Abstract
The spectra of signed matrices have played a fundamental role in social sciences, graph theory, and control theory. In this work, we investigate the computational problems of finding symmetric signings of matrices with natural spectral properties. Our results are the following:1. We characterize matrices that have an invertible signing: a symmetric matrix has an invertible symmetric signing if and only if the support graph of the matrix contains a perfect 2-matching. Further, we present an efficient algorithm to search for an invertible symmetric signing.2. We use the above-mentioned characterization to give an algorithm to find a minimum increase in the support of a given symmetric matrix so that it has an invertible symmetric signing.3. We show NP-completeness of the following problems: verifying whether a given matrix has a symmetric off-diagonal signing that is singular/has bounded eigenvalues. However, we also illustrate that the complexity could differ substantially for input matrices that are adjacency matrices of graphs.We use combinatorial techniques in addition to classic results from matching theory.
Highlights
The spectra of several graph-related matrices such as the adjacency and the Laplacian matrices have become fundamental objects of study in computer science
We use the above-mentioned characterization to give an algorithm to find a minimum increase in the support of a given symmetric matrix so that it has an invertible symmetric signing
In this work we consider the following computational problems: BOUNDEDEVALUESIGNING: Given a real symmetric matrix M and a real number λ, verify if there exists an off-diagonal symmetric signing s such that the largest eigenvalue λmax(M (s)) is at most λ
Summary
The spectra of several graph-related matrices such as the adjacency and the Laplacian matrices have become fundamental objects of study in computer science. In this work we consider the following computational problems: BOUNDEDEVALUESIGNING: Given a real symmetric matrix M and a real number λ, verify if there exists an off-diagonal symmetric signing s such that the largest eigenvalue λmax(M (s)) is at most λ. INCLUDESIGNING: Given a real symmetric matrix M and a real number λ, verify if there exists an off-diagonal symmetric signing s such that M (s) has λ as an eigenvalue. AVOIDSIGNING: Given a real symmetric matrix M and a real number λ, verify if there exists a symmetric signing s such that M (s) does not have λ as an eigenvalue. It suffices to focus on instances where λ is 0. INVERTIBLESIGNING: Given a real symmetric matrix M , verify if there exists a symmetric signing s such that M (s) is invertible (that is, non-singular)
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