A Hypergraph Dictatorship Test with Perfect Completeness

An Improved Dictatorship Test with Perfect Completeness

A Boolean function f : {0, 1} → {0, 1} is called a dictator if it depends on exactly one variable i.e f(x1, x2, . . . , xn) = xi for some i ∈ [n]. In this work, we study a k-query dictatorship test. Dictatorship tests are central in proving many hardness results for constraint satisfaction problems. The dictatorship test is said to have perfect completeness if it accepts any dictator function. ...

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 o...

Hardness of Constraint Satisfaction and Hypergraph Coloring Constructions of Probabilistically Checkable Proofs with Perfect Completeness

Voloshin's conjecture for C-perfect hypertrees

In the “mixed hypergraph” model, proper coloring requires that vertex subsets of one type (called C-edges) should contain two vertices of the same color, while the other type (D-edges) should not be monochromatic. Voloshin [Australas. J. Combin. 11 (1995), 25–45] introduced the concept of C-perfectness, which can be viewed as a dual kind of graph perfectness in the classical sense, and proposed...

3-bit Dictator Testing: 1 vs. 5/8

In the conclusion of his monumental paper on optimal inapproximability results, H̊astad [13]suggested that Fourier analysis of Dictator (Long Code) Tests does not seem universally appli-cable in the study of CSPs. His main open question was to determine if the technique couldresolve the approximability of satisfiable 3-bit constraint satisfaction problems. In particular,he as...