  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 .

Discrete Mathematics