Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik GmbH scholarly article en Bringmann, Karl; Kozma, László; Moran, Shay; Narayanaswamy, N. S. http://www.dagstuhl.de/lipics License
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URN: urn:nbn:de:0030-drops-63749
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Hitting Set for Hypergraphs of Low VC-dimension

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Abstract

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014). In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension 2, even if the VC-dimension of the dual set system is also 2. Thus, Hitting Set is very unlikely to be fixed-parameter tractable even in this arguably simple case. This answers an open question raised by King in 2010. For set systems whose (primal or dual) VC-dimension is 1, we show that Hitting Set is solvable in polynomial time. To bridge the gap in complexity between the classes of inputs with VC-dimension 1 and 2, we use a measure that is more fine-grained than VC-dimension. In terms of this measure, we identify a sharp threshold where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard. The tractable class that lies just under the threshold is a generalization of Edge Cover, and thus extends the domain of polynomial-time tractability of Hitting Set.

BibTeX - Entry

@InProceedings{bringmann_et_al:LIPIcs:2016:6374,
  author =	{Karl Bringmann and L{\'a}szl{\'o} Kozma and Shay Moran and N. S. Narayanaswamy},
  title =	{{Hitting Set for Hypergraphs of Low VC-dimension}},
  booktitle =	{24th Annual European Symposium on Algorithms (ESA 2016)},
  pages =	{23:1--23:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-015-6},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{57},
  editor =	{Piotr Sankowski and Christos Zaroliagis},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2016/6374},
  URN =		{urn:nbn:de:0030-drops-63749},
  doi =		{10.4230/LIPIcs.ESA.2016.23},
  annote =	{Keywords: hitting set, VC-dimension}
}

Keywords: hitting set, VC-dimension
Seminar: 24th Annual European Symposium on Algorithms (ESA 2016)
Issue date: 2016
Date of publication: 2016


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