1. Maximum flow problems II.
MA252 Combinatorial Optimization

2. 1 2 3 4
From 𝑠𝑡-paths to a flow 1 1 1 2 1 3 t 2 2 1 s 1 1 1 1

3. Conservation of flow 1 2 3 4 v t s value of flow net flow (excess)
𝑓 𝑥 𝑣 = 𝑤∈𝑉 𝑤𝑣∈𝐸 𝑥 𝑤𝑣 − 𝑤∈𝑉 𝑣𝑤∈𝐸 𝑥 𝑣𝑤
incoming =1+2+1=4
outgoing = =4
incoming
outgoing
=1+2+1
−1−3=0 1
incoming =4
outgoing =0 1
𝑓 𝑥 𝑡 =4 1 2 3 v t 2 2 1 s 1
incoming =0
outgoing =4 1
value
of flow 1 1
𝑓 𝑥 𝑠 =−4

4. Feasible flow to 𝑠𝑡-paths1 2 3 4
Feasible flow to 𝑠𝑡-paths
incoming >0
 outgoing >0
...
value of flow >0
 outgoing from 𝑠 >0
after finite
# of steps
we reach 𝑡 1 1 2 1 1 1
𝑓 𝑥 𝑡 =4
𝑓 𝑥 𝑡 =3
𝑓 𝑥 𝑡 =0
𝑓 𝑥 𝑡 =2 2 1 2 1 2 4 3 t s 2 2
value of flow =4  find 4 𝑠𝑡-paths together not exceeding capacities 1 1 1 1
𝑓 𝑥 𝑠 =0
𝑓 𝑥 𝑠 =−3
𝑓 𝑥 𝑠 =−4
𝑓 𝑥 𝑠 =−2 1 1
Question:
what if we
return to 𝑠?