Imperfect gaps in Gap-ETH and PCPs
Author(s)Vyas, Nikhil,S.M.Massachusetts Institute of Technology.
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.
Richard Ryan Williams.
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In this thesis we study the role of perfect completeness in probabilistically checkable proof systems (PCPs) and give a new way to transform a PCP with imperfect completeness to a PCP with perfect completeness, when the initial gap is a constant. In particular, we show that PCP[subscript c,s][r, q] [mathematical symbol] PCP[subscript 1,s'][r + 0(1), q+ 0 (r)] for c - s = [omega](1) which in turn implies that one can convert imperfect completeness to perfect in linear-sized PCPs for NTIME[0(n)] with a 0(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 imperfect), analogous to questions studied in parallel repetition  and pseudorandomness [141. 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; that is, MAX 3SAT(1 - [epsilon], 1 - [delta]) for [delta] > [epsilon] has 2⁰([superscript n])-time algorithms if and only if MAX 3SAT(1, 1 - [delta]) has 2⁰([superscript n])-time algorithms. We also relate the time complexities of these two problems in a more fine-grained way, to show that T₂ (n) </= T₁ (n(log log n)⁰(¹)), where T₁(n), T₂(n) denote the randomized time-complexity of approximating MAX 3SAT with perfect and imperfect completeness, respectively. This is joint work with Mitali Bafna.
Thesis: S.M., Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2019Cataloged from PDF version of thesis.Includes bibliographical references (pages 45-47).
DepartmentMassachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
Massachusetts Institute of Technology
Electrical Engineering and Computer Science.