2021/10/11 by Gordon, Spencer, Shu, Kevin
#Discrete Mathematics (cs.DM) #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2110.05644
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 .