I have 2 max-flow function f1,f2 that are different from one another (atleast on a single edge). I know they give each edge a natural even value.
I constrct 2 types of average functions:
A)$ g (e) = (f1(e)+f2(e))/2$
B) $h(e)=f1(e)*1/3 + f2(e)*2/3$
I want to prove or disproof their being max flow functions too.
My thoughts are to prove its true in both cases. I think for g I can look at the min-cut. I know the flow in the min-cut must be the same at that cut's capacity. So it means each edge in the cut is flowed like its capacity. Then the average is flowing the same for these edges... So g is a max flow too.
I feel the same argument shows h is max flow too. I believe it can be generelized for any fraction?
I would also like to hear whats the thoughts on my proof for g amd h? Am i right? Or did i mistake?