A Hypergraph Dictatorship Test with Perfect Completeness

TL;DR

This paper introduces an adaptive hypergraph dictatorship test that achieves perfect completeness while maintaining low query complexity, improving upon previous tests with inherent completeness loss.

Contribution

The paper presents a new adaptive hypergraph dictatorship test that attains perfect completeness without increasing query complexity.

Findings

Achieves perfect completeness in hypergraph dictatorship testing.

Maintains amortized query complexity of 1+O(log q / q).

Improves upon previous tests with inherent completeness loss.

Abstract

A hypergraph dictatorship test is first introduced by Samorodnitsky and Trevisan and serves as a key component in their unique games based $\PCP$ 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 $q\geq3$ queries and has amortized query complexity $1+O(\frac{\log q}{q})$ 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 $1+O(\frac{\log q}{q})$.