A new, efficient, configuration-driven algorithm utilizing the unitary group approach (UGA) was developed and implemented for the generalized van Vleck perturbation theory (GVVPT) variant of multireference perturbation theory. The computational speed has been improved by 1 or 2 orders of magnitude compared to the previous implementation based on the Table-CI technique. It is shown that the reformulation is applicable to both the second-order (GVVPT2) and third-order (GVVPT3) approximations. Calculations on model problems and on a chemically realistic description of cyclobutadiene are used to illustrate the performance of the method. The calculations on cyclobutadiene, using over 2.3 billion CSFs, provide results on geometric parameters and the barrier height of the automerization reaction in good agreement with established high accuracy results.