We present an algorithm for the multi-cut problem. This algorithm does the following: given a pipe system of volume F, where F is the optimal solution of the fractional relaxation of the linear program, we compute a multi-cut of cost O(log k) F. Since the optimal solution is at most the optimum of the linear program relaxation, we get an O(log k) approximation. This algorithm works as follows: at any step we call a procedure that picks a vertex si that is not disconnected from the correspondent sink ti and builds a ball around this vertex. What we do is basically to cut all the edges that are on the border of this ball. We prove that the cost of the edges on the border of the ball is related to the total volume of the pipe system inside the ball by our approximation ratio. So, all the vertices on the border of the ball will be part of the cut, then we remove all the edges of this cut from the graph and then we see if still some pair of terminals that is still connected. Moreover, we prove that no two terminals as (si,ti) are in the same ball. And this is because the radius of the ball is at most one-half and since all the terminals are at least away for a distance 1, they cannot be in the same ball.

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