Residual maximum flow is the remaining optimality gap ($f_{\mathrm{res}}^{\ast }=f^{\ast }-f$)

For an undirected capacitated network, write the net edge flow as $y_{ij}=-y_{ji}$, and the directed residual capacity as $c_{ij}-y_{ij}$. On every source–sink cut, residual capacity is the original cut capacity minus the current net flow value $f$. Taking the minimum over cuts and applying the max-flow min-cut theorem to each network gives the displayed remaining-flow identity.