MATH239
Assignment 10
Due: July 22 (by Noon)
1. (10 points)
Consider the following bipartite graph
G
with bipartition (
A,B
) where
A
=
{
1
,...,
9
}
and
B
=
{
a,b,.
..,i
}
.
i
2
3
4
5
6
7
8
9
a
b
c
d
e
f
g
h
1
i. Let
M
=
{
3
a,
4
b,
6
c,
7
d,
9
h,
8
i
}
. Using the bipartite matching algorithm, determine
the sets
X
0
,
X
, and
Y
from the XY construction. Find either an
M
augmenting path
or cover
C
with

C

=

M

.
ii. Let
M
=
{
1
a,
2
c,
5
d,
4
b,
6
f,
8
i,
9
h,
7
e
}
. Using the bipartite matching algorithm, deter
mine the sets
X
0
,
X
, and
Y
from the XY construction. Find either an
M
augmenting
path or cover
C
with

C

=

M

.
2. (10 points)
An
n
×
n
matrix
P
, with entries in
{
0
,
1
}
, is a
permutation matrix
if each row
and each column contains exactly one 1. Hence the identity matrix is a permutation matrix
and every permutation matrix can be obtained from the identity matrix by permuting its
rows and columns.
i. Write the following matrix as the sum of permutation matrices.