The P-matrix problem is co-NP-complete

Recently Rohn and Poljak proved that for interval matrices with rank-one radius matrices testing singularity is NP-complete. This paper will show that given any matrix family belonging to the class of matrix polytopes with hypercube domains and rank-one perturbation matrices, a class which contains the interval matrices, testing singularity reduces to testing whether a certain matrix is not a P-matrix. It follows from this result that the problem of testing whether a given matrix is a P-matrix is co-NP-complete.