Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions - LIRMM - Laboratoire d’Informatique, de Robotique et de Microélectronique de Montpellier
Journal Articles ACM Transactions on Algorithms Year : 2016

## Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions

Eun Jung Kim
Alexander Langer
• Function : Author
Christophe Paul
Felix Reidl
• Function : Author
Peter Rossmanith
• Function : Author
Ignasi Sau
Somnath Sikdar
• Function : Author

#### Abstract

We 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 a finite integer index and such that Yes-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 F be 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 ε F. As our second application, we present the first single-exponential algorithm to solve Planar-F-Deletion. Namely, our algorithm runs in time 2O(k) · n2, which is asymptotically optimal with respect to k. So far, single-exponential algorithms were only known for special cases of the family F.

#### Domains

Mathematics [math]

### Dates and versions

lirmm-01288472 , version 1 (15-03-2016)

### Identifiers

• HAL Id : lirmm-01288472 , version 1
• ARXIV :
• DOI :

### Cite

Eun Jung Kim, Alexander Langer, Christophe Paul, Felix Reidl, Peter Rossmanith, et al.. Linear Kernels and Single-Exponential Algorithms Via Protrusion Decompositions. ACM Transactions on Algorithms, 2016, 12 (2), pp.No. 21. ⟨10.1145/2797140⟩. ⟨lirmm-01288472⟩

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