<s>
A	O
fully	B-Algorithm
polynomial-time	I-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
FPTAS	B-Algorithm
)	O
is	O
an	O
algorithm	O
for	O
finding	O
approximate	O
solutions	O
to	O
function	O
problems	O
,	O
especially	O
optimization	O
problems	O
.	O
</s>
<s>
An	O
FPTAS	B-Algorithm
takes	O
as	O
input	O
an	O
instance	O
of	O
the	O
problem	O
and	O
a	O
parameter	O
ε>0	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
optimization	O
problems	O
,	O
the	O
correct	O
value	O
is	O
understood	O
to	O
be	O
the	O
value	O
of	O
the	O
optimal	O
solution	O
,	O
and	O
it	O
is	O
often	O
implied	O
that	O
an	O
FPTAS	B-Algorithm
should	O
produce	O
a	O
valid	O
solution	O
(	O
and	O
not	O
just	O
the	O
value	O
of	O
the	O
solution	O
)	O
.	O
</s>
<s>
Importantly	O
,	O
the	O
run-time	O
of	O
an	O
FPTAS	B-Algorithm
is	O
polynomial	O
in	O
the	O
problem	O
size	O
and	O
in	O
1/ε	O
.	O
</s>
<s>
This	O
is	O
in	O
contrast	O
to	O
a	O
general	O
polynomial-time	B-Algorithm
approximation	I-Algorithm
scheme	I-Algorithm
(	O
PTAS	O
)	O
.	O
</s>
<s>
The	O
term	O
FPTAS	B-Algorithm
may	O
also	O
be	O
used	O
to	O
refer	O
to	O
the	O
class	O
of	O
problems	O
that	O
have	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
FPTAS	B-Algorithm
is	O
a	O
subset	O
of	O
PTAS	O
,	O
and	O
unless	O
P	O
=	O
NP	O
,	O
it	O
is	O
a	O
strict	O
subset	O
.	O
</s>
<s>
All	O
problems	O
in	O
FPTAS	B-Algorithm
are	O
fixed-parameter	O
tractable	O
with	O
respect	O
to	O
the	O
standard	O
parameterization	O
.	O
</s>
<s>
Any	O
strongly	O
NP-hard	O
optimization	O
problem	O
with	O
a	O
polynomially	O
bounded	O
objective	O
function	O
cannot	O
have	O
an	O
FPTAS	B-Algorithm
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
if	O
P	O
does	O
not	O
equal	O
NP	O
,	O
knapsack	O
with	O
two	O
constraints	O
is	O
not	O
strongly	O
NP-hard	O
,	O
but	O
has	O
no	O
FPTAS	B-Algorithm
even	O
when	O
the	O
optimal	O
objective	O
is	O
polynomially	O
bounded	O
.	O
</s>
<s>
Woeginger	O
presented	O
a	O
general	O
scheme	O
for	O
converting	O
a	O
certain	O
class	O
of	O
dynamic	B-Algorithm
programs	I-Algorithm
to	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
If	O
V	O
is	O
in	O
O(X )	O
,	O
then	O
the	O
run-time	O
is	O
in	O
O(n Xb )	O
,	O
which	O
is	O
only	O
pseudo-polynomial	B-Algorithm
time	I-Algorithm
,	O
since	O
it	O
is	O
exponential	O
in	O
the	O
problem	O
size	O
which	O
is	O
in	O
O(log X )	O
.	O
</s>
<s>
Proximity	O
is	O
preserved	O
by	O
the	O
value	B-Algorithm
function	I-Algorithm
:	O
There	O
exists	O
an	O
integer	O
G	O
≥	O
0	O
(	O
which	O
is	O
a	O
function	O
of	O
the	O
value	B-Algorithm
function	I-Algorithm
g	O
and	O
the	O
degree	O
vector	O
d	O
)	O
,	O
such	O
that	O
for	O
any	O
r>1	O
,	O
and	O
for	O
any	O
two	O
state-vectors	O
s1	O
,	O
s2	O
,	O
the	O
following	O
holds	O
:	O
if	O
s1	O
is	O
(	O
d	O
,	O
r	O
)	O
-close	O
to	O
s2	O
,	O
then	O
:	O
g(s1 )	O
≤	O
rG	O
·	O
g(s2 )	O
(	O
in	O
minimization	O
problems	O
)	O
;	O
g(s1 )	O
≥	O
r(-G )	O
·	O
g(s2 )	O
(	O
in	O
maximization	O
problems	O
)	O
.	O
</s>
<s>
All	O
