  In the Popular Matching problem, we are given a bipartite graph $G = (A \cup
B, E)$ and for each vertex $v\in A\cup B$, strict preferences over the
neighbors of $v$. Given two matchings $M$ and $M'$, matching $M$ is more
popular than $M'$ if the number of vertices preferring $M$ to $M'$ is larger
than the number of vertices preferring $M'$ to $M$. A matching $M$ is called
popular if there is no matching $M'$ that is more popular than $M$.
  We consider a natural generalization of Popular Matching where every vertex
has a weight. Then, we call a matching $M$ more popular than matching $M'$ if
the weight of vertices preferring $M$ to $M'$ is larger than the weight of
vertices preferring $M'$ to $M$. For this case, we show that it is NP-hard to
find a popular matching. Our main result its a polynomial-time algorithm that
delivers a popular matching or a proof for it non-existence in instances where
all vertices on one side have weight $c > 3$ and all vertices on the other side
have weight 1.
