Genetic Algorithms applied to Problems of Forbidden Configurations

A simple matrix is a (0,1)-matrix with no repeated columns. For a (0,1)-matrix F, we say a (0,1)-matrix A avoids F (as a configuration) if there is no submatrix of A which is a row and column permutation of F. Let ‖A ‖ denote the number of columns of A. We define forb(m,F) = max{‖A ‖ : A is an m-rowed simple matrix that avoids F}. Define an extremal matrix as an m-rowed simple matrix A with that avoids F and ‖A ‖ = forb(m,F). We describe the use of Local Search Algorithms (in particular a Genetic Algorithm) for finding extremal matrices. We apply this technique to two forbidden configurations in turn, obtaining a guess for the structure of an m × forb(m,F) simple matrix avoiding F and then proving the guess is indeed correct. The Genetic Algorithm was also helpful in finding the proof