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 under biased assignments on a natural distribution. Under this hypothesis, the author strengthens the previous known hardness for approximating Minimum Unique Game, $5/4-ε$, by proving that Min 2-Lin-2 is hard to within $3/2-ε$ and strengthens the previous known hardness for approximating Small Set Expansion, $4/3-ε$, by proving that Min Bisection is hard to approximate within $3-ε$. 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 $ω_k(1)$ 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.

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