Semantics and commutativity of reductions at the USO level

Characterize the operations on Unique Sink Orientations induced by the reductions from PFixP to P-LCP and from SWS-Colorful-Tangent to SWS-2P-Colorful-Tangent, and determine whether these reductions commute with the Grid-USO to Cube-USO reduction described in Theorem 5.6.

Background

The paper establishes polynomial-time equivalences between algebraic, combinatorial, and geometric problems and provides a new reduction from Grid-USO to Cube-USO.

Understanding whether these distinct reductions correspond to the same underlying operation on USOs would unify the frameworks and potentially simplify future reductions.

References

On the levels of USOs, we do not know the exact operations that the reductions from PFixP to PLCP 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 (Theorem \ref{thm:generalizationIsUSO}), i.e., whether these reductions commute.

— Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents  (2402.07683 - Borzechowski et al., 2024) in Section 6 (Open Questions) — Semantics of the Grid-USO to Cube-USO reduction