Widest augmenting path (${\max }_{P:s⇝t}{\min }_{e\in P}{c}_{\mathrm{res}}(e)$)

An augmenting path maximizing the possible increase in flow, namely its smallest residual network edge capacity. If the remaining maximum flow is $F>0$ and the original undirected network has $m$ edges, a path has bottleneck at least $F/m$. Otherwise the vertices reachable through arcs of capacity at least $F/m$ define a cut of capacity strictly less than $F$, contradicting the max-flow min-cut theorem. Only one directed arc per original crossing edge contributes to that cut.

Starting from zero flow, a widest augmenting path increases the current flow by at least $F_k/m$, where $F_k$ is its remaining optimality gap. Therefore $F_{k+1}\le (1-1/m)F_k$. Integer capacities keep this gap integer, so a gap below one is zero. The method takes $O(m\log f^{\ast })$ augmentations for $f^{\ast }\ge 2$, or uniformly $O(m\log (1+f^{\ast }))$; zero maximum flow requires no augmentation.