Unique Games Conjecture

Establish that for every positive real number [?] there exists a positive integer r such that distinguishing label-cover instances that are unique games and have value at least 1-[?] from those having value at most [?] is NP-hard.

Background

The paper introduces the Unique Games Conjecture as the imperfect-completeness analogue of the PCP gap problem for unique games. It asserts NP-hardness of distinguishing nearly perfectly satisfiable unique-game instances from instances with arbitrarily small value, for bounded alphabet size.

The paper discusses substantial algorithmic and hardness evidence for this conjecture but does not prove it. Its main theorem instead establishes the perfect-completeness 4-to-1 case, leaving the Unique Games Conjecture itself unresolved.

References

Khot conjectured that allowing for imperfect completeness remedies that: \begin{conjecture}\label{conj:UGC} For any $>0$, there exists $r\in\mathbb{N}{+}$, such that $\mathsf{GapUG}_r(1-,)$ is $\mathbf{NP}$-hard. \end{conjecture}

— On the Hardness of $4$-to-$1$ Games with Perfect Completeness  (2610.12378 - Fei et al., 8 Oct 2026) in Conjecture 1 (labeled conj:UGC), Section 1.1, "Label-Cover, Unique Games and d-to-1 Games"