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.CCC.2019.32
URN: urn:nbn:de:0030-drops-108545
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2019/10854/
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Bafna, Mitali ; Vyas, Nikhil

Imperfect Gaps in Gap-ETH and PCPs

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LIPIcs-CCC-2019-32.pdf (0.5 MB)

Abstract

We study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a way to transform a PCP with imperfect completeness to one with perfect completeness, when the initial gap is a constant. We show that PCP_{c,s}[r,q] subseteq PCP_{1,s'}[r+O(1),q+O(r)] for c-s=Omega(1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NP with a O(log n) additive loss in the query complexity q. We show our result by constructing a "robust circuit" using threshold gates. These results are a gap amplification procedure for PCPs, (when completeness is not 1) analogous to questions studied in parallel repetition [Anup Rao, 2011] and pseudorandomness [David Gillman, 1998] and might be of independent interest.
We also investigate the time-complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the Gap-ETH conjecture without perfect completeness is equivalent to Gap-ETH with perfect completeness, i.e. MAX 3SAT(1-epsilon,1-delta), delta > epsilon has 2^{o(n)} algorithms if and only if MAX 3SAT(1,1-delta) has 2^{o(n)} algorithms. We also relate the time complexities of these two problems in a more fine-grained way to show that T_2(n) <= T_1(n(log log n)^{O(1)}), where T_1(n),T_2(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness respectively.

BibTeX - Entry

@InProceedings{bafna_et_al:LIPIcs:2019:10854,
  author =	{Mitali Bafna and Nikhil Vyas},
  title =	{{Imperfect Gaps in Gap-ETH and PCPs}},
  booktitle =	{34th Computational Complexity Conference (CCC 2019)},
  pages =	{32:1--32:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-116-0},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{137},
  editor =	{Amir Shpilka},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10854},
  URN =		{urn:nbn:de:0030-drops-108545},
  doi =		{10.4230/LIPIcs.CCC.2019.32},
  annote =	{Keywords: PCP, Gap-ETH, Hardness of Approximation}
}

Keywords: PCP, Gap-ETH, Hardness of Approximation
Collection: 34th Computational Complexity Conference (CCC 2019)
Issue Date: 2019
Date of publication: 16.07.2019

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