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Let the bipartite graph have parts $L,R$. Add a source $s$, a sink $t$, capacity-one edges $s\to u$ for $u\in L$ and $v\to t$ for $v\in R$, and capacity $M=|V|+1$ on each original edge directed $L\to R$. Use the max-flow min-cut theorem: maximum flow value equals minimum cut capacity. Also use Integrality of the Ford-Fulkerson algorithm: integral capacities admit an integral maximum flow, found by integral augmenting paths.

An integral flow selects a matching in a graph, because each left and right vertex carries at most one unit. Conversely every matching gives a unit flow along its selected $s\to u\to v\to t$ paths. Hence maximum flow value equals maximum matching cardinality.

A minimum cut has capacity at most $|L|$, whereas crossing even one original edge would cost $M>|L|$. For its source side $Z$, no edge of the original graph runs from $L\cap Z$ to $R\setminus Z$. Thereforeis a vertex cover, and its cardinality is exactly the cut capacity. Conversely, given any cover $U$, use $Z=\{s\}\cup (L\setminus U)\cup (R\cap U)$. No original edge crosses this cut, because both its endpoints would otherwise be outside the cover; its capacity is $|U|$. Thus minimum cut capacity equals minimum cover size, proving Kőnig's theorem for bipartite matching:For the algorithm, repeatedly find augmenting paths in the residual network, then take $Z$ to be the vertices reachable from $s$. Each integral augmentation increases flow by at least one; the total value is at most $\min (|L|,|R|)$. Each reachability search takes $O(|V|+|E|)$ time, so this procedure takes $O(|V|(|V|+|E|))$ time. The network uses only polynomially many edges and integer capacities, yielding the requested polynomial-time algorithm and an explicit cover.

Assume the capacities are integers and begin with the zero flow. Inductively, if every edge flow is an integer, then every forward residual capacity $c(u,v)-f(u,v)$ and reverse residual capacity $f(u,v)$ is an integer. The bottleneck on an augmenting path is therefore a positive integer. Augmenting by it changes every affected edge flow by an integer, so all edge flows remain integral.

Each augmentation raises the flow value by at least one. On the other hand, every feasible flow satisfiesand the right side is a finite integer because the network is finite. There can therefore be only finitely many augmentations. When the algorithm stops, no augmenting path remains, and the source-reachable cut in the residual network has capacity equal to the current flow value. The max-flow min-cut theorem proves that the terminating flow is maximum. This proves both termination and the Integrality of the Ford-Fulkerson algorithm.

In a finite flow network, a maximum flow exists because the feasible flow set contains the zero flow and is a compact set: the capacity constraints bound every coordinate, and they and flow conservation define a closed set. The strength of a flow is a continuous function, so the extreme value theorem gives an attained maximum. An augmenting path in the residual network would permit positive augmentation, contradicting maximality. The source-reachable vertices in the residual network therefore define a cut of a flow network whose outgoing original edges are saturated and incoming original edges carry zero flow. Flow conservation makes the strength of a flow equal to this cut capacity, proving the max-flow min-cut theorem.

Existence and strong duality for finite real flow network edge capacities do not imply finite termination for every choice of augmenting paths in the Ford-Fulkerson algorithm. With irrational capacities, some choices can continue forever. With integer capacities, Integrality of the Ford-Fulkerson algorithm instead forces finite termination; rational capacities reduce to that case by a common denominator. Choosing shortest augmenting paths as in the Edmonds–Karp algorithm avoids dependence on capacity magnitudes.^[ref]