License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2021.8
URN: urn:nbn:de:0030-drops-153918
URL: https://drops.dagstuhl.de/opus/volltexte/2021/15391/
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Bathie, Gabriel ; Bousquet, Nicolas ; Pierron, Théo

(Sub)linear Kernels for Edge Modification Problems Towards Structured Graph Classes

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LIPIcs-IPEC-2021-8.pdf (0.7 MB)


Abstract

In a (parameterized) graph edge modification problem, we are given a graph G, an integer k and a (usually well-structured) class of graphs 𝒢, and ask whether it is possible to transform G into a graph G' ∈ 𝒢 by adding and/or removing at most k edges. Parameterized graph edge modification problems received considerable attention in the last decades.
In this paper, we focus on finding small kernels for edge modification problems. One of the most studied problems is the Cluster Editing problem, in which the goal is to partition the vertex set into a disjoint union of cliques. Even if this problem admits a 2k kernel [Cao and Chen, 2012], this kernel does not reduce the size of most instances. Therefore, we explore the question of whether linear kernels are a theoretical limit in edge modification problems, in particular when the target graphs are very structured (such as a partition into cliques for instance). We prove, as far as we know, the first sublinear kernel for an edge modification problem. Namely, we show that Clique + Independent Set Deletion, which is a restriction of Cluster Deletion, admits a kernel of size O(k/log k).
We also obtain small kernels for several other edge modification problems. We prove that Split Addition (and the equivalent Split Deletion) admits a linear kernel, improving the existing quadratic kernel of Ghosh et al. [Ghosh et al., 2015]. We complement this result by proving that Trivially Perfect Addition admits a quadratic kernel (improving the cubic kernel of Guo [Guo, 2007]), and finally prove that its triangle-free version (Starforest Deletion) admits a linear kernel, which is optimal under ETH.

BibTeX - Entry

@InProceedings{bathie_et_al:LIPIcs.IPEC.2021.8,
  author =	{Bathie, Gabriel and Bousquet, Nicolas and Pierron, Th\'{e}o},
  title =	{{(Sub)linear Kernels for Edge Modification Problems Towards Structured Graph Classes}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{8:1--8:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-216-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{214},
  editor =	{Golovach, Petr A. and Zehavi, Meirav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/15391},
  URN =		{urn:nbn:de:0030-drops-153918},
  doi =		{10.4230/LIPIcs.IPEC.2021.8},
  annote =	{Keywords: kernelization, graph editing, split graphs, (sub)linear kernels}
}

Keywords: kernelization, graph editing, split graphs, (sub)linear kernels
Collection: 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)
Issue Date: 2021
Date of publication: 22.11.2021


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