2 Search Results for "Thiruvenkatachari, Devanathan"

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX-RANDOM.2019.9
Improved 3LIN Hardness via Linear Label Cover

Authors: Prahladh Harsha, Subhash Khot, Euiwoong Lee, and Devanathan Thiruvenkatachari

Published in: LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Abstract
We prove that for every constant c and epsilon = (log n)^{-c}, there is no polynomial time algorithm that when given an instance of 3-LIN with n variables where an (1 - epsilon)-fraction of the clauses are satisfiable, finds an assignment that satisfies atleast (1/2 + epsilon)-fraction of clauses unless NP subseteq BPP. The previous best hardness using a polynomial time reduction achieves epsilon = (log log n)^{-c}, which is obtained by the Label Cover hardness of Moshkovitz and Raz [J. ACM, 57(5), 2010] followed by the reduction from Label Cover to 3-LIN of Håstad [J. ACM, 48(4):798 - 859, 2001]. Our main idea is to prove a hardness result for Label Cover similar to Moshkovitz and Raz where each projection has a linear structure. This linear structure of Label Cover allows us to use Hadamard codes instead of long codes, making the reduction more efficient. For the hardness of Linear Label Cover, we follow the work of Dinur and Harsha [SIAM J. Comput., 42(6):2452 - 2486, 2013] that simplified the construction of Moshkovitz and Raz, and observe that running their reduction from a hardness of the problem LIN (of unbounded arity) instead of the more standard problem of solving quadratic equations ensures the linearity of the resultant Label Cover.
Cite as

Prahladh Harsha, Subhash Khot, Euiwoong Lee, and Devanathan Thiruvenkatachari. Improved 3LIN Hardness via Linear Label Cover. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{harsha_et_al:LIPIcs.APPROX-RANDOM.2019.9,
  author =	{Harsha, Prahladh and Khot, Subhash and Lee, Euiwoong and Thiruvenkatachari, Devanathan},
  title =	{{Improved 3LIN Hardness via Linear Label Cover}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)},
  pages =	{9:1--9:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-125-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{145},
  editor =	{Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.9},
  URN =		{urn:nbn:de:0030-drops-112245},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2019.9},
  annote =	{Keywords: probabilistically checkable proofs, PCP, composition, 3LIN, low soundness error}
}
Document
DOI: 10.4230/LIPIcs.FSTTCS.2017.15
An Improved Dictatorship Test with Perfect Completeness

Authors: Amey Bhangale, Subhash Khot, and Devanathan Thiruvenkatachari

Published in: LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

Abstract
A Boolean function f:{0,1}^n\->{0,1} is called a dictator if it depends on exactly one variable i.e f(x_1, x_2, ..., x_n) = x_i for some i in [n]. In this work, we study a k-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. The dictatorship test is said to have perfect completeness if it accepts any dictator function. The soundness of a test is the maximum probability with which it accepts any function far from a dictator. Our main result is a k-query dictatorship test with perfect completeness and soundness (2k + 1)/(2^k), where k is of the form 2^t -1 for any integer t > 2. This improves upon the result of [Tamaki-Yoshida, Random Structures & Algorithms, 2015] which gave a dictatorship test with soundness (2k + 3)/(2^k).
Cite as

Amey Bhangale, Subhash Khot, and Devanathan Thiruvenkatachari. An Improved Dictatorship Test with Perfect Completeness. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 15:1-15:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bhangale_et_al:LIPIcs.FSTTCS.2017.15,
  author =	{Bhangale, Amey and Khot, Subhash and Thiruvenkatachari, Devanathan},
  title =	{{An Improved Dictatorship Test with Perfect Completeness}},
  booktitle =	{37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)},
  pages =	{15:1--15:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-055-2},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{93},
  editor =	{Lokam, Satya and Ramanujam, R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.15},
  URN =		{urn:nbn:de:0030-drops-83854},
  doi =		{10.4230/LIPIcs.FSTTCS.2017.15},
  annote =	{Keywords: Property Testing, Distatorship Test, Fourier Analysis}
}
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