In max-flow problems, like in matching problems, augmenting paths are paths where the amount of flow between the source and sink can be increased.
The majority of realistic matching problems are much more complex than those presented above.
A common bipartite graph matching algorithm is the Hungarian maximum matching algorithm, which finds a maximum matching by finding augmenting paths.
More formally, the algorithm works by attempting to build off of the current matching, by the shortest augmenting path making sure that no constraints are violated.
Graph matching algorithms often use specific properties in order to identify sub-optimal areas in a matching, where improvements can be made to reach a desired goal.