Question

How can I implement a bipartite matching algorithm (probably based on a max-flow algorithm) in C or C++ ?

To be specific, I have this input in a file:
(1,3)
(1,5)
(2,5)

(M,F) –> where M represents id of MALE and F is id of FEMALE.

I need to find the maximum number of matches and show matched couples.
Like:
matches: 1&3 , 2&5

I have read in some books I can base this problem on a “maximum flow in a network” algorithm, but I couldn’t find any specific info other than the sentence “this problem can be solved by …. algorithm”.
I have little knowledge about max-flow, and dont know how to implement it either…

Answer 1

Yes, bipartite matching can be reduced to maximum flow:

1. You’re given sets of nodes M and F. Add a directed edge from a node m in M to a node f in F if you’ve got the pair (m, f) in your file.

2. Add a single node S with a directed edge from S to every node in M (this is your “super-source” node).

3. Add a single node T with a directed edge from every node in F to T (this is your “super-sink” node).

4. Now, you need to find the maximum flow (with all your edges of weight 1) from S to T.

So what the heck is maximum flow? A flow from S to T is a set of edges such that for each node (except S and T), the weight of its in-flux edges is the same as the weight of its out-flux edges. Imagine that your graph is a series of pipes, and you’re pouring water in the system at S and letting it out at T. At every node in between, the amount of water going in has to be the same as the amount of water coming out.

Try to convince yourself that a flow corresponds to a matching of your original sets. (Given a flow, how to you get a matching? Given a matching, how to you get a flow?)

Finally, to find the maximum flow in a graph, you can use the Ford-Fulkerson algorithm. The above wikipedia page gives a good description of it, with pseudo-code.