### 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.

Original language | English (US) |
---|---|

Title of host publication | 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 |

Editors | Joost-Pieter Katoen, Pinar Heggernes, Peter Rossmanith |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771177 |

DOIs | |

State | Published - Aug 2019 |

Event | 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 - Aachen, Germany Duration: Aug 26 2019 → Aug 30 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
---|---|

Volume | 138 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019 |
---|---|

Country | Germany |

City | Aachen |

Period | 8/26/19 → 8/30/19 |

### Fingerprint

### Keywords

- Matchings
- Matrix Signing
- Spectral Graph Theory

### ASJC Scopus subject areas

- Software

### Cite this

*44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019*[81] (Leibniz International Proceedings in Informatics, LIPIcs; Vol. 138). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.MFCS.2019.81