Max-Flow Problems

Basic definitions

1. Basic graph definitions
2. Basic flow definitions

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 feasible [math]\displaystyle{ (s,t) }[/math]-flow that has maximum flow value among all feasible [math]\displaystyle{ (s,t) }[/math]-flows.

Known algorithms

1. Ford-Fulkerson
2. Edmonds-Karp
3. Ahuja-Orlin
4. Dinic
5. Preflow-push
6. FIFO preflow-push
7. Preflow-push with excess scaling

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 is often called maximum flow with edge demands. It may be reduced to solving two instances of the standard version as follows:
   1. First, we construct a new graph [math]\displaystyle{ G'=(V',A') }[/math] as follows: We add a super-source [math]\displaystyle{ s' }[/math] and a super-target [math]\displaystyle{ t' }[/math] to [math]\displaystyle{ V }[/math]. Next, for every node [math]\displaystyle{ v\in V }[/math], we add an arc [math]\displaystyle{ (s,v) }[/math] and an arc [math]\displaystyle{ (v,t) }[/math] to [math]\displaystyle{ A }[/math]. Finally we add an arc [math]\displaystyle{ (t,s) }[/math] to [math]\displaystyle{ A }[/math], if not in [math]\displaystyle{ A }[/math] yet.
   2. For all arcs [math]\displaystyle{ a\in A }[/math], we set [math]\displaystyle{ \ell'(a):=0 }[/math]. For each node [math]\displaystyle{ v\in V }[/math], we set

      [math]\displaystyle{ u'(s,v):=\sum_{(w,v)\in A}\ell(w,v) }[/math] and [math]\displaystyle{ u'(v,t):=\sum_{(v,w)\in A} \ell(v,w) }[/math].

      For all arcs [math]\displaystyle{ a\in A }[/math], we set [math]\displaystyle{ u'(a):=u(a)-\ell(a) }[/math]. Finally, we set [math]\displaystyle{ u'(t,s):=+\infty }[/math].

   3. It is easy to see that there is a feasible [math]\displaystyle{ (s,t) }[/math]-flow [math]\displaystyle{ f }[/math] in [math]\displaystyle{ G }[/math] with respect to [math]\displaystyle{ \ell }[/math] and [math]\displaystyle{ u }[/math] if, and only if, all arcs [math]\displaystyle{ (s',\cdot) }[/math] and [math]\displaystyle{ (\cdot,t') }[/math] are saturated. This the case if, and only if, the maximum flow [math]\displaystyle{ f' }[/math] in [math]\displaystyle{ G }[/math] from [math]\displaystyle{ s' }[/math] to [math]\displaystyle{ t' }[/math] with respect to [math]\displaystyle{ \ell' }[/math] and [math]\displaystyle{ u' }[/math] yields a flow of value [math]\displaystyle{ \sum_{a\in A}\ell(a) }[/math].
   4. More specifically, for [math]\displaystyle{ (v,w)\in A }[/math], we set [math]\displaystyle{ f(v,w):=f'(v,w)+\ell(v,w) }[/math].
2. More than one source and more than one target can be reduced to the standard version by adding a super-source node [math]\displaystyle{ s }[/math], a super-target node [math]\displaystyle{ t }[/math], an arc [math]\displaystyle{ (s,v) }[/math] with "infinite" capacity for each source [math]\displaystyle{ v }[/math], and an arc [math]\displaystyle{ (v,t) }[/math] with "infinite" capacity for each target [math]\displaystyle{ v }[/math].
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).
4. The max-flow problem asks for an optimal steady-state flow. However, in many applications, a certain amount of flow is to be send as soon as possible from the source to the target. It is easy to see that, if the amount of flow is sufficiently large, then an optimal solution is constant most of the time, and the maximum steady-state flow is this constant flow.