Successive shortest paths

Abstract view

Invariant:

1. The capacity constraints are fulfilled, that is, [math]\displaystyle{ 0\leq f(a)\leq u(a) }[/math] for all [math]\displaystyle{ a\in A }[/math].
2. The balance discrepancy of each node [math]\displaystyle{ v\in V }[/math] is underestimating, that is,
   1. If [math]\displaystyle{ b(v)\gt 0 }[/math], then [math]\displaystyle{ \sum_{w:(v,w)\in A}f(v,w)-\sum_{w:(w,v)\in A}f(w,v)\leq b(v) }[/math].
   2. If [math]\displaystyle{ b(v)\lt 0 }[/math], then [math]\displaystyle{ \sum_{w:(v,w)\in A}f(v,w)-\sum_{w:(w,v)\in A}f(w,v)\geq b(v) }[/math].
   3. If [math]\displaystyle{ b(v)=0 }[/math], then [math]\displaystyle{ \sum_{w:(v,w)\in A}f(v,w)-\sum_{w:(w,v)\in A}f(w,v)=b(v) }[/math].

Variant: The total balance discrepancy strictly decreases, that is, the value

[math]\displaystyle{ \sum_{v\in V}\left|\sum_{w:(v,w)\in A}f(v,w)-\sum_{w:(w,v)\in A}f(w,v)-b(v)\right| }[/math].

Induction basis

Abstract view: Start with the zero flow.

Proof: Obvious.