Refuting Unique Game Conjecture

TL;DR

This paper refutes the Unique Game Conjecture by demonstrating that certain $k$-CSP gap problems can be solved in polynomial time using a modified Hast's Algorithm BiLin combined with SDP, under specific conditions.

Contribution

It introduces a novel approach that refutes the Unique Game Conjecture by solving specific $k$-CSP gap problems efficiently under new conditions.

Findings

Refutes the Unique Game Conjecture assuming P ≠ NP.

Shows polynomial-time solvability of certain $k$-CSP gap problems.

Utilizes a modified BiLin algorithm with SDP for large $k$.

Abstract

In this short note, the author shows that the gap problem of some $k$-CSPs with the support of its predicate the ground of a balanced pairwise independent distribution can be solved by a modified version of Hast's Algorithm BiLin that calls Charikar\&Wirth's SDP algorithm for two rounds in polynomial time, when $k$ is sufficiently large, the support of its predicate is combined by the grounds of three biased homogeneous distributions and the three biases satisfy certain conditions. To conclude, the author refutes Unique Game Conjecture, assuming $P\ne NP$.