Before revision:  the new algorithm can be viewed as a fine-grained reduction to O(n) Max-Flow computations on n-node graphs . Thus, under the assumption that Max-Flow can be computed in almost-linear time, our time bound improves further to n^{2.5+o(1) .
After revision:  the new algorithm can be viewed as a fine-grained reduction to O(n) Max-Flow computations on n-node graphs .