Complexity of the P-matrix Linear Complementarity Problem (P-LCP)

Determine the computational complexity of the P-matrix Linear Complementarity Problem (P-LCP). Establish whether P-LCP admits a polynomial-time algorithm for all inputs, or prove appropriate hardness results consistent with known implications (e.g., oracle NP-hardness would imply NP = co-NP).

Background

The linear complementarity problem (LCP) asks for nonnegative vectors w and z satisfying w = Mz + q and w^T z = 0. Under the P-matrix promise (all principal minors of M are positive), the LCP always has a unique solution for every q, and this promise version is referred to as P-LCP. Many central optimization problems reduce to P-LCP, including linear programming and strictly convex quadratic programming, making its complexity status highly consequential.

Despite decades of paper, no polynomial-time algorithm is known for P-LCP, and standard hardness would imply major class collapses (NP = co-NP). Clarifying the complexity of P-LCP remains a central open problem in total search/TFNP-related areas and the UEOPL landscape.