Kuhn's Algorithm

Kuhn's algorithm repeatedly finds augmenting paths until none exist. Start with an empty matching, then for each unmatched vertex in $L$, try to find an augmenting path using DFS. This is distinct from the Hungarian algorithm, which solves weighted assignment problems.

This runs in $O(VE)$ time: you do $O(V)$ iterations of DFS, each taking $O(E)$ time. This is fast enough for most problems with $V \leq 1000$.

Space complexity is $O(V)$ for the data structures used.