A fully polynomial parameterized algorithm for counting the number of reachable vertices in a digraph

We consider the problem of counting number vertices reachable from each vertex in a digraph G, which is equal to computing all out-degrees transitive closure G. The current (theoretically) fastest algorithms run quadratic time; however, Borassi has shown that this not solvable truly subquadratic time unless Strong Exponential Time Hypothesis fails [Borassi, 2016 [13]]. In paper, we present an O(f3n)-time exact algorithm, where n G and f feedback edge Our algorithm thus runs for digraphs f=O(n13−ϵ) any ϵ>0, i.e., edges plus O(n13−ϵ), fully polynomial fixed parameter tractable, notion was first introduced by Fomin et al. (2018) [22]. also show same result holds vertex-weighted digraphs, task compute total weights vertex.