\documentclass[12pt]{article}
\usepackage{amsmath,mathrsfs,bbm}
\usepackage{amssymb}
\textwidth=4.825in
\overfullrule=0pt
\thispagestyle{empty}

\begin{document}
\noindent
%
%
{\bf R.P. Anstee and Miguel Raggi}
%
%

\medskip
\noindent
%
%
{\bf Genetic Algorithms Applied to Problems of  Forbidden Configurations}
%
%

\vskip 5mm
\noindent
%
%
%
%
A \emph{simple} matrix is a (0,1)-matrix with no repeated columns. For
a (0,1)-matrix $F$, we say a (0,1)-matrix $A$ \emph{avoids} $F$ (as a
\emph{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 \emph{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.

\end{document}

.
