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A fully polynomial parameterized algorithm for counting the number of reachable vertices in a digraph
Information Processing Letters ( IF 0.7 ) Pub Date : 2021-05-11 , DOI: 10.1016/j.ipl.2021.106137
Naoto Ohsaka

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(n13ϵ) for any ϵ>0, i.e., the number of edges is n plus O(n13ϵ), 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.



中文翻译:

用于计算有向图中可达顶点数量的完全多项式参数化算法

我们考虑对有向图G中每个顶点可到达的顶点进行计数的问题,该问题等于计算G的传递闭包的所有向外度。当前(理论上)最快的算法在二次时间内运行;然而,Borassi指出,除非强指数时间假说失败,否则这个问题在真正的次二次时代是无法解决的[Borassi,2016 [13]]。在本文中,我们提出了一个ØF3ñ-time精确算法,其中Ñ是在顶点的数目ģ˚F是反馈边沿数ģ。因此,对于F=Øñ1个3-ϵ 对于任何 ϵ>0,即边数为nØñ1个3-ϵ,并且是完全多项式固定参数易处理的,其概念最早是由Fomin等人引入的。(2018)[22]。我们还显示出相同的结果适用于顶点加权有向图,其中的任务是计算从每个顶点可到达的顶点的总权重。

更新日期:2021-05-13
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