Component Order Connectivity in Directed Graphs

TL;DR

This paper investigates the parameterized complexity of the Directed Component Order Connectivity problem in directed graphs, providing new algorithms and complexity bounds, especially for semicomplete digraphs, and improving previous results.

Contribution

It introduces a fixed-parameter algorithm for DCOC on semicomplete digraphs with complexity $O^*(2^{16k})$, and refines bounds for general digraphs, advancing understanding of the problem's computational limits.

Findings

DCOC on semicomplete digraphs solvable in $O^*(2^{16k})$ time.

No $O^*(2^{o(k)})$ algorithm exists unless ETH fails.

Improved upper bounds for DCOC with parameters $ ext{ell}+k$ on general digraphs.

Abstract

A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of $D,$ there is at least one arc between $x$ and $y.$ \viol{Thus, a tournament is a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC) problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and $\ell$, we are to decide whether there is a subset $X$ of $V$ of size $k$ such that the largest strong connectivity component in $D-X$ has at most $\ell$ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $\ell=1.$ We study parametered complexity of DCOC for general and semicomplete digraphs with the following parameters: $k, \ell,\ell+k$ and $n-\ell$. In particular, we prove that DCOC with parameter $k$ on semicomplete digraphs can be solved in time $O^*(2^{16k})$ but not in time $O^*(2^{o(k)})$ unless the Exponential Time Hypothesis (ETH) fails. \gutin{The upper bound $O^*(2^{16k})$ implies the upper bound $O^*(2^{16(n-\ell)})$ for the parameter $n-\ell.$ We complement the latter by showing that there is no algorithm of time complexity $O^*(2^{o({n-\ell})})$ unless ETH fails.} Finally, we improve \viol{(in dependency on $\ell$)} the upper bound of G{\"{o}}ke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $\ell+k$ on general digraphs from $O^*(2^{O(k\ell\log (k\ell))})$ to $O^*(2^{O(k\log (k\ell))}).$ Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $O^*(2^{o(k\log \ell)})$ unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $O^*(2^{o(k\log k)}).$