Why Max Flow Works

In the flow network, integer max flow sends at most $1$ unit through each vertex (by capacity constraints). A flow of $f$ units means $f$ edge-disjoint paths from $s$ to $t$, each passing through one vertex in $L$ and one in $R$.

This set of paths corresponds exactly to a matching of size $f$. Since max flow finds the largest such flow, it finds the maximum matching. Algorithms like Dinic or Ford-Fulkerson solve this in $O(V^2 E)$ or faster.

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