Hitting minors on bounded treewidth graphs. III. Lower bounds

TL;DR

This paper establishes tight lower bounds on the parameterized complexity of the ${ m F}$-M-DELETION problem on graphs with bounded treewidth, showing superexponential lower bounds for certain graph collections, thus advancing the understanding of its computational limits.

Contribution

It provides the first superexponential lower bounds for ${ m F}$-M-DELETION with specific collections ${ m F}$, and completes a dichotomy classification for connected graphs.

Findings

Lower bounds of $2^{ ext{Omega}(tw)}$ for connected graphs of size at least two.

Superexponential lower bounds of $2^{ ext{Omega}(tw imes ext{log} tw)}$ for certain graph collections.

A tight dichotomy on the complexity of $ ext{H}$-M-DELETION for connected graphs H.

Abstract

For a finite collection of graphs ${\cal F}$, the ${\cal F}$-M-DELETION problem consists in, given a graph $G$ and an integer $k$, decide whether there exists $S \subseteq V(G)$ with $|S| \leq k$ such that $G \setminus S$ does not contain any of the graphs in ${\cal F}$ as a minor. We are interested in the parameterized complexity of ${\cal F}$-M-DELETION when the parameter is the treewidth of $G$, denoted by $tw$. Our objective is to determine, for a fixed ${\cal F}$, the smallest function $f_{{\cal F}}$ such that ${\cal F}$-M-DELETION can be solved in time $f_{{\cal F}}(tw) \cdot n^{O(1)}$ on $n$-vertex graphs. We provide lower bounds under the ETH on $f_{{\cal F}}$ for several collections ${\cal F}$. We first prove that for any ${\cal F}$ containing connected graphs of size at least two, $f_{{\cal F}}(tw)= 2^{\Omega(tw)}$, even if the input graph $G$ is planar. Our main contribution consists of superexponential lower bounds for a number of collections ${\cal F}$, inspired by a reduction of Bonnet et al.~[IPEC, 2017]. In particular, we prove that when ${\cal F}$ contains a single connected graph $H$ that is either $P_5$ or is not a minor of the banner (that is, the graph consisting of a $C_4$ plus a pendent edge), then $f_{{\cal F}}(tw)= 2^{\Omega(tw \cdot \log tw)}$. This is the third of a series of articles on this topic, and the results given here together with other ones allow us, in particular, to provide a tight dichotomy on the complexity of $\{H\}$-M-DELETION, in terms of $H$, when $H$ is connected.