3-Bit Dictator Testing: 1 vs. 5/8

In the conclusion of his monumental paper on optimal inapproximability results, Håstad [13] suggested that Fourier analysis of Dictator (Long Code) Tests does not seem universally applicable in the study of CSPs. His main open question was to determine if the technique could resolve the approximability of satisfiable 3-bit constraint satisfaction problems. In particular, he asked if the “Not Two” (NTW) predicate is non-approximable beyond the random assignment threshold of 5/8 on satisfiable instances. Around the same time, Zwick [30] showed that all satisfiable 3-CSPs are 5/8-approximable, and conjectured that the 5/8 is optimal. In this work we show that Fourier analysis techniques can produce a Dictator Test based on NTW with completeness 1 and soundness 5/8. Our test’s analysis uses the Bonami-Gross-Beckner hypercontractive inequality. We also show a soundness lower bound of 5/8 for all 3-query Dictator Tests with perfect completeness. This lower bound for Property Testing is proved in part via a semidefinite programming algorithm of Zwick [30]. Our work precisely determines the 3-query “Dictatorship Testing gap”. Although this represents progress on Zwick’s conjecture, current PCP “outer verifier” technology is insufficient to convert our Dictator Test into an NP-hardness-of-approximation result.