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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.IPEC.2018.7
URN: urn:nbn:de:0030-drops-102088
URL: https://drops.dagstuhl.de/opus/volltexte/2019/10208/
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Luo, Junjie ; Molter, Hendrik ; Suchý, Ondrej

A Parameterized Complexity View on Collapsing k-Cores

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LIPIcs-IPEC-2018-7.pdf (0.6 MB)

Abstract

We study the NP-hard graph problem Collapsed k-Core where, given an undirected graph G and integers b, x, and k, we are asked to remove b vertices such that the k-core of remaining graph, that is, the (uniquely determined) largest induced subgraph with minimum degree k, has size at most x. Collapsed k-Core was introduced by Zhang et al. [AAAI 2017] and it is motivated by the study of engagement behavior of users in a social network and measuring the resilience of a network against user drop outs. Collapsed k-Core is a generalization of r-Degenerate Vertex Deletion (which is known to be NP-hard for all r >=0) where, given an undirected graph G and integers b and r, we are asked to remove b vertices such that the remaining graph is r-degenerate, that is, every its subgraph has minimum degree at most r.
We investigate the parameterized complexity of Collapsed k-Core with respect to the parameters b, x, and k, and several structural parameters of the input graph. We reveal a dichotomy in the computational complexity of Collapsed k-Core for k <=2 and k >= 3. For the latter case it is known that for all x >= 0 Collapsed k-Core is W[P]-hard when parameterized by b. We show that Collapsed k-Core is W[1]-hard when parameterized by b and in FPT when parameterized by (b+x) if k <=2. Furthermore, we show that Collapsed k-Core is in FPT when parameterized by the treewidth of the input graph and presumably does not admit a polynomial kernel when parameterized by the vertex cover number of the input graph.

BibTeX - Entry

@InProceedings{luo_et_al:LIPIcs:2019:10208,
  author =	{Junjie Luo and Hendrik Molter and Ondrej Such{\'y}},
  title =	{{A Parameterized Complexity View on Collapsing k-Cores}},
  booktitle =	{13th International Symposium on Parameterized and Exact  Computation (IPEC 2018)},
  pages =	{7:1--7:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-084-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{115},
  editor =	{Christophe Paul and Michal Pilipczuk},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10208},
  URN =		{urn:nbn:de:0030-drops-102088},
  doi =		{10.4230/LIPIcs.IPEC.2018.7},
  annote =	{Keywords: r-Degenerate Vertex Deletion, Feedback Vertex Set, Fixed-Parameter Tractability, Kernelization Lower Bounds, Graph Algorithms, Social Network Analysis}
}

Keywords: r-Degenerate Vertex Deletion, Feedback Vertex Set, Fixed-Parameter Tractability, Kernelization Lower Bounds, Graph Algorithms, Social Network Analysis
Collection: 13th International Symposium on Parameterized and Exact Computation (IPEC 2018)
Issue Date: 2019
Date of publication: 05.02.2019

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