The isomorphic version of Brualdies nestedness is in P

TL;DR

This paper introduces an efficient method to compute the minimal discrepancy of ecological bipartite networks by leveraging isomorphism classes and reducing the problem to weighted perfect matching problems.

Contribution

It defines a metric for minimal discrepancy within isomorphic classes and provides a reduction to solve it via multiple weighted perfect matching problems.

Findings

Reduction to at most n weighted perfect matching problems

Efficient computation of isomorphic discrepancy in bipartite networks

Application to ecological network analysis

Abstract

The discrepancy BR for an $m \times n$ $0,1$-matrix from Brualdi and Sanderson \cite{Brualdi1998} counts the minimum number of $1$'s which need to be shifted in each row to the left to achieve its Ferrers matrix, i.e. each row consists of consecutive $1$'s followed by consecutive $0$'s. For ecological bipartite networks BR describes how nested a set of relationships is. Since different labeled matrices can be isomorphic but possess different discrepancies, we define a metric determining the minimum discrepancy in an isomorphic class. We give a reduction to $k\leq n$ minimum weighted perfect matching problems.