The isomorphic version of Brualdies nestedness is in P

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 $k\le n$ minimum weighted perfect matching problems.