Computational complexity of P-matrix Linear Complementarity (P-LCP)

Determine whether the P-matrix linear complementarity problem (P-LCP)—the problem of computing the unique solution (w, z) to LCP(M, q) when M is a P-matrix—admits a polynomial-time algorithm, or otherwise establish appropriate hardness results clarifying its computational complexity status.

Background

The P-matrix LCP asks for vectors w, z ≥ 0 with w = Mz + q and wT z = 0, under the promise that M is a P-matrix, which guarantees a unique solution for every q. Many optimization and game-theoretic problems reduce to P-LCP, underscoring the importance of its algorithmic complexity.

Despite extensive paper, it remains unresolved whether P-LCP is solvable in polynomial time; moreover, known hardness implications would have significant complexity-theoretic consequences if established.

References

However, the complexity status of the P remains a major open question. The P problem is not known to be polynomial-time solvable, but NP-hardness (in the sense that an oracle for it could be used to solve SAT in polynomial time) would imply NP=co-NP.

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