<s>
In	O
applied	O
mathematics	O
,	O
Graver	B-Algorithm
bases	I-Algorithm
enable	O
iterative	O
solutions	O
of	O
linear	O
and	O
various	O
nonlinear	O
integer	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
in	O
polynomial	O
time	O
.	O
</s>
<s>
The	O
algorithmic	O
theory	O
of	O
Graver	B-Algorithm
bases	I-Algorithm
and	O
its	O
application	O
to	O
integer	B-Algorithm
programming	I-Algorithm
is	O
described	O
by	O
Shmuel	O
Onn	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
of	O
consists	O
of	O
the	O
vectors	O
(	O
2	O
,	O
1	O
,	O
0	O
)	O
,	O
(	O
0	O
,	O
1	O
,	O
2	O
)	O
,	O
(	O
1	O
,	O
0	O
,	O
1	O
)	O
,	O
(	O
1	O
,	O
1	O
,	O
1	O
)	O
and	O
their	O
negations	O
.	O
</s>
<s>
Integer	B-Algorithm
programming	I-Algorithm
is	O
the	O
problem	O
of	O
optimizing	O
a	O
linear	O
or	O
nonlinear	O
objective	O
function	O
over	O
the	O
set	O
of	O
integer	O
points	O
satisfying	O
a	O
system	O
of	O
linear	O
inequalities	O
.	O
</s>
<s>
However	O
,	O
given	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
of	O
,	O
the	O
problem	O
with	O
linear	O
and	O
various	O
nonlinear	O
objective	O
functions	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
as	O
explained	O
next	O
.	O
</s>
<s>
The	O
most	O
studied	O
case	O
,	O
treated	O
thoroughly	O
in	O
,	O
is	O
that	O
of	O
linear	B-Algorithm
integer	I-Algorithm
programming	I-Algorithm
,	O
</s>
<s>
However	O
,	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
of	O
has	O
the	O
following	O
property	O
.	O
</s>
<s>
Or	O
there	O
exists	O
a	O
vector	O
g	O
in	O
,	O
such	O
that	O
x+g	O
is	O
feasible	O
and	O
(	O
i.e.	O
,	O
x	O
can	O
be	O
improved	O
by	O
adding	O
to	O
it	O
an	O
element	O
of	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
)	O
.	O
</s>
<s>
Therefore	O
,	O
given	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
,	O
the	O
ILP	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
using	O
the	O
following	O
simple	O
iterative	O
algorithm	O
.	O
</s>
<s>
Note	O
that	O
this	O
matrix	O
is	O
generally	O
not	O
totally	B-Algorithm
unimodular	I-Algorithm
.	O
</s>
<s>
Nonetheless	O
,	O
it	O
was	O
shown	O
in	O
that	O
for	O
every	O
fixed	O
l	O
and	O
m	O
,	O
its	O
Graver	B-Algorithm
basis	I-Algorithm
can	O
be	O
computed	O
in	O
time	O
which	O
is	O
polynomial	O
inn	O
.	O
</s>
<s>
Consider	O
the	O
integer	O
multi-commodity	B-Algorithm
flow	I-Algorithm
problem	I-Algorithm
of	O
routing	O
k	O
types	O
of	O
integer	O
commodities	O
from	O
m	O
suppliers	O
to	O
n	O
consumers	O
subject	O
to	O
supply	O
,	O
consumption	O
,	O
and	O
capacity	O
constraints	O
,	O
so	O
as	O
to	O
minimize	O
the	O
sum	O
of	O
linear	O
or	O
congestion-dependent	O
convex	O
costs	O
on	O
the	O
arcs	O
.	O
</s>
<s>
The	O
many	O
applications	O
of	O
the	O
algorithmic	O
theory	O
of	O
Graver	B-Algorithm
bases	I-Algorithm
also	O
include	O
:	O
</s>
<s>
stochastic	O
integer	B-Algorithm
programming	I-Algorithm
,	O
</s>
<s>
N-fold	O
integer	B-Algorithm
programming	I-Algorithm
,	O
</s>
<s>
N-fold	O
4-block	O
decomposable	O
integer	B-Algorithm
programming	I-Algorithm
,	O
</s>
<s>
In	O
some	O
of	O
these	O
applications	O
the	O
relevant	O
Graver	B-Algorithm
basis	I-Algorithm
cannot	O
be	O
computed	O
in	O
polynomial	O
time	O
,	O
but	O
can	O
be	O
accessed	O
in	O
an	O
indirect	O
way	O
that	O
allows	O
to	O
use	O
it	O
in	O
polynomial	O
time	O
.	O
</s>
<s>
It	O
was	O
