What is an algorithm to find a minimum vertex cover on a bipartite graph with weighted vertices?
The weighted vertex cover problem can be formulated as an Integer Program (see http://en.wikipedia.org/wiki/Vertex_cover). When the input graph is bipartite, the constraint matrix of this IP is totally unimodular. Hence this IP can be solved in polynomial time.

For more details of total unimodular matrices and the corresponding algorithms, see the excellent (three volume) book by Alexander Schrijver.