From USO clashes to nonpositive principal minors in P-matrix LCP

Develop a polynomial-time procedure that, given a clash—two distinct hypercube vertices u and v such that (s(u) ⊕ s(v)) ∧ (u ⊕ v) = 0^n—in the outmap produced by reducing a linear complementarity problem with matrix M to a unique sink orientation, constructs a nonpositive principal minor of M, thereby converting the USO violation certificate into the standard certificate used in the non-promise P-matrix LCP formulation.

Background

The paper studies a non-promise version of unique sink orientation (USO) sink-finding where the algorithm must either find a sink or produce an efficiently verifiable violation of the USO property (a clash). The P-matrix Linear Complementarity Problem (PLCP) also has a non-promise version where one must either produce a solution or a nonpositive principal minor of the matrix.

While PLCP reduces to USO, the certificate notions differ: USO violations are clashes, whereas PLCP violations are nonpositive principal minors. The authors highlight that, although a clash proves the matrix is not a P-matrix, there is no known method to transform a clash into the PLCP’s standard certificate.