Improve the approximation ratio for minimum matching vertex-cutsets

Design a polynomial-time algorithm with an approximation ratio strictly less than 2 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 establishes that determining whether a connected graph, excluding K_{2n} and K_{n,n}, has a matching vertex-cutset of size at most k is NP-complete. It also presents a 2-approximation algorithm running in O(nm^2) for finding a minimum matching vertex-cutset. The first explicitly stated problem asks whether this approximation guarantee can be improved to a factor strictly below 2 while retaining polynomial running time.