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

TL;DR

This paper introduces a fully polynomial fixed parameter tractable algorithm for counting reachable vertices in a digraph, achieving truly subquadratic time for graphs with small feedback edge numbers, improving efficiency over previous methods.

Contribution

The paper presents the first fully polynomial fixed parameter tractable algorithm for counting reachable vertices, with subquadratic time complexity based on feedback edge number.

Findings

Algorithm runs in O(f^3 n) time, where f is feedback edge number.

Achieves truly subquadratic time for graphs with small feedback edge number.

Extends results to vertex-weighted digraphs for total reachable vertex weights.

Abstract

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.