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A Family of Dictatorship Tests with Perfect Completeness for 2-to-2 Label Cover

We give a family of dictatorship tests with perfect completeness and low-soundness for 2–to–2 constraints. The associated 2–to–2 conjecture has been the basis of some previous inapproximability results with perfect completeness. However, evidence towards the conjecture in the form of integrality gaps even against weak semidefinite programs has been elusive. Our result provides some indication of the expressiveness and non-triviality of 2–to–2 constraints, given the close connections between dictatorship tests and satisfiability and approximability of CSPs.