transition	O
functions	O
f	O
in	O
F	O
and	O
the	O
value	B-Algorithm
function	I-Algorithm
g	O
can	O
be	O
evaluated	O
in	O
polytime	O
.	O
</s>
<s>
For	O
every	O
extremely-benevolent	O
problem	O
,	O
the	O
dynamic	O
program	O
can	O
be	O
converted	O
into	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
The	O
FPTAS	B-Algorithm
runs	O
similarly	O
to	O
the	O
DP	O
,	O
but	O
in	O
each	O
step	O
,	O
it	O
trims	O
the	O
state	O
set	O
into	O
a	O
smaller	O
set	O
Tk	O
,	O
that	O
contains	O
exactly	O
one	O
state	O
in	O
each	O
r-box	O
.	O
</s>
<s>
The	O
algorithm	O
of	O
the	O
FPTAS	B-Algorithm
is	O
:	O
</s>
<s>
The	O
run-time	O
of	O
the	O
FPTAS	B-Algorithm
is	O
polynomial	O
in	O
the	O
total	O
number	O
of	O
possible	O
states	O
in	O
each	O
Ti	O
,	O
which	O
is	O
at	O
most	O
the	O
total	O
number	O
of	O
r-boxes	O
,	O
which	O
is	O
at	O
most	O
R	O
,	O
which	O
is	O
polynomial	O
in	O
n	O
,	O
log(X )	O
,	O
and	O
.	O
</s>
<s>
The	O
main	O
lemma	O
for	O
proving	O
the	O
correctness	O
of	O
the	O
FPTAS	B-Algorithm
is	O
:	O
For	O
every	O
step	O
k	O
in	O
0	O
,...,	O
n	O
,	O
for	O
every	O
state	O
ss	O
in	O
Sk	O
,	O
there	O
is	O
a	O
state	O
st	O
in	O
Tk	O
that	O
is	O
(	O
d	O
,	O
rk	O
)	O
-close	O
to	O
ss	O
.	O
</s>
<s>
Since	O
proximity	O
is	O
preserved	O
by	O
the	O
value	B-Algorithm
function	I-Algorithm
,	O
g( t*	O
)	O
≥	O
r(-Gn )	O
·	O
g( s*	O
)	O
for	O
a	O
maximization	O
problem	O
.	O
</s>
<s>
Here	O
are	O
some	O
examples	O
of	O
extremely-benevolent	O
problems	O
,	O
that	O
have	O
an	O
FPTAS	B-Algorithm
by	O
the	O
above	O
theorem	O
.	O
</s>
<s>
Multiway	B-Algorithm
number	I-Algorithm
partitioning	I-Algorithm
(	O
equivalently	O
,	O
Identical-machines	B-Algorithm
scheduling	I-Algorithm
)	O
with	O
the	O
goal	O
of	O
minimizing	O
the	O
largest	O
sum	O
is	O
extremely-benevolent	O
.	O
</s>
<s>
The	O
result	O
extends	O
to	O
Uniform-machines	B-Algorithm
scheduling	I-Algorithm
and	O
Unrelated-machines	B-Algorithm
scheduling	I-Algorithm
whenever	O
the	O
number	O
of	O
machines	O
is	O
fixed	O
(	O
this	O
is	O
required	O
because	O
R	O
-	O
the	O
number	O
of	O
r-boxes	O
-	O
is	O
exponential	O
in	O
b	O
)	O
.	O
</s>
<s>
The	O
same	O
DP	O
can	O
be	O
used	O
,	O
but	O
this	O
time	O
with	O
value	B-Algorithm
function	I-Algorithm
g(s )	O
=	O
(	O
s1-s2	O
)	O
2	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
problem	O
does	O
not	O
have	O
an	O
FPTAS	B-Algorithm
unless	O
P	O
=	O
NP	O
,	O
since	O
an	O
FPTAS	B-Algorithm
could	O
be	O
used	O
to	O
decide	O
in	O
polytime	O
whether	O
the	O
optimal	O
value	O
is	O
0	O
.	O
</s>
<s>
Simple	O
dynamic	B-Algorithm
programs	I-Algorithm
add	O
to	O
the	O
above	O
formulation	O
the	O
following	O
components	O
:	O
</s>
<s>
Proximity	O
is	O
preserved	O
by	O
the	O
value	B-Algorithm
function	I-Algorithm
:	O
There	O
exists	O
an	O
integer	O
G	O
≥	O
0	O
(	O
a	O
function	O
of	O
the	O
value	B-Algorithm
function	I-Algorithm
g	O
and	O
the	O
degree	O
vector	O
d	O
)	O
,	O
such	O
that	O
for	O
any	O
r>1	O
,	O
and	O
for	O
any	O
two	O
state-vectors	O
s1	O
