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Spectral Aspects of Symmetric Matrix Signings

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



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Author Details

Charles Carlson
  • University of Colorado Boulder, Boulder, USA
Karthekeyan Chandrasekaran
  • University of Illinois, Urbana-Champaign, USA
Hsien-Chih Chang
  • Duke University, Durham, USA
Naonori Kakimura
  • Keio University, Yokohama, Japan
Alexandra Kolla
  • University of Colorado Boulder, Boulder, USA

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Charles Carlson, Karthekeyan Chandrasekaran, Hsien-Chih Chang, Naonori Kakimura, and Alexandra Kolla. Spectral Aspects of Symmetric Matrix Signings. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 81:1-81:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.MFCS.2019.81

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 signing that is singular or 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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Discrete mathematics
Keywords
  • Spectral Graph Theory
  • Matrix Signing
  • Matchings

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