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Reduction from α-Grid-USO to Grid-USO

Determine whether the α-Grid-USO search problem—finding a vertex in a grid USO with a specified number of outgoing edges in each dimension—admits a polynomial-time reduction to the standard Grid-USO sink-finding problem.

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Background

The α-Grid-USO problem generalizes the standard Grid-USO sink-finding problem by specifying exact outdegrees per dimension rather than requiring a sink.

Establishing a reduction would clarify whether α-Grid-USO is computationally no harder than Grid-USO and could enable leveraging known results for Grid-USO.

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)