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

In the conclusion of his monumental paper on optimal inapproximability results, H̊astad [13]suggested that Fourier analysis of Dictator (Long Code) Tests does not seem universally appli-cable in the study of CSPs. His main open question was to determine if the technique couldresolve 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 assignmentthreshold of 5/8 on satisfiable instances. Around the same time, Zwick [30] showed that allsatisfiable 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 onNTW with completeness 1 and soundness 5/8. Our test’s analysis uses the Bonami-Gross-Becknerhypercontractive inequality. We also show a soundness lower bound of 5/8 for all 3-query Dic-tator Tests with perfect completeness. This lower bound for Property Testing is proved in partvia a semidefinite programming algorithm of Zwick [30]. Our work precisely determines the 3-query “Dictatorship Testing gap”. Although this rep-resents progress on Zwick’s conjecture, current PCP “outer verifier” technology is insufficientto convert our Dictator Test into an NP-hardness-of-approximation result. ∗Supported in part by NSF CAREER grant CCF-0747250.