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 aself improving operation in some hard instances. It is interestingto note that this self improving operation does not work for allinstances. Moreover, based on this approach we deduce that theproblem does not admit constant approximation algorithms forconnected planar cubic bipartite graphs.
