We develop a systematic computational approach, which includes the genetic algorithm (GA) search method, to the minimal bromate reaction system, to automate the determination of its reaction mechanism and rate coefficients. We take the 10 species of the system as given, and then determine all possible bimolecular elementary reaction steps and all possible combinations of these steps that fulfill the given overall reaction. The optimization criteria used in the GA are chosen to be the following: The reaction mechanism must have an unstable focus, or must have oscillations: has specified ranges of periods of oscillations: and has specified ranges of inflow conditions. With rate coefficients given by Field, Koros, and Noyes(1), we find reaction mechanisms with four and five elementary reaction steps that show oscillations but do not fit well the experimental [Br-](0) - [BrO3-](0) oscillation domain. We then determine, by GA, a set of rate coefficients in one particular five-step reaction mechanism that gives very good agreement with all comparable experiments, The Noyes, Field, Thompson model mechanism(2) for this reaction has seven elementary reaction steps. The first five, with our GA determined rate coefficients, are sufficient for the good agreement with experiment; the last two are not necessary. We believe that the example developed here in detail shows the utility and promise of GA methods in chemical kinetics.