  In this paper, we present two approximation algorithms for the directed
multi-multiway cut and directed multicut problems. The so called region growing
paradigm \cite{1} is modified and used for these two cut problems on directed
graphs. By using this paradigm, we give for each problem an approximation
algorithm such that both algorithms have the approximate factor $O(k)$ the same
as the previous works done on these problems. However, the previous works need
to solve $k$ linear programming, whereas our algorithms require only one linear
programming. Therefore, our algorithms improve the running time of the previous
algorithms.
