当前位置: X-MOL 学术Discret. Optim. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Spectral aspects of symmetric matrix signings
Discrete Optimization ( IF 1.1 ) Pub Date : 2020-04-30 , DOI: 10.1016/j.disopt.2020.100582
Charles Carlson , Karthekeyan Chandrasekaran , Hsien-Chih Chang , Naonori Kakimura , Alexandra Kolla

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.



中文翻译:

对称矩阵符号的频谱方面

有符号矩阵的光谱在社会科学,图论和控制论中起着基本作用。在这项工作中,我们调查发现具有自然光谱特性的矩阵的对称符号的计算问题。我们的结果如下:

1.我们表征具有可逆符号的矩阵:当且仅当矩阵的支持图包含完美2匹配时,对称矩阵才具有可逆对称符号。此外,我们提出了一种有效的算法来搜索可逆对称签名。

2.我们使用上述特征来给出一种算法,以找到对给定对称矩阵的支持的最小增加,从而使其具有可逆的对称符号。

3.我们展示了以下问题的NP完备性:验证给定矩阵是否具有对称/非对角正负对称对称特征值。但是,我们还说明,对于图的邻接矩阵,输入矩阵的复杂度可能会有很大差异。

除了匹配理论的经典结果外,我们还使用组合技术。

更新日期:2020-04-30
down
wechat
bug