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

Approximation Algorithms for Multi-Multiway Cut and Multicut Problems on Directed Graphs