Abstract

We prove improved inapproximability results for hypergraph coloring using the low-degree polynomial code (aka the “short code” of Barak et al. [SIAM J. Comput., 44 (2015), pp. 1287--1324]) and the techniques proposed by Dinur and Guruswami [Israel J. Math., 209 (2015), pp. 611--649] to incorporate this code for inapproximability results. In particular, we prove quasi NP-hardness of the following problems on $n$-vertex hypergraphs: coloring a 2-colorable 8-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors; coloring a 4-colorable 4-uniform hypergraph with $2^{2^{\Omega(\sqrt{\log \log n})}}$ colors; and coloring a 3-colorable 3-uniform hypergraph with $(\log n)^{\Omega(1/\log\log\log n)}$ colors. For the first two cases, the hardness results obtained are superpolynomial in what was previously known, and in the last case it is an exponential improvement. In fact, prior to this result, $(\log n)^{O(1)}$ colors was the strongest quantitative bound on the number of colors ruled out by inapproximability results for $O(1)$-colorable hypergraphs, and $(\log\log n)^{O(1)}$ for $O(1)$-colorable, 3-uniform hypergraphs.

Keywords

  1. hardness of approximation
  2. hypergraph coloring
  3. short code

MSC codes

  1. 68QXX
  2. 68Q17
  3. 05C15

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Information & Authors

Information

Published In

cover image SIAM Journal on Computing
SIAM Journal on Computing
Pages: 132 - 159
ISSN (online): 1095-7111

History

Submitted: 13 November 2014
Accepted: 9 September 2016
Published online: 22 February 2017

Keywords

  1. hardness of approximation
  2. hypergraph coloring
  3. short code

MSC codes

  1. 68QXX
  2. 68Q17
  3. 05C15

Authors

Affiliations

Funding Information

US-Israel BSF : 2008293
Packard Fellowship
Google India PhD Fellowship
European Research Council http://dx.doi.org/10.13039/501100000781 : 226203
National Science Foundation http://dx.doi.org/10.13039/100000001 : CCF-1115525

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