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.APPROX-RANDOM.2014.433
URN: urn:nbn:de:0030-drops-47144
URL: https://drops.dagstuhl.de/opus/volltexte/2014/4714/
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Wenner, Cenny

Parity is Positively Useless

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Abstract

We give the first examples of non-trivially positively-useless predicates subject only to P != NP. In particular, for every constraint function Q : {-1,1}^4 -> R, we construct Contraint-Satisfaction-Problem (CSP) instances without negations which have value at least 1-eps when evaluted for the arity-four odd-parity predicate, yet it is NP-hard to find a solution with value significantly better than a random biased assignment when evaluated for Q. More generally, we show that all parities except one are positively useless.

Although we are not able to exhibit a single protocol producing hard instances when evaluated for every Q, we show that two protocols do the trick. The first protocol is the classical one used by Håstad with a twist. We extend the protocol to multilayered Label Cover and employ a particular distribution over layers in order to limit moments of table biases. The second protocol is a modification of Chan's multi-question protocol where queried tuples of Label Cover vertices are randomized in such a way that the tables can be seen as being independently sampled from a common distribution and in effect having identical expected biases. We believe that our techniques may prove useful in further analyzing the approximability of CSPs without negations.

BibTeX - Entry

@InProceedings{wenner:LIPIcs:2014:4714,
  author =	{Cenny Wenner},
  title =	{{Parity is Positively Useless}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)},
  pages =	{433--448},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-74-3},
  ISSN =	{1868-8969},
  year =	{2014},
  volume =	{28},
  editor =	{Klaus Jansen and Jos{\'e} D. P. Rolim and Nikhil R. Devanur and Cristopher Moore},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2014/4714},
  URN =		{urn:nbn:de:0030-drops-47144},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2014.433},
  annote =	{Keywords: Approximation hardness, approximation resistance, parity, usefulness, negations, monotone, constraint satisfaction problems, smoothness, multilayer, L}
}

Keywords: Approximation hardness, approximation resistance, parity, usefulness, negations, monotone, constraint satisfaction problems, smoothness, multilayer, L
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)
Issue Date: 2014
Date of publication: 04.09.2014

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