Dice Question Streamline Icon: https://streamlinehq.com

“⋉” (alpha) variant of the 2-to-1 Conjecture

Prove the “⋉” variant of the 2-to-1 conjecture: establish that the 2-to-1 conjecture with perfect completeness remains true even when all label cover constraints are the specific “⋉” (alpha/fish-shaped) constraints.

Information Square Streamline Icon: https://streamlinehq.com

Background

To obtain hardness for 3-colorability and related independent set problems under perfect completeness, the paper references a strengthened form of the 2-to-1 conjecture in which all constraints have a particular structured form (the “⋉” constraints).

This variant, originating in prior work, focuses on constraint families that preserve strong completeness while enabling robust hardness reductions for graph coloring problems.

References

Conjecture[“$\ltimes$” variant of the $2$-to-$1$ conjecture] $\Cref{conj:2-to-1}$ is true even assuming that all constraints in the label cover instance are “$\ltimes$” constraints.

— Rounding Large Independent Sets on Expanders (2405.10238 - Bafna et al., 16 May 2024) in Appendix: Hardness of Finding Independent Sets in k-colorable Expanders (Section B)