Complexity of linear mappings induced by graphs

Jose Zamora

Abstract

What is the computational complexity of the following problem:

Given a set X of cardinality N=2^n and k bijections f_1,f_2,...,f_k: X --> X, does there exist a bijection g: X --> GF(2)^n such that each composed mapping g^{-1} o f_i o g is linear (affine)?

The problem is motivated by locally bijective graph homomorphisms (also known as graph covers). We study the case k=1, and give polynomial algorithm under some assuptions. The algorithm is based on a procedure to find a representative of any conjugacy class of GL(n,2).