C&O 355: Lecture 21 Notes
Nicholas Harvey
http://www.math.uwaterloo.ca/~harvey/
1
Minimum
s

t
Cuts
In Lecture 20, we looked at the minimum
s

t
cut problem. Let’s now look at a variant of this
problem which introduces capacities on the arcs.
Problem 1.1.
Let
G
= (
V, A
) be a directed graph. Let
s
and
t
be particular vertices in
V
. For
each arc
a
∈
A
, there is a capacity
c
a
≥
0. A
cut
(or
s

t
cut) is a set
F
⊆
A
such that there is
no
s

t
path in
G
\
F
. The goal is to find a cut
F
that minimizes its capacity, namely
∑
a
∈
F
c
a
.
Let
P
be the set of all simple directed paths from
s
to
t
. An integer program formulation of
this problem is:
min
c
T
y
(IPMC)
s.t.
X
a
∈
p
y
a
≥
1
∀
p
∈ P
y
∈ {
0
,
1
}
A linear program relaxation is:
min
c
T
y
(LPMC)
s.t.
X
a
∈
p
y
a
≥
1
∀
p
∈ P
y
≥
0
Last time we proved the following theorem.
Theorem 1.2.
Assume
c
a
= 1 for all
a
∈
A
. Then there is an optimal solution
z
such that,
for some
U
⊆
V
, we have
z
a
=
(
1
(if
a
∈
δ
+
(
U
))
0
(otherwise).
That is,
z
is the characteristic vector of the set
δ
+
(
U
).
The previous theorem is actually true for any capacities
c
≥
0, and the same proof works.
Our proof used the dual of LPMC, complementary slackness, and Strong LP Duality. In this
lecture we will prove a theorem very similar to Theorem 1.2, using a very different approach.
We will again consider the dual of LPMC, which is:
1
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max
X
p
∈P
x
p
(LPPF)
s.t.
X
p
:
a
∈
p
x
p
≤
c
a
∀
a
∈
A
x
≥
0
We will prove the following theorem.
Theorem 1.3.
For any nonnegative, integral arc capacities
c
, there is an integral optimal
solution to LPPF.
2
Network Flows
We will think of the solutions to LPPF as specifying
pathflows
. For every directed
s

t
path,
it specifies a “flow value”
x
p
, such that every arc
a
has at most
c
a
units of flow passing through
it. These objects are studied in the area known as “network flows”, and the course C&O 351
focuses on this area.
However, LPMF is not the usual linear program to consider. It is more common to look at
what I’ll call
arcflows
, which are feasible solutions to the linear program LPAF, defined next.
For any
u
∈
V
, define
δ
+
(
u
) =
{
(
u, v
) : (
u, v
)
∈
A
}
δ

(
u
) =
{
(
v, u
) : (
v, u
)
∈
A
}
So
δ
+
(
u
) is the set of outgoing arcs from
u
, and
δ

(
u
) is the set of incoming arcs to
u
. The
arcflow linear program is:
max
X
a
∈
δ
+
(
s
)
z
a

X
a
∈
δ

(
s
)
z
a
(LPAF)
s.t.
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 Fall '09
 Harvey
 Optimization, objective value, LPMC

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