Genetic Algorithms Applied to Problems of Forbidden Configurations

Abstract

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 $\mathrm{forb}(m,F)=\max\{\|{A}\|\ : A$ 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 $\|{A}\|=\mathrm{forb}(m,F)$. 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 $m\times\mathrm{forb}(m,F)$ simple matrix avoiding $F$ and then proving the guess is indeed correct. The Genetic Algorithm was also helpful in finding the proof.