The seriation problem in the presence of a double Fiedler value

TL;DR

This paper investigates the spectral seriation problem when the Fiedler value is not simple, analyzing its impact on solutions and proposing methods to handle this case with supporting experiments.

Contribution

It extends spectral seriation algorithms to cases with a double Fiedler value, providing new insights and computational approaches.

Findings

Double Fiedler value affects admissible solutions

Proposed methods effectively handle non-simple Fiedler values

Numerical experiments validate the approaches

Abstract

Seriation is a problem consisting of seeking the best enumeration order of a set of units whose interrelationship is described by a bipartite graph, that is, a graph whose nodes are partitioned in two sets and arcs only connect nodes in different groups. An algorithm for spectral seriation based on the use of the Fiedler vector of the Laplacian matrix associated to the problem was developed by Atkins et al., under the assumption that the Fiedler value is simple. In this paper, we analyze the case in which the Fiedler value of the Laplacian is not simple, discuss its effect on the set of the admissible solutions, and study possible approaches to actually perform the computation. Examples and numerical experiments illustrate the effectiveness of the proposed methods.