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Commutation of reductions across P-LCP, P-Lin-Bellman, and USO frameworks

Ascertain whether the reductions P-LCP ↔ P-Lin-Bellman and SWS-Colorful-Tangent → SWS-2P-Colorful-Tangent correspond, under the USO abstraction, to the same transformation as the Grid-USO → Cube-USO reduction, i.e., determine whether these reductions commute.

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Background

The paper establishes polynomial-time equivalences between P-LCP and P-Lin-Bellman, and between SWS-Colorful-Tangent and its binary variant, as well as a reduction from Grid-USO to Cube-USO. It is unclear whether these transformations represent the same underlying operation when viewed through the USO lens.

A commuting relationship would reveal a unified combinatorial structure underlying algebraic (LCP/Bellman) and geometric (colorful tangents) formulations, potentially enabling deeper cross-framework insights.

References

On the levels of USOs, we do not know the exact operations that the reductions from P to P and SWS-Colorful-Tangent to SWS-2P-Colorful-Tangent perform. It would be very interesting to analyze whether these reductions actually perform the same operation as the Grid-USO to Cube-USO reduction (\Cref{thm:generalizationIsUSO}), i.e., whether these reductions commute.

— Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents (2402.07683 - Borzechowski et al., 12 Feb 2024) in Section 6 (Open Questions)