Spectral aspects of symmetric matrix signings

Charles Carlson, Karthekeyan Chandrasekaran, Hsien Chih Chang, Naonori Kakimura, Alexandra Kolla

Research output: Contribution to journalArticle

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.

Original languageEnglish
Article number100582
JournalDiscrete Optimization
Volume37
DOIs
Publication statusPublished - 2020 Aug

Keywords

  • Matchings
  • Matrix signing
  • Spectral graph theory

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Computational Theory and Mathematics
  • Applied Mathematics

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    Carlson, C., Chandrasekaran, K., Chang, H. C., Kakimura, N., & Kolla, A. (2020). Spectral aspects of symmetric matrix signings. Discrete Optimization, 37, [100582]. https://doi.org/10.1016/j.disopt.2020.100582