  We study the two-stage vertex-weighted online bipartite matching problem of
Feng, Niazadeh, and Saberi (SODA 2021) in a setting where the algorithm has
access to a suggested matching that is recommended in the first stage. We
evaluate an algorithm by its robustness $R$, which is its performance relative
to that of the optimal offline matching, and its consistency $C$, which is its
performance when the advice or the prediction given is correct. We characterize
for this problem the Pareto-efficient frontier between robustness and
consistency, which is rare in the literature on advice-augmented algorithms,
yet necessary for quantifying such an algorithm to be optimal. Specifically, we
propose an algorithm that is $R$-robust and $C$-consistent for any $(R,C)$ with
$0 \leq R \leq \frac{3}{4}$ and $\sqrt{1-R} + \sqrt{1-C} = 1$, and prove that
no other algorithm can achieve a better tradeoff.
