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.