On the rank of Z_2-matrices with free entries on the diagonal

TL;DR

This paper introduces algorithms to determine the minimal rank of $\, ext{Z}_2$-matrices with free diagonal entries and provides bounds, with applications to graph embeddings on non-orientable surfaces.

Contribution

It presents polynomial-time algorithms for deciding the minimal rank and approximating it for $\, ext{Z}_2$-matrices with free diagonal entries, a novel computational approach.

Findings

Polynomial algorithm for deciding if $R(M) \, extless= k$

Polynomial algorithm for approximating $R(M)$ within a factor of 2

Applications to graph drawings on non-orientable surfaces

Abstract

For an $n \times n$ matrix $M$ with entries in $\mathbb{Z}_2$ denote by $R(M)$ the minimal rank of all the matrices obtained by changing some numbers on the main diagonal of $M$. We prove that for each non-negative integer $k$ there is a polynomial in $n$ algorithm deciding whether $R(M) \leq k$ (whose complexity may depend on $k$). We also give a polynomial in $n$ algorithm computing a number $m$ such that $m/2 \leq R(M) \leq m$. These results have applications to graph drawings on non-orientable surfaces.