On the computational equivalence of co-NP refutations of a matrix being a P-matrix

TL;DR

This paper investigates the computational complexity of co-NP refutations for P-matrices, demonstrating that various known witnesses for non-P-matrices are computationally equivalent and can be converted efficiently.

Contribution

It establishes the polynomial-time equivalence of different co-NP witnesses for non-P-matrices, addressing an open question in the literature.

Findings

Various witnesses for non-P-matrices are computationally equivalent.

Polynomial-time conversion exists between different types of witnesses.

Answers an open question from prior research (arXiv:1811.03841).

Abstract

A P-matrix is a square matrix $X$ such that all principal submatrices of $X$ have positive determinant. Such matrices appear naturally in instances of the linear complementarity problem, where these are precisely the matrices for which the corresponding linear complementarity problem has a unique solution for any input vector. Testing whether or not a square matrix is a P-matrix is co-NP complete, so while it is possible to exhibit polynomially-sized witnesses for the fact that a matrix is not a P-matrix, it is believed that there is no efficient way to prove that a given matrix is a P-matrix. We will show that several well known witnesses for the fact that a matrix is not a P-matrix are computationally equivalent, so that we are able to convert between them in polynomial time, answering a question raised in arXiv:1811.03841 .