Hitting minors on bounded treewidth graphs. III. Lower bounds

Abstract

For a finite collection of graphs F, the F-M-Deletion problem consists in, given a graph G and an integer k, decide whether there exists S ⊆ V (G) with |S| ≤ 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. We prove that f F (tw) = 2 2 O(tw·log tw) for every collection F, that f F (tw) = 2 O(tw·log tw) if all the graphs in F are connected and at least one of them is planar, and that f F (tw) = 2 O(tw) if in addition the input graph G is planar or embedded in a surface. When F contains a single connected planar graph H, we obtain a tight dichotomy about the asymptotic complexity of {H}-M-Deletion. Namely, we prove that f {H} (tw) = 2 Θ(tw) if H is a minor of the banner (that is, the graph consisting of a C 4 plus a pendent edge) that is different from P 5 , and that f {H} (tw) = 2 Θ(tw·log tw) otherwise. All the lower bounds hold under the ETH. We also consider the version of the problem where the graphs in F are forbidden as topological minors, and prove similar results, except that, in the algorithms, instead of requiring F to contain a planar graph, we need it to contain a subcubic planar graph. We also prove that, for this problem, f {K1,i} (tw) = 2 Θ(tw) for every i ≥ 1, while for the minor version it holds that f {K1,i} (tw) = 2 Θ(tw·log tw) for every i ≥ 4. Extended abstracts containing some of the results of this article appeared in the Proc. of the 12th International Symposium on Parameterized and Exact Computation (IPEC 2017) [3] and in the Proc. of the 13th International Symposium on Parameterized and Exact Computation (IPEC 2018) [4]. Work supported by French projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).

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

“…Also, the algorithm in [5] is written in terms of topological minors, that is, it computes a minimum-size set of vertices S ⊆ V (G) whose removal leaves a graph without any of the graphs in a fixed collection F as a topological minor; we denote |S| =: tm F (G). It is easy to see that computing this parameter suffices for computing m F (G), since, as observed in [4,Lemma 4], for every proper collection F and every graph G, it holds that m F (G) = tm F (G), where F is the family containing every topological minor minimal graph among those that contain some graph in F as a minor; note that F has size bounded by a small function of F (see Observation 3).…”

“…It is also interesting to consider the version of the problem where the graphs in F are forbidden as topological minors; we call this problem F-TM-Deletion. While the lower bounds that we presented in this article also hold for F-TM-Deletion (with the exception of K 1,i for i ≥ 4; see [4]), the algorithm in time O * (2 O(tw · log tw) ) for every connected collection F does not work for topological minors. In this direction, the algorithm in time O * (2 O(tw · log tw) ) in [5] for F-M-Deletion (when F is connected and contains a planar graph) also works for F-TM-Deletion if we additionally require F to contain a subcubic planar graph (in order to bound the treewidth of the representatives).…”

“…In fact, in the whole algorithm (see Figure 3) the connectivity of the graphs in F is only used at the very end, when we apply the dynamic programming algorithm of [5] based on representatives. This algorithm uses the connectivity of F in the "base case", namely to guarantee that the representative of a graph G without boundary is the empty boundaried graph if and only if G does not contain any of the graphs in F as a minor (see [4,Lemma 7]). We think that this is a technical hurdle, rather than an intrinsic one, and we believe that the condition on the connectivity of F can be dropped, that is, that there exists an algorithm to solve F-M-Deletion in time O * (2 O(tw · log tw) ) for every collection F. As an evidence towards this, note that the minor obstructions for being embeddable on a surface of Euler genus at most g contain disconnected graphs if g ≥ 2 (for instance, the disjoint union of two K 5 's is an obstruction for being embeddable on the torus [39]), and that Kociumaka and Pilipczuk [35] presented an algorithm running in time O * (2 O((tw +g)·log(tw +g)) ) for deleting a minimum number of vertices to obtain a graph embeddable on a surface of Euler genus at most g.…”

“…In fact, we realized a posteriori that the "easy" cases can be succinctly described in terms of the chair and the banner by taking a look at Figure 26. Note that the "easy" graphs can be equivalently characterized as those that are minors of the banner, with the exception 4 For the sake of transparency, we would like to mention that these lower bounds are also permanently available in arXiv [4,Section 5.2], where lower bounds for the topological minor version are given as well. In this article, we present in Section 7 only the bounds for the minor version, and we highlight along the reduction the main differences with respect to the framework given in [6].…”

“…Nevertheless, there is some intuitive reason for which excluding the chair or the banner constitutes the horizon on the existence of single-exponential algorithms. Namely, focusing on the banner, every connected component (with at least five vertices) of a graph that excludes the banner as a minor is either a cycle (of any length) or a tree in which some vertices have been replaced by triangles; both such types of components can be maintained by a dynamic programming algorithm in single-exponential time [4]. A similar situation occurs when excluding the chair.…”