Membership of α-Grid-USO in Promise-UEOPL

Ascertain whether α-Grid-USO (searching for a vertex with a specified number of outgoing edges per dimension in a grid USO) lies within the promise class Promise-UEOPL.

Background

While α-Ham Sandwich is known to lie in Promise-UEOPL, the placement of α-Grid-USO within this class remains unsettled.

Clarifying the class membership would advance the understanding of the relationship between geometric and combinatorial UEOPL-type problems.

References

Note that α-Grid-USO is not known to reduce to regular Grid-USO, nor is it known to lie in Promise-UEOPL (compared to α-Ham Sandwich which does lie in Promise-UEOPL).

— Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents (2402.07683 - Borzechowski et al., 12 Feb 2024) in Remark 5.1 (rem:alphaHam2alphaGrid) in Section 5 (Colorful Tangents, P, and USO)