<s>
Balanced	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
is	O
a	O
variant	O
of	O
multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
in	O
which	O
there	O
are	O
constraints	O
on	O
the	O
number	O
of	O
items	O
allocated	O
to	O
each	O
set	O
.	O
</s>
<s>
An	O
example	O
application	O
is	O
identical-machines	B-Algorithm
scheduling	I-Algorithm
where	O
each	O
machine	O
has	O
a	O
job-queue	O
that	O
can	O
hold	O
at	O
most	O
k	O
jobs	O
.	O
</s>
<s>
In	O
the	O
standard	O
three-field	B-Algorithm
notation	I-Algorithm
for	I-Algorithm
optimal	I-Algorithm
job	I-Algorithm
scheduling	I-Algorithm
problems	I-Algorithm
,	O
the	O
problem	O
of	O
minimizing	O
the	O
largest	O
sum	O
is	O
sometimes	O
denoted	O
by	O
"	O
P|#	O
≤	O
k|Cmax	O
"	O
.	O
</s>
<s>
It	O
is	O
a	O
variant	O
of	O
the	O
partition	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Coffman	O
,	O
Frederickson	O
and	O
Lueker	O
present	O
a	O
restricted	O
version	O
of	O
the	O
LPT	B-Algorithm
algorithm	I-Algorithm
(	O
called	O
RLPT	O
)	O
,	O
in	O
which	O
inputs	O
are	O
assigned	O
in	O
pairs	O
.	O
</s>
<s>
Lueker	O
presents	O
a	O
variant	O
of	O
the	O
LDM	B-Algorithm
algorithm	I-Algorithm
(	O
called	O
the	O
pairwise	O
differencing	O
method	O
(	O
PDM	O
)	O
)	O
.	O
</s>
<s>
Yakir	O
presents	O
a	O
balanced	O
variant	O
of	O
the	O
LDM	B-Algorithm
algorithm	I-Algorithm
for	O
m	O
=	O
2	O
,	O
called	O
BLDM	O
.	O
</s>
<s>
Mertens	O
presents	O
a	O
complete	O
anytime	B-Algorithm
algorithm	I-Algorithm
for	O
balanced	O
two-way	O
partitioning	O
.	O
</s>
<s>
Deciding	O
whether	O
there	O
exists	O
such	O
a	O
partition	O
with	O
equal	O
sums	O
is	O
exactly	O
the	O
3-partition	B-Algorithm
problem	I-Algorithm
,	O
which	O
is	O
known	O
to	O
be	O
strongly	O
NP-hard	O
.	O
</s>
<s>
Kellerer	O
and	O
Woeginger	O
adapt	O
the	O
LPT	B-Algorithm
algorithm	I-Algorithm
to	O
triplet	O
partitioning	O
(	O
where	O
there	O
are	O
at	O
most	O
3*m	O
items	O
,	O
and	O
each	O
subset	O
should	O
contain	O
at	O
most	O
3	O
items	O
)	O
.	O
</s>
<s>
Primal-dual	O
algorithm	O
(	O
a	O
combination	O
of	O
LPT	O
and	O
MultiFit	B-Algorithm
)	O
:	O
approximation	O
ratio	O
at	O
most	O
.	O
</s>
<s>
The	O
approximation	O
ratio	O
of	O
the	O
modified	O
list	B-Algorithm
scheduling	I-Algorithm
is	O
1/2	O
for	O
the	O
unconstrained	O
variant	O
,	O
but	O
it	O
is	O
0	O
for	O
the	O
constrained	O
variant	O
(	O
it	O
can	O
be	O
arbitrarily	O
bad	O
)	O
.	O
</s>
<s>
The	O
approximation	O
ratio	O
of	O
the	O
modified	O
LPT	B-Algorithm
algorithm	I-Algorithm
is	O
at	O
most	O
2	O
.	O
</s>
<s>
Both	O
these	O
algorithms	O
are	O
ordinal	B-General_Concept
–	O
they	O
partition	O
the	O
items	O
based	O
only	O
on	O
the	O
order	O
between	O
them	O
rather	O
than	O
their	O
exact	O
values	O
.	O
</s>
<s>
They	O
prove	O
that	O
any	O
ordinal	B-General_Concept
algorithm	O
has	O
ratio	O
at	O
most	O
for	O
maximizing	O
the	O
smallest	O
sum	O
.	O
</s>
<s>
For	O
any	O
fixed	O
k	O
,	O
any	O
ordinal	B-General_Concept
algorithm	O
has	O
ratio	O
at	O
most	O
the	O
smallest	O
root	O
of	O
the	O
equation	O
.	O
</s>
<s>
There	O
are	O
some	O
general	O
relations	O
between	O
approximations	O
to	O
the	O
balanced	B-Algorithm
partition	I-Algorithm
problem	I-Algorithm
and	O
the	O
standard	O
(	O
unconstrained	O
)	O
partition	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Chen	O
,	O
He	O
and	O
Yao	O
prove	O
that	O
the	O
problem	O
is	O
NP-hard	O
even	O
for	O
k	O
=3	O
(	O
for	O
k	O
=	O
2	O
it	O
can	O
be	O
solved	O
efficiently	O
by	O
finding	O
a	O
maximum	B-Algorithm
weight	I-Algorithm
matching	I-Algorithm
)	O
.	O
</s>
<s>
They	O
then	O
present	O
an	O
algorithm	O
called	O
Kernel-LPT	O
(	O
KLPT	O
)	O
:	O
it	O
assigns	O
a	O
kernel	O
to	O
each	O
subset	O
,	O
and	O
then	O
runs	O
the	O
modified	O
LPT	B-Algorithm
algorithm	I-Algorithm
(	O
puts	O
each	O
item	O
into	O
the	O
subset	O
with	O
the	O
smallest	O
sum	O
among	O
those	O
that	O
have	O
fewer	O
than	O
k	O
items	O
)	O
.	O
</s>
<s>
Wu	O
and	O
Yao	O
presented	O
the	O
layered	O
LPT	B-Algorithm
algorithm	I-Algorithm
–	O
a	O
variant	O
of	O
the	O
LPT	B-Algorithm
algorithm	I-Algorithm
.	O
</s>
