We present a Branch-and-Cut algorithm where the volume algorithm is applied
instead of the traditionally used dual simplex algorithm to the linear
programming relaxations in the root node of the search tree. This means that
we use fast approximate solutions to these linear programs instead of exact
but slower solutions. We present computational results with the Steiner tree
and Max-Cut problems. We show evidence that one can solve these problems
much faster with the volume algorithm based Branch-and-Cut code than with a
dual simplex based one. We discuss when the volume based approach might be
more efficient than the simplex based approach.