  In the weighted bipartite matching problem, the goal is to find a
maximum-weight matching in a bipartite graph with nonnegative edge weights. We
consider its online version where the first vertex set is known beforehand, but
vertices of the second set appear one after another. Vertices of the first set
are interpreted as items, and those of the second set as bidders. On arrival,
each bidder vertex reveals the weights of all adjacent edges and the algorithm
has to decide which of those to add to the matching. We introduce an optimal,
$e$-competitive truthful mechanism under the assumption that bidders arrive in
random order (secretary model).
  It has been shown that the upper and lower bound of e for the original
secretary problem extends to various other problems even with rich
combinatorial structure, one of them being weighted bipartite matching. But
truthful mechanisms so far fall short of reasonable competitive ratios once
respective algorithms deviate from the original, simple threshold form. The
best known mechanism for weighted bipartite matching by Krysta and V\"ocking
(ICALP 2012) offers only a ratio logarithmic in the number of online vertices.
We close this gap, showing that truthfulness does not impose any additional
bounds. The proof technique is new in this surrounding, and based on the
observation of an independency inherent to the mechanism. The insights provided
hereby are interesting in their own right and appear to offer promising tools
for other problems, with or without truthfulness.
