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arXiv · 2110.05644

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

Spencer Gordon·Kevin Shu
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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 .

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Spencer Gordon, Kevin Shu. 2021-10-11. On the computational equivalence of co-NP refutations of a matrix being a P-matrix. https://arxiv.org/abs/2110.05644

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