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.2022.23
URN: urn:nbn:de:0030-drops-173795
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17379/
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Mizutani, Yosuke ; Sullivan, Blair D.

Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph

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LIPIcs-IPEC-2022-23.pdf (0.9 MB)

Abstract

We present fixed-parameter tractable (FPT) algorithms for two problems, Maximum Happy Set (MaxHS) and Densest k-Subgraph (DkS) - also known as Maximum Edge Happy Set. Given a graph G and an integer k, MaxHS asks for a set S of k vertices such that the number of happy vertices with respect to S is maximized, where a vertex v is happy if v and all its neighbors are in S. We show that MaxHS can be solved in time ?(2^mw ⋅ mw ⋅ k² ⋅ |V(G)|) and ?(8^cw ⋅ k² ⋅ |V(G)|), where mw and cw denote the modular-width and the clique-width of G, respectively. This answers the open questions on fixed-parameter tractability posed in [Asahiro et al., 2021].
The DkS problem asks for a subgraph with k vertices maximizing the number of edges. If we define happy edges as the edges whose endpoints are in S, then DkS can be seen as an edge-variant of MaxHS. In this paper we show that DkS can be solved in time f(nd)⋅|V(G)|^?(1) and ?(2^{cd}⋅ k² ⋅ |V(G)|), where nd and cd denote the neighborhood diversity and the cluster deletion number of G, respectively, and f is some computable function. This result implies that DkS is also fixed-parameter tractable by twin cover number.

BibTeX - Entry

@InProceedings{mizutani_et_al:LIPIcs.IPEC.2022.23,
  author =	{Mizutani, Yosuke and Sullivan, Blair D.},
  title =	{{Parameterized Complexity of Maximum Happy Set and Densest k-Subgraph}},
  booktitle =	{17th International Symposium on Parameterized and Exact Computation (IPEC 2022)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-260-0},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{249},
  editor =	{Dell, Holger and Nederlof, Jesper},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17379},
  URN =		{urn:nbn:de:0030-drops-173795},
  doi =		{10.4230/LIPIcs.IPEC.2022.23},
  annote =	{Keywords: parameterized algorithms, maximum happy set, densest k-subgraph, modular-width, clique-width, neighborhood diversity, cluster deletion number, twin cover}
}

Keywords: parameterized algorithms, maximum happy set, densest k-subgraph, modular-width, clique-width, neighborhood diversity, cluster deletion number, twin cover
Collection: 17th International Symposium on Parameterized and Exact Computation (IPEC 2022)
Issue Date: 2022
Date of publication: 14.12.2022

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