A Fully Polynomial Parameterized Algorithm for Counting the Number of Reachable Vertices in a Digraph
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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 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=𝒪(n^1/3-ϵ) for any ϵ > 0, i.e., the number of edges is n plus 𝒪(n^1/3-ϵ), 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.
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