A complexity dichotomy for hitting connected minors on bounded treewidth graphs: the chair and the banner draw the boundary

Abstract

For a fixed connected graph H, the {H}-M-Deletion problem asks, given a graph G, for the minimum number of vertices that intersect all minor models of H in G. It is known that this problem can be solved in time f (tw) · n O(1) , where tw is the treewidth of G. We determine the asymptotically optimal function f (tw), for each possible choice of H. Namely, we prove that, under the ETH, f (tw) = 2 Θ(tw) if H is a contraction of the chair or the banner, and f (tw) = 2 Θ(tw · log tw) otherwise. Prior to this work, such a complete characterization was only known when H is a planar graph with at most five vertices. For the upper bounds, we present an algorithm in time 2 Θ(tw · log tw) · n O(1) for the more general problem where all minor models of connected graphs in a finite family F need to be hit. We combine several ingredients such as the machinery of boundaried graphs in dynamic programming via representatives, the Flat Wall Theorem, Bidimensionality, the irrelevant vertex technique, treewidth modulators, and protrusion replacement. In particular, this algorithm vastly generalizes a result of Jansen et al. [SODA 2014] for the particular case F = {K 5 , K 3,3 }. For the lower bounds, our reductions are based on a generic construction building on the one given by the authors in [IPEC 2018], which uses the framework introduced by Lokshtanov et al. [SODA 2011] to obtain superexponential lower bounds.

Hitting Minors on Bounded Treewidth Graphs. I. General Upper Bounds

“…In all these papers, FPT-algorithms parameterized by treewidth play a fundamental role. The results that we presented in this paper have already been used in [4], which in turn has been strongly used in the algorithms in [46].…”

“…Namely, we prove that for any collection F containing only connected graphs of size at least two, F-Deletion cannot be solved in time 2 o(tw) • n O (1) , even if the input graph G is planar, and we also provide superexponential lower bounds for a number of collections F. In particular, we prove a lower bound of 2 o(tw•log tw) • n O (1) when F contains a single graph 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), with the exception of K 1,i for the topological minor version. These lower bounds, together with the ad hoc single-exponential algorithms given in [6] and the algorithm described in item 2 above, cover all the cases of F-M-Deletion where F consists of a single connected planar graph H, yielding a tight dichotomy in terms of H. In a recent article [4], we presented an algorithm for F-M-Deletion in time O * (2 O(tw•log tw) ) for any collection F, yielding together with the lower bounds in [6] a dichotomy for F-M-Deletion where F consists of a single connected (not necessarily planar) graph H.…”

Hitting minors on bounded treewidth graphs. II. Single-exponential algorithms

“…Namely, we prove that for any connected 3 F, F-Deletion cannot be solved in time O * (2 o(tw) ), even if the input graph G is planar, and we provide superexponential lower bounds for a number of collections F. In particular, we prove a lower bound of O * (2 o(tw•log tw) ) when F contains a single connected graph that is either P 5 or is not a minor of the banner, with the exception of K 1,i for the topological minor version. These lower bounds, together with the ad hoc single-exponential algorithms given in this article and the general algorithms described in [6], cover all the cases of F-M-Deletion where F contains a single connected planar graph H, yielding a dichotomy in terms of H. In the fourth article of this series [5] (whose full version is [4]), we presented an algorithm for F-M-Deletion in time O * (2 O(tw•log tw) ) for any collection F, yielding together with the lower bounds in [7] and the results of the current article a dichotomy for F-M-Deletion where F consists of a single connected (non-necessarily planar) graph H. Namely, as stated in [4], if H is a connected graph on at least two vertices, then the {H}-M-Deletion problem can be solved in time…”

Hitting minors on bounded treewidth graphs. III. Lower bounds

“…Note that the cases H = P 2 [16,22], H = P 3 [1,28], and H = C 3 [10,17] were already known (nevertheless, for completeness we provide in [8] a simple algorithm when H = P 3 ). In the fourth article of this series [6] (whose full version is [5]), we present an algorithm for F-M-Deletion in time O * (2 O(tw•log tw) ) for any collection F.…”

“…The lower bounds presented in this article, together with the algorithms given in [5,6,8], cover all the cases of F-M-Deletion where F contains a single connected graph, as discussed in Section 4. Namely, we obtain the following theorem.…”