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Reduction from α-Grid-USO to Grid-USO

Determine whether α-Grid-USO (the problem of finding a vertex in a grid USO with a prescribed number of outgoing edges in each dimension) admits a polynomial-time reduction to Grid-USO (finding a sink), thereby establishing computational equivalence between the two search tasks.

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Background

Grid USOs are unique sink orientations on products of complete graphs; Grid-USO asks for the global sink. The paper defines α-Grid-USO: given target counts per dimension, find the unique vertex with that number of outgoing edges per dimension (guaranteed to exist and be unique).

Although α-Grid-USO naturally generalizes sink-finding, it is currently unknown whether the α-constraint search reduces to standard sink search. Establishing this reduction would extend the “two choices are enough” paradigm to the grid USO framework with prescribed outdegree targets.

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)