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Imperfect Gaps in Gap-ETH and PCPs

Authors Mitali Bafna, Nikhil Vyas

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Author Details

Mitali Bafna
  • Harvard University, Cambridge, MA, USA
Nikhil Vyas
  • MIT, Cambridge, MA, USA

Funding

  • Bafna, Mitali: Supported by NSF Grant CCF-1565641 and CCF-1715187.
  • Vyas, Nikhil: Supported by NSF Grant CCF-1741615.

Acknowledgements

We would like to thank our advisors Madhu Sudan and Ryan Williams for helpful discussions. We are also grateful to all the reviewers, for detailed comments on the paper.

Cite AsGet BibTex

Mitali Bafna and Nikhil Vyas. Imperfect Gaps in Gap-ETH and PCPs. In 34th Computational Complexity Conference (CCC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 137, pp. 32:1-32:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.CCC.2019.32

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.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Theory of computation → Complexity theory and logic
Keywords
  • PCP
  • Gap-ETH
  • Hardness of Approximation

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