2-to-1 Conjecture with perfect completeness

Prove the 2-to-1 conjecture with perfect completeness: for every ε > 0, establish the existence of an alphabet size R such that, given a label cover instance ψ over alphabet size R with all constraints being 2-to-2 constraints, it is NP-hard to distinguish between ψ being fully satisfiable and the case where no assignment satisfies more than an ε fraction of constraints.

Background

The paper’s hardness reductions for finding large independent sets in colorable expanders rely on known conjectures from the literature. The 2-to-1 conjecture with perfect completeness asserts NP-hardness of distinguishing fully satisfiable instances from those with only small satisfiable fractions under specific structured constraints.

This conjecture serves as a foundational assumption for deriving hardness results in graph problems with strong completeness guarantees, including colorability and independent set size barriers.

References

Conjecture [$2$-to-$1$ conjecture with perfect completeness] For every $\eps > 0$, there exists some $R \in \N$ such that given a label cover instance $\psi$ with alphabet size $R$ such that all constraints are $2$-to-$2$ constraints, it is $\NP$-hard to decide between 1) $\psi$ is satisfiable, 2) no assignment satisfies more than $\eps$ fraction of the constraints in $\psi$.

— Rounding Large Independent Sets on Expanders  (2405.10238 - Bafna et al., 2024) in Appendix: Hardness of Finding Independent Sets in k-colorable Expanders (Section B)

For $d 2$ and $d'=1$, he conjectured that the assertion of~\Cref{thm:PCP} should hold as is. More precisely: \begin{conjecture}\label{conj:d-to-1} For any $>0$ and any integer $d 2$, there exists a positive integer $r$ such that ${d}{1}_r(1,)$ 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 2 (labeled conj:d-to-1), Section 1.1, "Label-Cover, Unique Games and d-to-1 Games"