A hypergraph dictatorship test with perfect completeness

Abstract

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based construction. Such a test has oracle access to a collection of functions and determines whether all the functions are the same dictatorship, or all their low degree influences are o(1). Their test makes q ≥ 3 queries, has amortized query complexity 1 + O(log q/q), but has an inherent loss of perfect completeness. In this paper we give an (adaptive) hypergraph dictatorship test that achieves both perfect completeness and amortized query complexity 1 + O(log q/q). © 2009 Springer.