Linear Kernels and Single-Exponential Algorithms via Protrusion Decompositions

Abstract

International audienceWe present a linear-time algorithm to compute a decomposition scheme for graphs G that have a set X ⊆ V(G), called a treewidth-modulator, such that the treewidth of G − X is bounded by a constant. Our decomposition, called a protrusion decomposition, is the cornerstone in obtaining the following two main results. Our first result is that any parameterized graph problem (with parameter k) that has finite integer index and such that positive instances have a treewidth-modulator of size O(k) admits a linear kernel on the class of H-topological-minor-free graphs, for any fixed graph H. This result partially extends previous meta-theorems on the existence of linear kernels on graphs of bounded genus and H-minor-free graphs.Let Fbe a fixed finite family of graphs containing at least one planar graph. Given an n-vertex graph G and a non-negative integer k, Planar F- Deletion asks whether G has a set X ⊆ V(G) such that |X|⩽k and G − X is H-minor-free for every H∈F. As our second application, we present the first single-exponential algorithm to solve Planar F- Deletion. Namely, our algorithm runs in time 2 O(k)·n 2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F

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

“…The following theorem is a reformulation using our notation of one of the main results of Kim et al [34], which is stronger than what we need uin the sense that also applies to graphs excluding a topological minor. Theorem 23.…”

“…In order to obtain our algorithm of time O * (2 O(tw · log tw) ) for every connected collection F, our approach can be streamlined as follows. We use the machinery of boundaried graphs, equivalence relations, and representatives originating in the seminal work of Bodlaender et al [9] and subsequently used, for instance, in [5,21,23,34]. Let h be a constant depending only on the collection F (to be defined in the formal description of the algorithm) and let t be a positive integer that is at most the treewidth of the input graph plus one.…”

“…Once we have a treewidth modulator of size O(t) of a representative R, all that remains is to pipeline it with known techniques to compute an appropriate protrusion decomposition [34] (Lemma 24) and to reduce protrusions to smaller equivalent ones of size bounded by a function of h -we use the version given in [5] adapted to the F-M-Deletion problem-(Lemma 25), implying that |V (R)| = O h (t) for every every R ∈ R It should be noted that all the items above do not need to be converted into an algorithm, they are just used in the analysis: the conclusion is that if R ∈ R (t) h is a minimum-sized representative, then |V (R)| = O h (t), as otherwise some reduction rule could be applied to it (either by removing an irrelevant vertex or by protrusion replacement), thus obtaining an equivalent representative of smaller size and contradicting its minimality. Our main result can be formally stated as follows.…”

“…Once we have the protrusion decomposition given by Lemma 24, all that remains is to replace the protrusions by equivalent ones of size depending only on the collection F. The protrusion replacement technique, which is nowadays part of the basic toolbox of parameterized complexity, originated in the meta-theorem of Bodlaender et al [9], whose objective was to produce linear kernels for a wide family of problems on graphs of bounded genus. This technique was later extended to graphs excluding a fixed minor by Fomin et al [21] and then to graphs excluding a fixed topological minor by Kim et al [34]. We could directly apply the results of Fomin et al [21] to the protrusion decomposition of a representative given by Lemma 24, hence reducing each protrusion to an equivalent one of size O F (1), yielding an equivalent representative of size O F (t).…”

“…We could directly apply the results of Fomin et al [21] to the protrusion decomposition of a representative given by Lemma 24, hence reducing each protrusion to an equivalent one of size O F (1), yielding an equivalent representative of size O F (t). However, the drawback of the results in [21] (and also in [9,34]) is that they do not provide explicit bounds on the hidden constants. In order to be able to do so (cf.…”