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

Abstract

For a fixed collection of graphs F, the F-M-DELETION problem consists in, given a graph G and an integer k, decide whether there exists a subset S of V(G) of size at most k such that G-S does not contain any of the graphs in F as a minor. We are interested in the parameterized complexity of F-M-DELETION when the parameter is the treewidth of G, denoted by tw. Our objective is to determine, for a fixed F}, the smallest function f_F such that F-M-DELETION can be solved in time f_F(tw)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 F are connected and at least one of them is planar, then  f_F(w) = 2^{O(wlog w)}. When F is a singleton containing a clique, a cycle, or a path on i vertices, we prove the following asymptotically tight bounds:

- f_{K_4}(w) = 2^{Theta(wlog w)}.
- f_{C_i}(w) = 2^{Theta(w)} for every i<5, and f_{C_i}(w) = 2^{Theta(wlog w)} for every i>4.
- f_{P_i}(w) = 2^{Theta(w)} for every i<5, and f_{P_i}(w) = 2^{Theta(wlog w)} for every i>5.

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 F are forbidden as topological minors, and prove essentially the same set of results holds.