A Fully Polynomial Parameterized Algorithm for Counting the Number of Reachable Vertices in a Digraph

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.