ICALP 2013: Automata, Languages, and Programming pp 613-624

Linear Kernels and Single-Exponential Algorithms via Protrusion Decompositions

• Eun Jung Kim
• Alexander Langer
• Christophe Paul
• Felix Reidl
• Peter Rossmanith
• Ignasi Sau
• Somnath Sikdar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7965)

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 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 $$\mathcal{F}$$ 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$$\mathcal{F}$$- Deletion asks whether G has a set X ⊆ V(G) such that $$|X|\leqslant k$$ and G − X is H-minor-free for every $$H\in \mathcal{F}$$. As our second application, we present the first single-exponential algorithm to solve Planar$$\mathcal{F}$$- 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 $$\mathcal{F}$$.

Keywords

parameterized complexity linear kernels algorithmic meta-theorems sparse graphs single-exponential algorithms graph minors

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Authors and Affiliations

• Eun Jung Kim
• 1
• Alexander Langer
• 2
• Christophe Paul
• 3
• Felix Reidl
• 2
• Peter Rossmanith
• 2
• Ignasi Sau
• 3
• Somnath Sikdar
• 2
2. 2.Theoretical Computer Science, Department of Computer ScienceRWTH Aachen UniversityGermany
3. 3.CNRS, LIRMMMontpellierFrance