Let N $=($G $=($V$,$E$),$c$,$s$,$t$)$ be a flow network and f a flow which establishes antisymmetries and flow conservation. For a vertex that is not $\{$s$,$t$\}$ we define:

Given the following problem:

Input: a flow network N $=($G $=($V$,$E$),$c$,$s$,$t$),$ and a function k:V$/\{$s$,$t$\}->$ R on the vertices of the graph.

Find: a flow function f on the edges of the graph which is a maximum flow, antisymmetric and Conservation of flows so that:

Describe an algorithm for the given problem and analyze its running time.