Embedding nearly decomposable matrices into certain staircase matrices

Abstract

We consider the problem of embedding an n× n nearly decomposable (0,1) matrix A into a staircase matrix with L diagonals, where L is as small as possible. We obtain an upper bound for L depending on A and use this bound to obtain a lower bound for the minimum value of the permanent over F( A), the face of the n× n doubly stochastic matrices determined by A. We show that some n× n nearly decomposable matrices have L=Ω(√ n).