  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 under
biased assignments on a natural distribution. Under this hypothesis, the author
strengthens the previous known hardness for approximating Minimum Unique Game,
$5/4-\epsilon$, by proving that Min 2-Lin-2 is hard to within $3/2-\epsilon$
and strengthens the previous known hardness for approximating Small Set
Expansion, $4/3-\epsilon$, by proving that Min Bisection is hard to approximate
within $3-\epsilon$. In addition, the author discusses the limitation of this
method to show that it can strengthen the hardness for approximating Minimum
Unique Game to $2-\kappa$ where $\kappa$ is a small absolute positive, but is
short of proving $\omega_k(1)$ hardness for Minimum Unique Game (or Small Set
Expansion), by assuming a generalization of this hypothesis on Unbalanced Max
k-CSP with Samorodnitsky-Trevisan hypergraph predicate.
