The last type of graph problem we will investigate is *maximum flow* problems. Intuitively, these problems can be thought of as starting with a *source* faucet that can provide as much water as possible. The faucet is connected to a *sink* drain that can handle any amount of water through a *network* of pipes. The pipes are of different sizes, i.e. have different *capacities* which limit the *flow* through certain branches of the network. The problem is to determine what is the *maximal flow*, i.e. the most water, that can travel through the network from the faucet to the drain.

We shall now formalize the *maximum flow* problem in graph terms, state some key definitions for network flows, and prove a very important theorem that will serve as the foundation for the algorithms.

Given a *flow network* *G*(*V*, *E*) which is a *connected, directed* graph with each edge (*u*,*v*) having a *capacity* *c*(*u*,*v*) ≥ 0 (assuming *c*(*u*,*v*) = 0 if (*u*,*v*) ∉ *E*), a *source* vertex *s*, and a *sink* vertex *t*: a *flow* is a function between all pairs of vertices that satisfies

f(u,v) ≤c(u,v) for allu,v(capacity constraint)f(u,v) = -f(v,u) for allu,v(skew-symmetry)- Σ
f(u,v) = 0 for allu,vexcepts,t(flow conservation, i.e. flow in = flow out)

The *value* of a flow is defined as

which is the total amount out of the source (which by flow conservation must also equal the total amount into the sink).

**Problem**

Given *G*, *s*, and *t* find the *maximal flow value*.

*Multiple Source/Sinks*

The problem of multiple sources and/or sinks can be easily recast as a single source, single sink problem by creating "super" source and sink nodes with edges of *infinite* capacity between the "super" nodes and the network source/sinks as illustrated below.

A *residual network* is an *induced graph* that contains edges that can admit more flow, i.e. the edges that are not at capacity. The *residual capacity* of any edge is given by

Thus the residual network consists of the edges for which *c*_{f}(*u*,*v*) > 0. Note that for an edge to appear in the residual network, either (*u*,*v*) or (*v*,*u*) must be in the original network.

An *augmenting path* is a *simple path* from *s* ↝ *t* in the residual network. In other words it is an entire path which can admit additional flow from the source to the sink (all edges along the path have residual capacity). The *residual capacity of an augmenting path* is the maximum amount of additional flow that can be admitted along an augmenting path - which is limited by the *edge* with *minimum* residual capacity.

A *cut* is a partition of *V* into sets *S* and *T* such that *s* ∈ *S* and *T* = *V* - *S* has *t* ∈ *T*. This is similar to MST cuts except that the source must be in one partition and the sink in the other partition (and the graph is directed).

The *net flow* across the cut is *f*(*S*,*T*) and the *capacity* of the cut is *c*(*S*,*T*).

A *minimum cut* is a cut with *minimum capacity*. It can be shown that the *net flow* across any cut is equal to the total flow of the network (again by flow conservation), and furthermore the value of any flow is upper bounded by the capacity of any cut.

Using the above definitions, the following theorem can be proven:

If

fis a flow in a networkGwith sourcesand sinkt, then the following three statements are equivalent:

fis amaximum flowinG(with value |f^{*}|)- The
residual networkcontainsnoaugmenting paths.- |
f| =c(S,T) for some cut (S,T) ofG

By the max-flow, min-cut theorem, a maximal flow can be found by continually augmenting flow along paths with residual capacity, i.e.

1. Initialize f to 0 2. while there exists an augmenting path p 3. augment the flow along p by the residual capacityc_{f}(p)

While this algorithm is not particularly useful for implementation, we can observe that for integer capacities the algorithm runs in O(*E*|*f* ^{*}|) since each iteration checks at most *E* edges (e.g. BFS) and increases the flow by at least 1.