### 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

**Spectral aspects of symmetric matrix signings.** / Carlson, Charles; Chandrasekaran, Karthekeyan; Chang, Hsien Chih; Kakimura, Naonori; Kolla, Alexandra.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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, 44th International Symposium on Mathematical Foundations of Computer Science, MFCS 2019, Aachen, Germany, 8/26/19. https://doi.org/10.4230/LIPIcs.MFCS.2019.81

}

TY - GEN

T1 - Spectral aspects of symmetric matrix signings

AU - Carlson, Charles

AU - Chandrasekaran, Karthekeyan

AU - Chang, Hsien Chih

AU - Kakimura, Naonori

AU - Kolla, Alexandra

PY - 2019/8

Y1 - 2019/8

N2 - 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.

AB - 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.

KW - Matchings

KW - Matrix Signing

KW - Spectral Graph Theory

UR - http://www.scopus.com/inward/record.url?scp=85071753068&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85071753068&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.MFCS.2019.81

DO - 10.4230/LIPIcs.MFCS.2019.81

M3 - Conference contribution

AN - SCOPUS:85071753068

T3 - Leibniz International Proceedings in Informatics, LIPIcs

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

A2 - Katoen, Joost-Pieter

A2 - Heggernes, Pinar

A2 - Rossmanith, Peter

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -