Improve the efficiency of the 2-approximation algorithm

Design a more efficient 2-approximation algorithm for the minimum matching vertex-cutset problem on connected simple graphs that are neither complete graphs K_{2n} nor complete bipartite graphs K_{n,n}.

Background

The paper gives a 2-approximation algorithm for the minimum matching vertex-cutset problem with running time O(nm^2), using minimum vertex cuts, maximum matching, and related matching procedures. The second explicitly stated problem seeks a 2-approximation algorithm with improved computational efficiency, without requiring an improved approximation ratio.