Promise-UEOPL membership of α-Grid-USO

Ascertain whether α-Grid-USO lies in the promise search class Promise-UEOPL. Provide an explicit containment proof or a counterexample establishing that α-Grid-USO is outside Promise-UEOPL.

Background

Promise-UEOPL captures problems with uniquely structured potential-line witnesses under a promise ensuring well-posedness. Several USO-related problems, including Grid-USO, fall within Promise-UEOPL.

For α-Grid-USO, uniqueness of the target vertex is guaranteed combinatorially, but its formal placement in Promise-UEOPL remains unknown. Resolving this classification would situate α-Grid-USO within the broader TFNP/UEOPL hierarchy and clarify its relationship to established USO search problems.

References

Note that $\alpha$-Grid-USO is not known to reduce to regular Grid-USO, nor is it known to lie in #1{Promise-UEOPL} (compared to $\alpha$-Ham Sandwich which does lie in #1{Promise-UEOPL}).

— Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents (2402.07683 - Borzechowski et al., 12 Feb 2024) in Remark “alphaHam2alphaGrid” (Section 5, Colorful Tangents, P, and USO)