<s>
In	O
mathematics	O
,	O
a	O
balanced	B-Algorithm
matrix	I-Algorithm
is	O
a	O
0-1	O
matrix	B-Architecture
(	O
a	O
matrix	B-Architecture
where	O
every	O
entry	O
is	O
either	O
zero	O
or	O
one	O
)	O
that	O
does	O
not	O
contain	O
any	O
square	B-Algorithm
submatrix	I-Algorithm
of	O
odd	O
order	O
having	O
all	O
row	O
sums	O
and	O
all	O
column	O
sums	O
equal	O
to	O
2	O
.	O
</s>
<s>
Balanced	O
matrices	O
are	O
studied	O
in	O
linear	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
The	O
importance	O
of	O
balanced	O
matrices	O
comes	O
from	O
the	O
fact	O
that	O
the	O
solution	O
to	O
a	O
linear	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
is	O
integral	O
if	O
its	O
matrix	B-Architecture
of	O
coefficients	O
is	O
balanced	O
and	O
its	O
right	O
hand	O
side	O
or	O
its	O
objective	O
vector	O
is	O
an	O
all-one	O
vector	O
.	O
</s>
<s>
In	O
particular	O
,	O
if	O
one	O
searches	O
for	O
an	O
integral	O
solution	O
to	O
a	O
linear	B-Algorithm
program	I-Algorithm
of	O
this	O
kind	O
,	O
it	O
is	O
not	O
necessary	O
to	O
explicitly	O
solve	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
,	O
but	O
it	O
suffices	O
to	O
find	O
an	O
optimal	O
vertex	O
solution	O
of	O
the	O
linear	B-Algorithm
program	I-Algorithm
itself	I-Algorithm
.	O
</s>
<s>
As	O
an	O
example	O
,	O
the	O
following	O
matrix	B-Architecture
is	O
a	O
balanced	B-Algorithm
matrix	I-Algorithm
:	O
</s>
<s>
Equivalent	O
to	O
the	O
definition	O
,	O
a	O
0-1	O
matrix	B-Architecture
is	O
balanced	O
if	O
and	O
only	O
if	O
it	O
does	O
not	O
contain	O
a	O
submatrix	O
that	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
any	O
odd	O
cycle	O
(	O
a	O
cycle	O
graph	O
of	O
odd	O
order	O
)	O
.	O
</s>
<s>
Therefore	O
,	O
the	O
only	O
three	O
by	O
three	O
0-1	O
matrix	B-Architecture
that	O
is	O
not	O
balanced	O
is	O
(	O
up	O
to	O
permutation	O
of	O
the	O
rows	O
and	O
columns	O
)	O
the	O
following	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
a	O
cycle	O
graph	O
of	O
order	O
3	O
:	O
</s>
<s>
The	O
following	O
matrix	B-Architecture
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
a	O
cycle	O
graph	O
of	O
order	O
5	O
:	O
</s>
<s>
The	O
above	O
characterization	O
implies	O
that	O
any	O
matrix	B-Architecture
containing	O
or	O
(	O
or	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
any	O
other	O
odd	O
cycle	O
)	O
as	O
a	O
submatrix	O
,	O
is	O
not	O
balanced	O
.	O
</s>
<s>
Every	O
balanced	B-Algorithm
matrix	I-Algorithm
is	O
a	O
perfect	B-Algorithm
matrix	I-Algorithm
.	O
</s>
<s>
A	O
0-1	O
matrix	B-Architecture
is	O
called	O
totally	O
balanced	O
if	O
it	O
does	O
not	O
contain	O
a	O
square	B-Algorithm
submatrix	I-Algorithm
having	O
no	O
repeated	O
columns	O
and	O
all	O
row	O
sums	O
and	O
all	O
column	O
sums	O
equal	O
to	O
2	O
.	O
</s>
<s>
Equivalently	O
,	O
a	O
matrix	B-Architecture
is	O
totally	O
balanced	O
if	O
and	O
only	O
if	O
it	O
does	O
not	O
contain	O
a	O
submatrix	O
that	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
any	O
cycle	O
(	O
no	O
matter	O
if	O
of	O
odd	O
or	O
even	O
order	O
)	O
.	O
</s>
<s>
This	O
characterization	O
immediately	O
implies	O
that	O
any	O
totally	O
balanced	B-Algorithm
matrix	I-Algorithm
is	O
balanced	O
.	O
</s>
<s>
Moreover	O
,	O
any	O
0-1	O
matrix	B-Architecture
that	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
is	O
also	O
balanced	O
.	O
</s>
<s>
The	O
following	O
matrix	B-Architecture
is	O
a	O
balanced	B-Algorithm
matrix	I-Algorithm
as	O
it	O
does	O
not	O
contain	O
any	O
submatrix	O
that	O
is	O
the	O
incidence	B-Algorithm
matrix	I-Algorithm
of	O
an	O
odd	O
cycle	O
:	O
</s>
<s>
Since	O
this	O
matrix	B-Architecture
is	O
not	O
totally	B-Algorithm
unimodular	I-Algorithm
(	O
its	O
determinant	O
is	O
-2	O
)	O
,	O
0-1	O
totally	B-Algorithm
unimodular	I-Algorithm
matrices	O
are	O
a	O
proper	O
subset	O
of	O
balanced	O
matrices	O
.	O
</s>
<s>
For	O
example	O
,	O
balanced	O
matrices	O
arise	O
as	O
the	O
coefficient	O
matrix	B-Architecture
in	O
special	O
cases	O
of	O
the	O
set	O
partitioning	O
problem	O
.	O
</s>
<s>
If	O
a	O
0-1	O
matrix	B-Architecture
A	O
has	O
SC(s )	O
≤1	O
for	O
all	O
rows	O
s=	O
1	O
,...,	O
m	O
,	O
then	O
A	O
has	O
a	O
unique	O
subsequence	O
,	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
and	O
therefore	O
also	O
balanced	O
.	O
</s>
<s>
As	O
a	O
more	O
general	O
notion	O
,	O
a	O
matrix	B-Architecture
where	O
every	O
entry	O
is	O
either	O
0	O
,	O
1	O
or	O
-1	O
is	O
called	O
balanced	O
if	O
in	O
every	O
submatrix	O
with	O
two	O
nonzero	O
entries	O
per	O
row	O
and	O
column	O
,	O
the	O
sum	O
of	O
the	O
entries	O
is	O
a	O
multiple	O
of	O
4	O
.	O
</s>
