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

•We give an FPT algorithm for counting the number of reachable vertices in a digraph.•Our algorithm runs in truly subquadratic time if the feedback edge number is O(n13−ϵ).•The same result holds for vertex-weighted digraphs. We consider the problem of counting the number of vertices reachable from e...

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Vydáno v:Information processing letters Ročník 171; s. 106137
Hlavní autor: Ohsaka, Naoto
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier B.V 01.10.2021
Témata:
FPT in P
Graph algorithms
Parameterized algorithms
Reachability
Transitive closure
Transitive closure
Parameterized algorithms
Graph algorithms
FPT in P
Reachability
ISSN:0020-0190, 1872-6119
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Shrnutí:•We give an FPT algorithm for counting the number of reachable vertices in a digraph.•Our algorithm runs in truly subquadratic time if the feedback edge number is O(n13−ϵ).•The same result holds for vertex-weighted digraphs. 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 O(f3n)-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(n13−ϵ) for any ϵ>0, i.e., the number of edges is n plus O(n13−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.
ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2021.106137