<s>
An	O
integer	B-Algorithm
programming	I-Algorithm
problem	I-Algorithm
is	O
a	O
mathematical	O
optimization	O
or	O
feasibility	B-Application
program	O
in	O
which	O
some	O
or	O
all	O
of	O
the	O
variables	O
are	O
restricted	O
to	O
be	O
integers	O
.	O
</s>
<s>
In	O
many	O
settings	O
the	O
term	O
refers	O
to	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
(	O
ILP	O
)	O
,	O
in	O
which	O
the	O
objective	O
function	O
and	O
the	O
constraints	O
(	O
other	O
than	O
the	O
integer	B-Algorithm
constraints	I-Algorithm
)	O
are	O
linear	O
.	O
</s>
<s>
Integer	B-Algorithm
programming	I-Algorithm
is	O
NP-complete	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
special	O
case	O
of	O
0-1	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
,	O
in	O
which	O
unknowns	O
are	O
binary	O
,	O
and	O
only	O
the	O
restrictions	O
must	O
be	O
satisfied	O
,	O
is	O
one	O
of	O
Karp	O
's	O
21	O
NP-complete	O
problems	O
.	O
</s>
<s>
If	O
some	O
decision	O
variables	O
are	O
not	O
discrete	O
,	O
the	O
problem	O
is	O
known	O
as	O
a	O
mixed-integer	B-Algorithm
programming	I-Algorithm
problem	O
.	O
</s>
<s>
In	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
,	O
the	O
canonical	O
form	O
is	O
distinct	O
from	O
the	O
standard	O
form	O
.	O
</s>
<s>
An	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
in	O
canonical	O
form	O
is	O
expressed	O
thus	O
(	O
note	O
that	O
it	O
is	O
the	O
vector	O
which	O
is	O
to	O
be	O
decided	O
)	O
:	O
</s>
<s>
The	O
blue	O
lines	O
together	O
with	O
the	O
coordinate	O
axes	O
define	O
the	O
polyhedron	O
of	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
,	O
which	O
is	O
given	O
by	O
the	O
inequalities	O
without	O
the	O
integrality	O
constraint	O
.	O
</s>
<s>
The	O
following	O
is	O
a	O
reduction	O
from	O
minimum	O
vertex	O
cover	O
to	O
integer	B-Algorithm
programming	I-Algorithm
that	O
will	O
serve	O
as	O
the	O
proof	O
of	O
NP-hardness	O
.	O
</s>
<s>
Define	O
a	O
linear	B-Algorithm
program	I-Algorithm
as	O
follows	O
:	O
</s>
<s>
Given	O
that	O
the	O
constraints	O
limit	O
to	O
either	O
0	O
or	O
1	O
,	O
any	O
feasible	O
solution	O
to	O
the	O
integer	B-Algorithm
program	I-Algorithm
is	O
a	O
subset	O
of	O
vertices	O
.	O
</s>
<s>
Additionally	O
given	O
some	O
vertex	O
cover	O
C	O
,	O
can	O
be	O
set	O
to	O
1	O
for	O
any	O
and	O
to	O
0	O
for	O
any	O
thus	O
giving	O
us	O
a	O
feasible	O
solution	O
to	O
the	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Zero-one	O
linear	B-Algorithm
programming	I-Algorithm
(	O
or	O
binary	O
integer	B-Algorithm
programming	I-Algorithm
)	O
involves	O
problems	O
in	O
which	O
the	O
variables	O
are	O
restricted	O
to	O
be	O
either	O
0	O
or	O
1	O
.	O
</s>
<s>
There	O
are	O
two	O
main	O
reasons	O
for	O
using	O
integer	O
variables	O
when	O
modeling	O
problems	O
as	O
a	O
linear	B-Algorithm
program	I-Algorithm
:	O
</s>
<s>
These	O
considerations	O
occur	O
frequently	O
in	O
practice	O
and	O
so	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
