34:1--34:45, 2018]. 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.

翻译：我们考虑了一个问题, 将每个顶端都能达到的脊椎数计算成 $G$, 这等于计算中途关闭$G$的所有外度。 目前( 理论上的) 最快的算法在二次时间运行; 但是, Boassi 已经表明, 在真正的次赤道时间里, 这个 probl m 是不能溶解的, 除非强烈的 光学时间假曲[ Inf. proc. proc. proc., 116(10): 628- 630, 2016] 。 在本文中, 我们提出了一个 $\ mathcal{O}( f_ 3n) 的时间精确算, 美元是 $G$, 美元是反馈的边缘 $。 因此, 我们的算法运行在 $\ mathcal{O} (ncralcral) (n_xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx