Optimal Algorithms for Hitting (Topological) Minors on Graphs of Bounded Treewidth

Abstract

For a fixed 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. Using and enhancing the machinery of boundaried graphs and small sets of representatives introduced by Bodlaender et al. [J ACM, 2016], we prove that when all the graphs in ${\cal F}$ are connected and at least one of them is planar, then $f_{{\cal F}}(w) = 2^{O (w \cdot\log w)}$. When ${\cal F}$ is a singleton containing a clique, a cycle, or a path on $i$ vertices, we prove the following asymptotically tight bounds: $\bullet$ $f_{\{K_4\}}(w) = 2^{\Theta (w \cdot \log w)}$. $\bullet$ $f_{\{C_i\}}(w) = 2^{\Theta (w)}$ for every $i \leq 4$, and $f_{\{C_i\}}(w) = 2^{\Theta (w \cdot\log w)}$ for every $i \geq 5$. $\bullet$ $f_{\{P_i\}}(w) = 2^{\Theta (w)}$ for every $i \leq 4$, and $f_{\{P_i\}}(w) = 2^{\Theta (w \cdot \log w)}$ for every $i \geq 6$. The lower bounds hold unless the Exponential Time Hypothesis fails, and the superexponential ones are inspired by a reduction of Marcin Pilipczuk [Discrete Appl Math, 2016]. The single-exponential algorithms use, in particular, the rank-based approach introduced by Bodlaender et al. [Inform Comput, 2015]. We also consider the version of the problem where the graphs in ${\cal F}$ are forbidden as topological minors, and prove that essentially the same set of results holds.