ESTABLISHING DETERMINANTAL INEQUALITIES FOR POSITIVE-DEFINITE MATRICES

Let A be an n x n matrix, and S be a subset of N = {1, 2,...,n}. A [S] denotes the principal submatrix of A which lies in the rows and columns indexed by S. If alpha = {alpha(1)...,alpha(p)} and beta = {beta(1)...,beta(q)} are two collections of subsets of N, the inequality alpha less than or equal to beta expresses that Pi(i=1)(p) det A[alpha(i)]less than or equal to Pi(i-1)(q) = det A[beta(i)], for all n x n positive-definite matrices A. Recently, Johnson and Barrett gave necessary and sufficient conditions for alpha less than or equal to beta. In their paper they showed that the necessary condition is not sufficient, and they raised the following questions: What is the computational complexity of checking the necessary condition? Is the sufficient condition also necessary? Here we answer the first question proving that checking the necessary condition is co-NP-complete. We also show that checking the sufficient condition is NP-complete, and we use this result to give their second question the following answer: If NP not equal co-NP, the sufficient condition is not necessary.