Generalized Feedback Vertex Set Problems on Bounded-Treewidth Graphs: Chordality is the Key to Single-Exponential Parameterized Algorithms

Abstract

It has long been known that Feedback Vertex Set can be solved in time 2 Opw log wq n Op1q on n-vertex graphs of treewidth w, but it was only recently that this running time was improved to 2 Opwq n Op1q , that is, to single-exponential parameterized by treewidth. We investigate which generalizations of Feedback Vertex Set can be solved in a similar running time. Formally, for a class P of graphs, the Bounded P-Block Vertex Deletion problem asks, given a graph G on n vertices and positive integers k and d, whether G contains a set S of at most k vertices such that each block of G´S has at most d vertices and is in P. Assuming that P is recognizable in polynomial time and satisfies a certain natural hereditary condition, we give a sharp characterization of when single-exponential parameterized algorithms are possible for fixed values of d:• if P consists only of chordal graphs, then the problem can be solved in time 2 Opwd 2 q n Op1q ,• if P contains a graph with an induced cycle of length ě 4, then the problem is not solvable in time 2 opw log wq n Op1q even for fixed d " , unless the ETH fails.We also study a similar problem, called Bounded P-Component Vertex Deletion, where the target graphs have connected components of small size rather than blocks of small size, and we present analogous results. For this problem, we also show that if d is part of the input and P contains all chordal graphs, then it cannot be solved in time f pwqn opwq for some function f , unless the ETH fails.Bounded-size components. Using a similar technique, we can obtain analogous results for a simpler problem, which we call Bounded P-Component Vertex Deletion, where we want to remove at most k vertices such that each connected component of the resulting graph has at most d vertices and belongs to P. If we have only the size constraint (i.e., P contains every graph), then this problem is known as Component Order Connectivity [9]. Let P be a class of graphs.Bounded P-Component Vertex Deletion Parameter: d, w Input: A graph G of treewidth at most w, and positive integers d and k. Question: Is there a set S of at most k vertices in G such that each connected component of G´S has at most d vertices and is in P?Drange, Dregi, and van 't Hof [9] studied the parameterized complexity of a weighted variant of the Component Order Connectivity problem; their results imply, in particular, that Component Order Connectivity can be solved in time 2 Opk log dq n, but is W r1s-hard parameterized by only k or d. The corresponding edge-deletion problem, parameterized by treewidth, was studied by Enright and Meeks [10]. For general classes P, we prove results that are analogous to those for Bounded P-Block Vertex Deletion. Theorem 1.3. Let P be a class of graphs that is hereditary, recognizable in polynomial time, and consists of only chordal graphs. Then Bounded P-Component Vertex Deletion can be solved in time 2 Opwd 2 q k 2 n on graphs with n vertices and treewidth w.Theorem 1.4. Let P be a hereditary class of graphs that is polynomial-time recognizable. I...

Hitting minors on bounded treewidth graphs. III. Lower bounds

“…This concludes the definition of the framework graph F , which is depicted in Figure 5 (a similar figure appears in [10]). Figure 5: The shape of the framework graph F assuming that k = 3, G contains only the three edges e 1 , e 2 , and e 3 , and σ is the cyclic permutation (e 1 , e 2 , e 3 ).…”

“…Namely, we start in Section 5.1 with the single-exponential lower bound for any connected F, and we observe that the hardness result still applies if the input graph is assumed to be planar. In Section 5.2 we present the superexponential lower bounds inspired by the reduction of Bonnet et al [10].…”

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

“…Moreover, we fix a cyclic permutation σ of the elements of E(G), agreeing that σ −1 (e) and σ(e) is the edge before and after e, respectively, in this cyclic ordering. For each e ∈ E(G), and each (i, j) ∈ [1, k] 2 , we add to F the edges This concludes the definition of the framework graph F , which is depicted in Figure 12 (we would like to note that a similar figure appears in [11]). Figure 12: The shape of the framework graph F assuming that k = 3, G contains only the three edges e 1 , e 2 , and e 3 , and σ is the cyclic permutation (e 1 , e 2 , e 3 ).…”

“…In particular, the former result applies to K 5 and all the connected graphs with at least six vertices. Our reductions are based on a generic framework that generalizes the one given in [6], which was inspired by a reduction of Bonnet et al [11]. These lower bounds also subsume the ones in [5], which were proved using a different reduction inspired by the one of Marcin Pilipczuk [41].…”

Parameterized Complexity of Safe Set

“…It is known that the problem admits a kernel of O(k ) vertices [34] (see also [12,23] for previous results on polynomial kernels). When parameterized by both the treewidth tw and the solution size k, the problem is W[1]-hard, and the ETH implies that there is no f (tw + k) · n o((tw+k)/ log(tw+k)) -time algorithm [6].…”