The elemental set algorithm involves performing many fits to a data set, each fit made to a subsample of size just large enough to estimate the parameters in the model. Elemental sets have been proposed as a computational device to approximate estimators in the areas of high breakdown regression and multivariate location/scale estimation, where exact optimization of the criterion function is computationally intractable. Although elemental set algorithms are used widely and for a variety of problems, the quality of the approximation they give has not been studied. This article shows that they provide excellent approximations for the least median of squares, least trimmed squares, and ordinary least squares criteria. It is suggested that the approach likely will be equally effective in the other problem areas in which exact optimization of a criterion is difficult or impossible.