The Hungarian algorithm finds a maximum independent edge set on a graph. The algorithm starts with any matching 
and constructs a tree via a breadth-first search to find an augmenting
path, namely a path 
that starts and finishes at unmatched vertices whose first
and last edges are not in 
and whose edges alternate being outside and inside 
. If the search succeeds, the symmetric
difference of 
and the edges in 
yields a matching with one more
edge than 
. That edge is added, and then another search is performed
for a new augmenting path. If the search is unsuccessful,
the algorithm terminates and 
must be the largest-size matching.
As an added bonus, the tree data provides a vertex cover.
If the tree search is unsuccessful, as it is at the end, then the size of the vertex cover is the same as the size of the matching, which proves that the final matching has the maximum size possible.
Algorithm for Maximum Matching in Bipartite Graphs."
SIAM J. Comput. 2, 225-231, 1975.
Wagon, S. "The Hungarian
Maximum Matching Algorithm."