Augmenting Paths

Given flow network G and flow f, an augmenting path p is a simple path from s to t in the residual network G_f.

The minimum net flow along path p in G_f indicates the amount flow can increase along this path in G.

Thus, define the residual capacity of path p as

$$c_f(p) \;=\; min\{c_f(u,v) \;\mid\; (u,v) \;{\rm is} \;{\rm on} \;p\}$$

Define flow f_p in G_f as

$$f_p(u,v) \;=\; \left\{ \begin{array}{cl} c_f(p) & {\rm if} \;... ...,u) \;{\rm on} \;p \\ 0 & {\rm otherwise} \end{array} \right.$$

$$\mid f_p \mid \;=\; c_f(p) \;>\; 0$$

Define flow sum f₁ + f₂ as .