<s>
In	O
the	O
bin	B-Algorithm
covering	I-Algorithm
problem	I-Algorithm
,	O
items	O
of	O
different	O
sizes	O
must	O
be	O
packed	O
into	O
a	O
finite	O
number	O
of	O
bins	O
or	O
containers	O
,	O
each	O
of	O
which	O
must	O
contain	O
at	O
least	O
a	O
certain	O
given	O
total	O
size	O
,	O
in	O
a	O
way	O
that	O
maximizes	O
the	O
number	O
of	O
bins	O
used	O
.	O
</s>
<s>
This	O
problem	O
is	O
a	O
dual	O
of	O
the	O
bin	O
packing	O
problem	O
:	O
in	O
bin	B-Algorithm
covering	I-Algorithm
,	O
the	O
bin	O
sizes	O
are	O
bounded	O
from	O
below	O
and	O
the	O
goal	O
is	O
to	O
maximize	O
their	O
number	O
;	O
in	O
bin	O
packing	O
,	O
the	O
bin	O
sizes	O
are	O
bounded	O
from	O
above	O
and	O
the	O
goal	O
is	O
to	O
minimize	O
their	O
number	O
.	O
</s>
<s>
The	O
problem	O
is	O
NP-hard	O
,	O
but	O
there	O
are	O
various	O
efficient	O
approximation	B-Algorithm
algorithms	I-Algorithm
:	O
</s>
<s>
An	O
asymptotic	O
PTAS	B-Algorithm
,	O
algorithms	O
with	O
bounded	O
worst-case	O
behavior	O
whose	O
expected	O
behavior	O
is	O
asymptotically-optimal	O
for	O
some	O
discrete	O
distributions	O
,	O
and	O
a	O
learning	O
algorithm	O
with	O
asymptotically	O
optimal	O
expected	O
behavior	O
for	O
all	O
discrete	O
distributions	O
.	O
</s>
<s>
An	O
asymptotic	O
FPTAS	B-Algorithm
.	O
</s>
<s>
Csirik	O
,	O
Johnson	O
and	O
Kenyon	O
present	O
an	O
asymptotic	O
PTAS	B-Algorithm
.	O
</s>
<s>
The	O
algorithm	O
solves	O
a	O
variant	O
of	O
the	O
configuration	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
,	O
with	O
variables	O
and	O
constraints	O
.	O
</s>
<s>
Jansen	O
and	O
Solis-Oba	O
present	O
an	O
asymptotic	O
FPTAS	B-Algorithm
.	O
</s>
<s>
Many	O
techniques	O
from	O
bin	B-Algorithm
covering	I-Algorithm
are	O
used	O
in	O
this	O
problem	O
too	O
.	O
</s>
