Problem with understanding two sided Matching Algorithm: maximium cardinality I am trying to understand the maximum cardinality problem in the context of stable matching algorithm. I am reading the following article at the link:

A Two-Sided Matching Decision Model Based on Uncertain Preference Sequences

The article says that:


  In general, we can categorize two-sided matching problem into three
  typical kinds of models in terms of different decision objectives:
  stable matching, maximum cardinality matching, and maximum weight
  matching. In the first model, the objective is to seek a stable matching
  solution, and we count a solution as stable matching only when there
  does not exist any alternative pairing (, ) in which  and  are
  individually better off than they would be with the element currently
  matched. Gale and Shapley put forward an approach, also named
  Gale-Shapley algorithm, to get a stable matching solution in the
  perspective of mathematics and game theory, which symbolizes the
  beginning of two-sided matching research and enlightens the subsequent
  scholars to pay more attention to this topic. In the second model, the
  objective is to seek a solution in which the number of matching pairs
  is maximized.


I am able to understand stable matching. I can’t understand how the number of matching pair is maximized. This may occur because we have 2 sets. One of boys and other of girls. One element in one set has more than
one matching in the other set. This might occur due to preference sequence. 
Am I right about maximum cardinality?

What I understand preference sequence as the order of preferences of elements of one set for the other. Due to maximum cardinality, it is possible  that element 
Of one set has same preferences for  multiple elements of the other set. 

Am I right about preference sequences?

Somebody please guide me.

Zulfi.
The standard definition of the stable marriage problem is, given $n$ men and $n$ women, find a stable matching that marries all of the men and women.  Consequently, by definition, everyone will be matched and the number of matches will be exactly $n$ in any solution.  So, no, what you mention cannot happen, if you use that as the definition of the stable marriage problem.

I'm not 100% sure what your text is referring to when it talks about maximum cardinality matching, but I suspect it's referring to find a maximum matching in a bipartite graph, which is completely separate problem.  In this problem, the matching is not required to be a stable matching and it has nothing to do with the stable marriage problem.