Conditional Dichotomy of Boolean Ordered Promise CSPs by Joshua Brakensiek, Venkatesan Guruswami, Sai Sandeep

“…We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [BKM21] which is a perfect completeness surrogate of the Unique Games Conjecture. Assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every $\epsilon>0$, it has polymorphisms where each coordinate has Shapley value at most $\epsilon$, else it is NP-hard. …”
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NP-Hardness of Approximately Solving Linear Equations Over Reals by Khot, Subhash, Moshkovitz Aaronson, Dana Hadar

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Invariance Principle on the Slice by Filmus, Yuval, Kindler, Guy, Mossel, Elchanan, Wimmer, Karl

“…The invariance principle has seen many applications in theoretical computer science, including the Majority is Stablest conjecture, which shows that the Goemans-Williamson algorithm for MAX-CUT is optimal under the Unique Games Conjecture. More generally, MOO's invariance principle works for any two vectors of hypercontractive random variables (x1, . . . , xn ), (Y1, . . . ,Yn ) such that (i) Matching moments: Xi and Yi have matching first and second moments and (ii) Independence: the variablesx1, . . . , xn are independent, as are Y1, . . . ,Yn. …”
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Hypercontractivity, sum-of-squares proofs, and their applications by Barak, Boaz, Brandao, Fernando G.S.L., Harrow, Aram W., Steurer, David, Zhou, Yuan, Kelner, Jonathan Adam

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