<s>
In	O
graph	O
theory	O
,	O
the	O
graph	B-Algorithm
bandwidth	I-Algorithm
problem	I-Algorithm
is	O
to	O
label	O
the	O
vertices	O
of	O
a	O
graph	O
with	O
distinct	O
integers	O
so	O
that	O
the	O
quantity	O
is	O
minimized	O
(	O
is	O
the	O
edge	O
set	O
of	O
)	O
.	O
</s>
<s>
The	O
weighted	O
graph	B-Algorithm
bandwidth	I-Algorithm
problem	I-Algorithm
is	O
a	O
generalization	B-Algorithm
wherein	O
the	O
edges	O
are	O
assigned	O
weights	O
and	O
the	O
cost	O
function	O
to	O
be	O
minimized	O
is	O
.	O
</s>
<s>
In	O
terms	O
of	O
matrices	O
,	O
the	O
(	O
unweighted	O
)	O
graph	B-Algorithm
bandwidth	I-Algorithm
is	O
the	O
minimal	O
bandwidth	B-Algorithm
of	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
which	O
is	O
an	O
adjacency	B-Algorithm
matrix	I-Algorithm
of	O
the	O
graph	O
.	O
</s>
<s>
The	O
bandwidth	B-Algorithm
may	O
also	O
be	O
defined	O
as	O
one	O
less	O
than	O
the	O
maximum	O
clique	O
size	O
in	O
a	O
proper	O
interval	O
supergraph	O
of	O
the	O
given	O
graph	O
,	O
chosen	O
to	O
minimize	O
its	O
clique	O
size	O
.	O
</s>
<s>
For	O
several	O
families	O
of	O
graphs	O
,	O
the	O
bandwidth	B-Algorithm
is	O
given	O
by	O
an	O
explicit	O
formula	O
.	O
</s>
<s>
The	O
bandwidth	B-Algorithm
of	O
a	O
path	O
graph	O
Pn	O
on	O
n	O
vertices	O
is1	O
,	O
and	O
for	O
a	O
complete	O
graph	O
Km	O
we	O
have	O
.	O
</s>
<s>
As	O
a	O
special	O
case	O
of	O
this	O
formula	O
,	O
the	O
star	O
graph	O
on	O
k+1	O
vertices	O
has	O
bandwidth	B-Algorithm
.	O
</s>
<s>
Chvatálová	O
showed	O
that	O
the	O
bandwidth	B-Algorithm
of	O
the	O
mn	O
square	O
grid	O
graph	O
,	O
that	O
is	O
,	O
the	O
Cartesian	O
product	O
of	O
two	O
path	O
graphs	O
on	O
and	O
vertices	O
,	O
is	O
equal	O
tomin{m,n}	O
.	O
</s>
<s>
The	O
bandwidth	B-Algorithm
of	O
a	O
graph	O
can	O
be	O
bounded	O
in	O
terms	O
of	O
various	O
other	O
graph	O
parameters	O
.	O
</s>
<s>
If	O
a	O
graph	O
G	O
has	O
bandwidth	B-Algorithm
k	O
,	O
then	O
its	O
pathwidth	O
is	O
at	O
most	O
k	O
,	O
and	O
its	O
tree-depth	O
is	O
at	O
most	O
klog( 	O
n/k	O
)	O
.	O
</s>
<s>
In	O
contrast	O
,	O
as	O
noted	O
in	O
the	O
previous	O
section	O
,	O
the	O
star	O
graph	O
Sk	O
,	O
a	O
structurally	O
very	O
simple	O
example	O
of	O
a	O
tree	O
,	O
has	O
comparatively	O
large	O
bandwidth	B-Algorithm
.	O
</s>
<s>
Both	O
the	O
unweighted	O
and	O
weighted	O
versions	O
are	O
special	O
cases	O
of	O
the	O
quadratic	B-Algorithm
bottleneck	I-Algorithm
assignment	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
bandwidth	B-Algorithm
problem	I-Algorithm
is	O
NP-hard	O
,	O
even	O
for	O
some	O
special	O
cases	O
.	O
</s>
<s>
approximation	B-Algorithm
algorithms	I-Algorithm
,	O
it	O
is	O
known	O
that	O
the	O
bandwidth	B-Algorithm
is	O
NP-hard	O
to	O
approximate	O
within	O
any	O
constant	O
,	O
and	O
this	O
even	O
holds	O
when	O
the	O
input	O
graphs	O
are	O
restricted	O
to	O
caterpillar	O
trees	O
with	O
maximum	O
hair	O
length	O
2	O
.	O
</s>
<s>
A	O
heuristic	B-Algorithm
algorithm	I-Algorithm
for	O
obtaining	O
linear	O
graph	O
layouts	O
of	O
low	O
bandwidth	B-Algorithm
is	O
the	O
Cuthill	B-Algorithm
–	I-Algorithm
McKee	I-Algorithm
algorithm	I-Algorithm
.	O
</s>
<s>
Fast	O
multilevel	O
algorithm	O
for	O
graph	B-Algorithm
bandwidth	I-Algorithm
computation	O
was	O
proposed	O
in	O
.	O
</s>
<s>
One	O
area	O
is	O
sparse	O
matrix/band	O
matrix	O
handling	O
,	O
and	O
general	O
algorithms	O
from	O
this	O
area	O
,	O
such	O
as	O
Cuthill	B-Algorithm
–	I-Algorithm
McKee	I-Algorithm
algorithm	I-Algorithm
,	O
may	O
be	O
applied	O
to	O
find	O
approximate	O
solutions	O
for	O
the	O
graph	B-Algorithm
bandwidth	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
In	O
this	O
context	O
,	O
graph	B-Algorithm
bandwidth	I-Algorithm
problem	I-Algorithm
models	O
the	O
problem	O
of	O
placement	O
of	O
a	O
set	O
of	O
standard	O
cells	O
in	O
a	O
single	O
row	O
with	O
the	O
goal	O
of	O
minimizing	O
the	O
maximal	O
propagation	O
delay	O
(	O
which	O
is	O
assumed	O
to	O
be	O
proportional	O
to	O
wire	O
length	O
)	O
.	O
</s>
