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CIS 413/513 Advanced Data Structures
Winter 2020
Assignment 3
due Wednesday, February 12, 2020
1. (question 4 from chap 10 of Er) Let G be a flow network that contains an opposing pair of
edges u → v and v → u, both with positive capacity. Let G0 be the flow network obtained
from G by decreasing the capacities of both these edges by min{c(u → v), c(v → u)}. In
other words:
• if c(u → v) > c(v → u), change the capacity of u → v to c(u → v)−c(v → u) and delete
v → u.
• if c(u → v) < c(v → u), change the capacity of v → u to c(v → u)−c(u → v) and delete
u → v.
• if c(u → v) = c(v → u), delete both u → v and v → u.
(a) Prove that every maximum (s, t)-flow in G0
is also a maximum (s, t)-flow in G. (Thus,
by simplifying every opposing pair of edges in G, we obtain a new reduced flow network
with the same maximum flow value as G.)
(b) Prove that every minimum (s, t)-cut in G is also a minimum (s, t)-cut in G0 and vice
versa.
(c) (for 513) Prove that there is at least one maximum (s, t)-flow in G that is not a
maximum (s, t)-flow in G0
.
1

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