On the Bipartite Unique Perfect Matching Problem

In this note, we give tighter bounds on the complexity of the bipartite unique perfect matching problem, bipartite-UPM. We show that the problem is in C=L and in NL⊕L, both subclasses of NC2. We also consider the (unary) weighted version of the problem. We show that testing uniqueness of the minimum-weight perfect matching problem for bipartite graphs is in ${\rm \bf L}^{{\rm \bf C_=L}}$ and in NL⊕L. Furthermore, we show that bipartite-UPM is hard for NL.

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