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Chapter 11
Min Cut
By
Sariel HarPeled
, October 1, 2007
±
To acknowledge the corn
 This purely American expression means to admit the losing of an argu
ment, especially in regard to a detail; to retract; to admit defeat. It is over a hundred years old. Andrew
Stewart, a member of Congress, is said to have mentioned it in a speech in 1828. He said that haystacks
and cornﬁelds were sent by Indiana, Ohio and Kentucky to Philadelphia and New York. Charles A.
Wickli
ﬀ
e, a member from Kentucky questioned the statement by commenting that haystacks and corn
ﬁelds could not walk. Stewart then pointed out that he did not mean literal haystacks and cornﬁelds, but
the horses, mules, and hogs for which the hay and corn were raised. Wickli
ﬀ
e then rose to his feet, and
said, "Mr. Speaker, I acknowledge the corn".
– Funk, Earle, A Hog on Ice and Other Curious Expressions.
11.1
Min Cut
11.1.1
Problem Deﬁnition
Let
G
=
(
V
,
E
) be undirected graph with
n
vertices and
m
edges. We are interested in
cuts
in
G
.
Deﬁnition 11.1.1
A
cut
in
G
is a partition of the vertices of
V
into two sets
S
and
V
\
S
, where
the edges of the cut are
V
\
S
S
(
S
,
V
\
S
)
=
±
uv
²
²
²
²
u
∈
S
,
v
∈
V
\
S
,
and
uv
∈
E
³
,
where
S
,
∅
and
V
\
S
,
∅
. We will refer to the number of edges in the
cut (
S
,
V
\
S
) as the
size of the cut
. For an example of a cut, see ﬁgure on
the right.
We are interested in the problem of computing the
minimum cut
(i.e.,
mincut
), that is, the cut
in the graph with minimum cardinality. Speciﬁcally, we would like to ﬁnd the set
S
⊆
V
such that
(
S
,
V
\
S
) is as small as possible, and
S
is neither empty nor
V
\
S
is empty.
±
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View Full Document 11.1.2
Some Deﬁnitions
We remind the reader of the following concepts. The
conditional probability
of
X
given
Y
is
Pr
±
X
=
x

Y
=
y
²
=
Pr
±
(
X
=
x
)
∩
(
Y
=
y
)
²
/
Pr
±
Y
=
y
²
. An equivalent, useful restatement of this is
that
Pr
±
(
X
=
x
)
∩
(
Y
=
y
)
²
=
Pr
³
X
=
x
´
´
´
´
Y
=
y
µ
·
Pr
±
Y
=
y
²
.
(11.1)
Two events
X
and
Y
are
independent
, if
Pr
±
X
=
x
∩
Y
=
y
²
=
Pr
[
X
=
x
]
·
Pr
±
Y
=
y
²
. In particular,
if
X
and
Y
are independent, then
Pr
³
X
=
x
´
´
´
´
Y
=
y
µ
=
Pr
[
X
=
x
].
The following is easy to prove by induction using Eq. (11.1).
Lemma 11.1.2
Let
E
1
, . . . ,
E
n
be n events which are not necessarily independent. Then,
Pr
±
∩
n
i
=
1
E
i
²
=
Pr
[
E
1
]
*
Pr
[
E
2
E
1
]
*
Pr
[
E
3
E
1
∩ E
2
]
*
. . .
*
Pr
³
E
n
´
´
´
´
E
1
∩
. . .
∩ E
n

1
µ
.
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This note was uploaded on 06/14/2009 for the course CS 473 taught by Professor Viswanathan during the Fall '08 term at University of Illinois at Urbana–Champaign.
 Fall '08
 Viswanathan
 Algorithms

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