Max-Flow Problems

Assumptions

For convenience, without loss of generality:

1. In all variations below, all numbers are integral: real numbers may be approximated by uniformly distant numbers with a negligible loss of precision.
2. There are no parallel arcs in directed graphs [math]\displaystyle{ G=(V,A) }[/math]: if there are parallel arcs, they may be replaced by one of them, with their upper bounds, lower bounds, and flow values summed up, respectively. In particular, an arc may be identified with the ordered pair [math]\displaystyle{ (v,w) }[/math]consisting of the arc's tail [math]\displaystyle{ v\in V }[/math] and head [math]\displaystyle{ w\in V }[/math].
3. For each arc [math]\displaystyle{ (v,w)\in A }[/math], there is also a reverse arc [math]\displaystyle{ (w,v)\in A }[/math]: if not, just add one with upper (and lower) bound zero.

Standard version

Input:

1. A directed graph [math]\displaystyle{ G=(V,A) }[/math].
2. A source node [math]\displaystyle{ s\in V }[/math] and a target (a.k.a. sink) node [math]\displaystyle{ t\in V }[/math].
3. A nonnegative upper bound (a.k.a. capacity) [math]\displaystyle{ u(a) }[/math] for each arc [math]\displaystyle{ a\in A }[/math].

Output: A maximum [math]\displaystyle{ (s,t) }[/math]-flow, that is, a nonnegative value [math]\displaystyle{ f(a) }[/math] for each arc [math]\displaystyle{ a\in A }[/math] such that the following two conditions are fulfilled:

1. Capacity constraint: For each arc [math]\displaystyle{ a\in A }[/math], it is [math]\displaystyle{ f(a)\leq u(a) }[/math].
2. Flow conservation condition: For each node [math]\displaystyle{ v\in V\setminus\{s,t\} }[/math], it is [math]\displaystyle{ \sum_{a=(v,w)\in A}f(a)=\sum_{a=(w,v)\in A}f(a) }[/math].

Generalizations

1. For each arc [math]\displaystyle{ a\in A }[/math], there is a lower bound [math]\displaystyle{ \ell(a) }[/math], and [math]\displaystyle{ f(a)\geq\ell(a) }[/math] is additionally required. The lower bounds need not be nonnegative, so the flow values need not be nonnegative, either. This version may be reduced to the standard version as follows:
   1. For each arc [math]\displaystyle{ (v,w)\in A }[/math] with a lower bound [math]\displaystyle{ \ell(v,w)\lt 0 }[/math], add [math]\displaystyle{ -ell(v,w) }[/math]to the upper bound of [math]\displaystyle{ (w,v) }[/math] (or, if [math]\displaystyle{ (w,v) }[/math]
2. More than one source and more than one target can be reduced to the standard version by adding a super-source node and a super-target node
3. Usually, the term generalized flow' is reserved for the specific generalization in which for each node [math]\displaystyle{ v\in V\setminus\{vs,t\} }[/math], the ratio of the total sum of all incoming flow an the total sum of all outgoing flow is given (in the standard version, this ratio is 1 due to the flow conservation condition).