Maximum flow problems II. MA252 Combinatorial Optimization

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

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 =1+3 =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

Feasible flow to 𝑠𝑡-paths 1 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?