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