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Reductions from USO-type problems to LCP-type problems

Determine whether there exists a polynomial-time reduction from Unique Sink Orientation sink-finding problems (such as Cube-USO or Grid-USO) to P-matrix Linear Complementarity formulations (e.g., P-LCP or P-matrix GLCP), thereby establishing an LCP-type encoding of USO-type problems, or prove that no such reduction exists under standard complexity assumptions.

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Background

Reductions are known from LCP-type problems to USO-type problems (e.g., P-LCP to Cube-USO and P-matrix GLCP to Grid-USO), but the reverse direction is not established.

A reduction from USO-type to LCP-type would unify frameworks and could allow transferring algorithmic advances and complexity classifications between these domains.

References

It should be noted that while P is known to reduce to cube USOs, and P is known to reduce to grid USOs, neither of our "two choices is enough" results imply each other. This is because there is no known reduction from a USO-type problem to an LCP-type problem.

— Two Choices are Enough for P-LCPs, USOs, and Colorful Tangents (2402.07683 - Borzechowski et al., 12 Feb 2024) in Section 1 (Introduction, Two choices are enough for USOs)