  We consider the max-size popular matching problem in a roommates instance G =
(V,E) with strict preference lists. A matching M is popular if there is no
matching M' in G such that the vertices that prefer M' to M outnumber those
that prefer M to M'. We show it is NP-hard to compute a max-size popular
matching in G. This is in contrast to the tractability of this problem in
bipartite graphs where a max-size popular matching can be computed in linear
time. We define a subclass of max-size popular matchings called strongly
dominant matchings and show a linear time algorithm to solve the strongly
dominant matching problem in a roommates instance.
  We consider a generalization of the max-size popular matching problem in
bipartite graphs: this is the max-weight popular matching problem where there
is also an edge weight function w and we seek a popular matching of largest
weight. We show this is an NP-hard problem and this is so even when w(e) is
either 1 or 2 for every edge e. We also show an algorithm with running time
O*(2^{n/4}) to find a max-weight popular matching matching in G = (A U B,E)$ on
n vertices.
