Spectral aspects of symmetric matrix signings

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

研究成果: Article査読

抄録

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.

本文言語English
論文番号100582
ジャーナルDiscrete Optimization
37
DOI
出版ステータスPublished - 2020 8

ASJC Scopus subject areas

  • 理論的コンピュータサイエンス
  • 計算理論と計算数学
  • 応用数学

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