In this note, we strengthen the inapproximation bound of O(logn) for the labeled perfect matching problem established in J.
Monnot, The Labeled perfect matching in bipartite graphs, Information Processing Letters96 (2005) 81–88, 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.