Preflow-push

Abstract view

Algorithmic problem: max-flow problem (standard version)

Type of algorithm: loop.

Definition:

1. For [math]\displaystyle{ a\in A }[/math], let [math]\displaystyle{ f(a) }[/math] be a real number (not necessarily foring a flow). For [math]\displaystyle{ v\in V\setminus\{s,t\} }[/math], the excess of [math]\displaystyle{ v }[/math] with respect to [math]\displaystyle{ f }[/math] is defined by [math]\displaystyle{ e_f(v):=\sum_{w:(w,v)\in A}f(w,v)-\sum_{w:(v,w)\in A}f(v,w) }[/math].
2. A node [math]\displaystyle{ v\in V\setminus\{s,t\} }[/math] is called active it its excess is positive.
3. Let [math]\displaystyle{ \Delta }[/math] denote the sum of the distance labels [math]\displaystyle{ d(v) }[/math] of all active nodes [math]\displaystyle{ v\in V\setminus\{s,t\} }[/math].

Invariant: Before and after each iteration:

1. For each arc [math]\displaystyle{ a\in A }[/math], it is [math]\displaystyle{ 0\leq f(a)\leq u(a) }[/math] .
2. For each node [math]\displaystyle{ v\in V }[/math], it is [math]\displaystyle{ e_f(v)\geq 0 }[/math].
3. The node labels [math]\displaystyle{ d }[/math] form a valid distance labeling, and it is [math]\displaystyle{ d(s)=|V| }[/math].

Variant: At least one

1. The label [math]\displaystyle{ d(v) }[/math] of at least one node [math]\displaystyle{ v\in V\setminus|{s,t\} }[/math] increases.
2. The push is saturating.
3. The value of [math]\displaystyle{ \Delta }[/math] decreases.

Induction basis

1. Set