On approximability of Satisfiable k-CSPs: I

We consider the P P -CSP problem for 3-ary predicates P P on satisfiable instances. We show that under certain conditions on P P and a $$(1,s)$$ ( 1 , s ) integrality gap instance of the P P -CSP problem, it can be translated into a dictatorship vs. quasirandomness test with perfect completeness and soundness s+ε s + ϵ , for every constant ε>0 ϵ > 0 . Compared to Ragahvendra (in: Proceedings of the fortieth annual ACM symposium on theory of computing (STOC), pp 245–254, 2008), we do not lose perfect completeness. This is particularly interesting as this test implies new hardness results on satisfiable constraint satisfaction problems, assuming the Rich 2-to-1 Games Conjecture by Braverman et al. (in: Lee JR (ed) Volume 185 of Leibniz international proceedings in informatics (LIPIcs), 27:1–27:20. Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, 2021b. https://drops.dagstuhl.de/opus/volltexte/2021/13566 ).Our result can be seen as the first step of a potentially long-term challenging program of characterizing optimal inapproximability of every satisfiable k k -ary CSP. At the heart of the reduction is our main analytical lemma for a class of 3-ary predicates, which is a generalization of a lemma by Mossel (Geom Funct Anal 19(6):1713–1756, 2010). The lemma and a further generalization of it that we conjecture may be of independent interest.