Largest Eigenvalue and Invertibility of Symmetric Matrix Signings.

The spectra of signed matrices have played a fundamental role in social sciences, graph theory and control theory. They have been key to understanding balance in social networks, to counting perfect matchings in bipartite graphs, and to analyzing robust stability of dynamic systems involving uncertainties. More recently, the results of Marcus et al. have shown that an efficient algorithm to find a signing of a given adjacency matrix that minimizes the largest eigenvalue could immediately lead to efficient construction of Ramanujan expanders. Motivated by these applications, we investigate natural spectral properties of signed matrices and address the computational problems of identifying signings with these spectral properties. Our main results are: (a) NP-completeness of three problems: verifying whether a given matrix has a signing that is positive semi-definite/singular/has bounded eigenvalues, (b) a polynomial-time algorithm to verify whether a given matrix has a signing that is invertible, and (c) a polynomial-time algorithm to find a minimum increase in support of a given symmetric matrix so that it has an invertible signing. We use combinatorial and spectral techniques; our main new tool is a combinatorial characterization of matrices with invertible signings that might be of independent interest. We use our characterization and classic structural results from matching theory to find a minimum increase in the support in order to obtain invertible signings.