Strengthened Hardness for Approximating Minimum Unique Game and Small Set Expansion

TL;DR

This paper introduces a new hypothesis-based approach to strengthen hardness results for approximating problems like Minimum Unique Game and Small Set Expansion, showing they are harder to approximate than previously proven.

Contribution

It presents a variation of Feige's Hypothesis that leads to stronger hardness of approximation results for specific problems, advancing the theoretical understanding of their computational difficulty.

Findings

Strengthens hardness for Min 2-Lin-2 to 3/2 - epsilon

Improves hardness for Min Bisection to 3 - epsilon

Discusses limitations in extending these results to larger gaps

Abstract

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.