,	O
s2	O
,	O
the	O
following	O
holds	O
:	O
</s>
<s>
For	O
every	O
benevolent	O
problem	O
,	O
the	O
dynamic	O
program	O
can	O
be	O
converted	O
into	O
an	O
FPTAS	B-Algorithm
similarly	O
to	O
the	O
one	O
above	O
,	O
with	O
two	O
changes	O
(	O
boldfaced	O
)	O
:	O
</s>
<s>
Here	O
are	O
some	O
examples	O
of	O
benevolent	O
problems	O
,	O
that	O
have	O
an	O
FPTAS	B-Algorithm
by	O
the	O
above	O
theorem	O
.	O
</s>
<s>
The	O
0-1	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
is	O
benevolent	O
.	O
</s>
<s>
The	O
value	B-Algorithm
function	I-Algorithm
g(s )	O
returns	O
s2	O
.	O
</s>
<s>
The	O
run-time	O
of	O
this	O
FPTAS	B-Algorithm
can	O
be	O
improved	O
to	O
operations	O
on	O
integers	O
.	O
</s>
<s>
Note	O
:	O
consider	O
In	O
the	O
2-weighted	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
where	O
each	O
item	O
has	O
two	O
weights	O
and	O
a	O
value	O
,	O
and	O
the	O
goal	O
is	O
to	O
maximize	O
the	O
value	O
such	O
that	O
the	O
sum	O
of	O
squares	O
of	O
the	O
total	O
weights	O
is	O
at	O
most	O
the	O
knapsack	O
capacity	O
:	O
.	O
</s>
<s>
Indeed	O
,	O
this	O
problem	O
does	O
not	O
have	O
an	O
FPTAS	B-Algorithm
unless	O
P	O
=	O
NP	O
.	O
</s>
<s>
The	O
same	O
is	O
true	O
for	O
the	O
two-dimensional	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
The	O
same	O
is	O
true	O
for	O
the	O
multiple	B-Algorithm
subset	I-Algorithm
sum	I-Algorithm
problem	I-Algorithm
:	O
the	O
quasi-dominance	O
relation	O
should	O
be	O
:	O
s	O
quasi-dominates	O
t	O
iff	O
max(s1,s2 )	O
≤	O
max(t1,t2 )	O
,	O
but	O
it	O
is	O
not	O
preserved	O
by	O
transitions	O
,	O
by	O
the	O
same	O
example	O
as	O
above	O
.	O
</s>
<s>
In	O
the	O
total	O
tardiness	O
problem	O
1||	O
,	O
the	O
dynamic	B-Algorithm
programming	I-Algorithm
formulation	O
of	O
Lawler	O
requires	O
to	O
update	O
all	O
states	O
in	O
the	O
old	O
state	O
space	O
some	O
B	O
times	O
,	O
where	O
B	O
is	O
of	O
the	O
order	O
of	O
X	O
(	O
the	O
maximum	O
input	O
size	O
)	O
.	O
</s>
<s>
The	O
state-trimming	O
technique	O
is	O
not	O
useful	O
,	O
but	O
another	O
technique	O
-	O
input-rounding	O
-	O
has	O
been	O
used	O
to	O
design	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
But	O
different	O
techniques	O
have	O
been	O
used	O
to	O
design	O
an	O
FPTAS	B-Algorithm
.	O
</s>
<s>
The	O
knapsack	B-Algorithm
problem	I-Algorithm
,	O
as	O
well	O
as	O
some	O
of	O
its	O
variants	O
:	O
</s>
<s>
0-1	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Unbounded	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Multi-dimensional	O
knapsack	B-Algorithm
problem	I-Algorithm
with	O
Delta-modular	O
constraints	O
.	O
</s>
<s>
Multi-objective	O
0-1	B-Algorithm
knapsack	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Parametric	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Symmetric	O
quadratic	O
knapsack	B-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
Count-subset-sum	O
(	O
#SubsetSum	O
)	O
-	O
finding	O
the	O
number	O
of	O
distinct	O
subsets	O
with	O
a	O
sum	O
of	O
at	O
most	O
C	O
.	O
</s>
