License: Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2018.15
URN: urn:nbn:de:0030-drops-90190
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2018/9019/
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Bhangale, Amey

NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors

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LIPIcs-ICALP-2018-15.pdf (0.5 MB)

Abstract

We give very short and simple proofs of the following statements: Given a 2-colorable 4-uniform hypergraph on n vertices,
1) It is NP-hard to color it with log^delta n colors for some delta>0.
2) It is quasi-NP-hard to color it with O({log^{1-o(1)} n}) colors.
In terms of NP-hardness, it improves the result of Guruswam, Håstad and Sudani [SIAM Journal on Computing, 2002], combined with Moshkovitz-Raz [Journal of the ACM, 2010], by an `exponential' factor. The second result improves the result of Saket [Conference on Computational Complexity (CCC), 2014] which shows quasi-NP-hardness of coloring a 2-colorable 4-uniform hypergraph with O(log^gamma n) colors for a sufficiently small constant 1 >> gamma>0.
Our result is the first to show the NP-hardness of coloring a c-colorable k-uniform hypergraph with poly-logarithmically many colors, for any constants c >= 2 and k >= 3.

BibTeX - Entry

@InProceedings{bhangale:LIPIcs:2018:9019,
  author =	{Amey Bhangale},
  title =	{{NP-Hardness of Coloring 2-Colorable Hypergraph with Poly-Logarithmically Many Colors}},
  booktitle =	{45th International Colloquium on Automata, Languages, and  Programming (ICALP 2018)},
  pages =	{15:1--15:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Ioannis Chatzigiannakis and Christos Kaklamanis and D{\'a}niel Marx and Donald Sannella},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/9019},
  URN =		{urn:nbn:de:0030-drops-90190},
  doi =		{10.4230/LIPIcs.ICALP.2018.15},
  annote =	{Keywords: Hypergraph coloring, Inapproximability, Schrijver graph}
}

Keywords: Hypergraph coloring, Inapproximability, Schrijver graph
Collection: 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)
Issue Date: 2018
Date of publication: 04.07.2018

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