Imperfect Gaps in GapETH and PCPs
Abstract
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 $\text{PCP}_{c,s}[r,q] \subseteq \text{PCP}_{1,1\Omega(1)}[r+O(1),q+O(r)]$, for $cs=\Omega(1)$. This implies that one can convert imperfect completeness to perfect in linearsized PCPs for $NTIME[O(n)]$ 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 imperfect), analogous to questions studied in parallel repetition and pseudorandomness. We also investigate the time complexity of approximating perfectly satisfiable instances of 3SAT versus those with imperfect completeness. We show that the GapETH conjecture without perfect completeness is equivalent to GapETH with perfect completeness, i.e. we show that Gap3SAT, where the gap is not around 1, has a subexponential algorithm, if and only if, Gap3SAT with perfect completeness has subexponential algorithms. We also relate the time complexities of these two problems in a more finegrained way, to show that $T_2(n) \leq T_1(n(\log\log n)^{O(1)})$, where $T_1(n),T_2(n)$ denote the randomized timecomplexity of approximating MAX 3SAT with perfect and imperfect completeness, respectively.
 Publication:

arXiv eprints
 Pub Date:
 July 2019
 DOI:
 10.48550/arXiv.1907.08185
 arXiv:
 arXiv:1907.08185
 Bibcode:
 2019arXiv190708185B
 Keywords:

 Computer Science  Computational Complexity
 EPrint:
 To appear in CCC 2019