shown	O
in	O
that	O
every	O
(	O
bounded	O
)	O
integer	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
is	O
precisely	O
equivalent	O
to	O
the	O
3mn	O
table	O
problem	O
discussed	O
above	O
for	O
some	O
m	O
and	O
n	O
and	O
some	O
line	O
sums	O
.	O
</s>
<s>
Thus	O
,	O
such	O
3mn	O
table	O
problems	O
are	O
universal	O
for	O
integer	B-Algorithm
programming	I-Algorithm
.	O
</s>
<s>
Moreover	O
,	O
for	O
each	O
fixed	O
m	O
,	O
the	O
resulting	O
class	O
of	O
integer	B-Algorithm
programs	I-Algorithm
can	O
be	O
solved	O
in	O
polynomial	O
time	O
by	O
the	O
iterative	O
Graver	B-Algorithm
basis	I-Algorithm
method	O
described	O
above	O
.	O
</s>
<s>
So	O
the	O
table	O
width	O
m	O
provides	O
a	O
parametrization	O
of	O
all	O
integer	B-Algorithm
programming	I-Algorithm
problems	I-Algorithm
.	O
</s>
<s>
Denote	O
by	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
of	O
the	O
matrix	O
defining	O
the	O
universal	O
3mn	O
table	O
problem	O
discussed	O
above	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
there	O
is	O
a	O
finite	O
Graver	O
complexity	O
function	O
g(m )	O
such	O
that	O
for	O
any	O
n	O
,	O
the	O
type	O
of	O
any	O
element	O
of	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
is	O
at	O
most	O
g(m )	O
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
is	O
the	O
union	O
of	O
the	O
suitably	O
embedded	O
copies	O
of	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
.	O
</s>
<s>
To	O
approximately	O
solve	O
in	O
practice	O
an	O
arbitrary	O
integer	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
,	O
proceed	O
as	O
follows	O
.	O
</s>
<s>
At	O
level	O
k	O
of	O
this	O
hierarchy	O
,	O
let	O
be	O
the	O
subset	O
of	O
consisting	O
only	O
of	O
those	O
elements	O
of	O
type	O
at	O
most	O
k	O
;	O
use	O
this	O
approximation	O
in	O
the	O
iterative	O
algorithm	O
as	O
long	O
as	O
possible	O
to	O
obtain	O
as	O
good	O
as	O
possible	O
feasible	O
point	O
for	O
the	O
integer	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
embedded	O
in	O
the	O
3mn	O
table	O
problem	O
;	O
if	O
the	O
objective	O
function	O
value	O
of	O
this	O
point	O
is	O
satisfactory	O
(	O
say	O
,	O
as	O
compared	O
to	O
the	O
value	O
of	O
the	O
linear	B-Algorithm
programming	I-Algorithm
relaxation	I-Algorithm
)	O
,	O
then	O
use	O
this	O
point	O
;	O
otherwise	O
increment	O
k	O
and	O
proceed	O
to	O
the	O
next	O
level	O
of	O
the	O
hierarchy	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
that	O
for	O
any	O
fixed	O
level	O
k	O
,	O
the	O
approximation	O
of	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
has	O
polynomial	O
cardinality	O
and	O
can	O
be	O
computed	O
in	O
that	O
much	O
time	O
.	O
</s>
<s>
In	O
a	O
much	O
faster	O
algorithm	O
is	O
presented	O
,	O
which	O
finds	O
at	O
each	O
iteration	O
the	O
best	O
improvement	O
x+qg	O
with	O
q	O
positive	O
integer	O
and	O
g	O
element	O
in	O
without	O
explicitly	O
constructing	O
the	O
Graver	B-Algorithm
basis	I-Algorithm
,	O
in	O
cubic	O
time	O
regardless	O
of	O
the	O
system	O
.	O
</s>
<s>
In	O
the	O
terminology	O
of	O
parameterized	B-General_Concept
complexity	I-General_Concept
,	O
this	O
implies	O
that	O
all	O
these	O
problems	O
suitably	O
parameterized	B-General_Concept
,	O
and	O
in	O
particular	O
lmn	O
table	O
problems	O
parametrized	O
by	O
l	O
and	O
m	O
,	O
are	O
fixed	O
parameter	O
tractable	O
.	O
</s>
