Question

Basic defintions:

Capacity constraint: For all u, v V, we require f(u, v) <= c(u, v).

Skew symmetry: For all u, v V, we require âf(u, v) = -f (v, u).

Flow conservation: For all u belongs to V – {s, t}, we require ( (sum of(v belongs to V)) f(u,v) ) = 0

Let f1 and f2 be flows in a flow network G = (V, E). The sum f1 +f2 is defined by
(f1 +f2)(u, v) = f1(u, v) + f2(u, v) for all (u, v) belongs to V. Of the three flow properties the following are satisfied by f1 + f2.

Capacity constraint: May clearly be violated.

Skew symmetry: We have:
(f1 + f2)(u, v) = f1(u, v) + f2(u, v) = -f1(v, u) – f2(v, u)
= -(f1(v, u) + f2(v, u)) = -(f1 + f2)(v, u)

My questions are below

1. How capacity contraint is violated in above?

2. What is flow conservation? and why sum of flow conservation is zero for vertices not including source and tank in u ? Request to help with simple example.

Thanks!

Answer 1

1. flow capacity is indeed violated. look at the following example: f1(u,v) = f2(u,v) = c(u,v) > 0. The constraint is kept for each f1,f2 because they are both not greater then c. However, look at f1+f2: f1+f2(u,v) = f1(u,v) + f2(u,v) = 2*c(u,v), and since for this example c(u,v) > 0, clearly f1+f2(u,v) > c(u,v), so the capacity constraint is not kept.

2. flow conservation is basically: for each vertex except s,t: the same amount of flow enters the vertex and leaves the vertex. So each v in V\{s,t} is not “creating” any flow, nor is consuming any flow: only s,t are allowed to do it.