Strengthened Hardness for Approximating Minimum Unique Game and Small Set Expansion

In this paper, the author puts forward a variation of Feige’s Hypothesis, which claims that it is hard on average refuting Unbalanced Max 3-XOR against biased assignments on a natural distribution. Under this hypothesis, the author strengthens the previous known hardness for approximating Minimum Unique Game, 5/4 − ǫ, by proving that Min 2-Lin-2 is hard to within 3/2 − ǫ and strengthens the previous known hardness for approximating Small Set Expansion, 4/3 − ǫ, by proving that Min Bisection is hard to approximate within 3 − ǫ. In addition, the author discusses the limitation of this method to show that it can strengthen the hardness for approximating Minimum Unique Game to 2− κ where κ is a small absolute positive, but is short of proving ωk(1) hardness for Minimum Unique Game or Small Set Expansion, by assuming a generalization of this hypothesis on Unbalanced Max k-CSP with balanced pairwise independent predicate.