A note on the hardness results for the labeled perfect matching problems in bipartite graphs

In this note, we strengthen the inapproximation bound of O(log n) for the labeled perfect matching problem established in J. Monnot (2005), using a self improving operation in some hard instances. It is interesting to note that this self improving operation does not work for all instances. Moreover, based on this approach we deduce that the problem does not admit constant approximation algorithms for connected planar cubic bipartite graphs.