can	O
be	O
used	O
in	O
many	O
applications	O
areas	O
,	O
some	O
of	O
which	O
are	O
briefly	O
described	O
below	O
.	O
</s>
<s>
Mixed-integer	B-Algorithm
programming	I-Algorithm
has	O
many	O
applications	O
in	O
industrial	O
productions	O
,	O
including	O
job-shop	O
modelling	O
.	O
</s>
<s>
In	O
some	O
cases	O
,	O
this	O
can	O
be	O
expressed	O
in	O
terms	O
of	O
a	O
linear	B-Algorithm
program	I-Algorithm
,	O
but	O
the	O
variables	O
must	O
be	O
constrained	O
to	O
be	O
integer	O
.	O
</s>
<s>
It	O
is	O
used	O
in	O
a	O
special	O
case	O
of	O
integer	B-Algorithm
programming	I-Algorithm
,	O
in	O
which	O
all	O
the	O
decision	O
variables	O
are	O
integers	O
.	O
</s>
<s>
This	O
problem	O
can	O
be	O
formulated	O
as	O
an	O
integer	B-Algorithm
linear	I-Algorithm
program	I-Algorithm
in	O
which	O
binary	O
variables	O
indicate	O
whether	O
a	O
frequency	O
is	O
assigned	O
to	O
an	O
antenna	O
.	O
</s>
<s>
The	O
naive	O
way	O
to	O
solve	O
an	O
ILP	O
is	O
to	O
simply	O
remove	O
the	O
constraint	O
that	O
x	O
is	O
integer	O
,	O
solve	O
the	O
corresponding	O
LP	O
(	O
called	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
of	O
the	O
ILP	O
)	O
,	O
and	O
then	O
round	O
the	O
entries	O
of	O
the	O
solution	O
to	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
.	O
</s>
<s>
While	O
in	O
general	O
the	O
solution	O
to	O
LP	B-Algorithm
relaxation	I-Algorithm
will	O
not	O
be	O
guaranteed	O
to	O
be	O
integral	O
,	O
if	O
the	O
ILP	O
has	O
the	O
form	O
such	O
that	O
where	O
and	O
have	O
all	O
integer	O
entries	O
and	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
,	O
then	O
every	O
basic	O
feasible	O
solution	O
is	O
integral	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
solution	O
returned	O
by	O
the	O
simplex	B-Algorithm
algorithm	I-Algorithm
is	O
guaranteed	O
to	O
be	O
integral	O
.	O
</s>
<s>
and	O
therefore	O
by	O
assumption	O
,	O
is	O
unimodular	B-Algorithm
and	O
so	O
.	O
</s>
<s>
Thus	O
,	O
if	O
the	O
matrix	O
of	O
an	O
ILP	O
is	O
totally	B-Algorithm
unimodular	I-Algorithm
,	O
rather	O
than	O
use	O
an	O
ILP	O
algorithm	O
,	O
the	O
simplex	B-Algorithm
method	I-Algorithm
can	O
be	O
used	O
to	O
solve	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
and	O
the	O
solution	O
will	O
be	O
integer	O
.	O
</s>
<s>
When	O
the	O
matrix	O
is	O
not	O
totally	B-Algorithm
unimodular	I-Algorithm
,	O
there	O
are	O
a	O
variety	O
of	O
algorithms	O
that	O
can	O
be	O
used	O
to	O
solve	O
integer	B-Algorithm
linear	I-Algorithm
programs	I-Algorithm
exactly	O
.	O
</s>
<s>
One	O
class	O
of	O
algorithms	O
are	O
cutting	B-Algorithm
plane	I-Algorithm
methods	I-Algorithm
which	O
work	O
by	O
solving	O
the	O
LP	B-Algorithm
relaxation	I-Algorithm
and	O
then	O
adding	O
linear	O
constraints	O
that	O
drive	O
the	O
solution	O
towards	O
being	O
integer	O
without	O
excluding	O
any	O
integer	O
feasible	O
points	O
.	O
</s>
<s>
Another	O
class	O
of	O
algorithms	O
are	O
variants	O
of	O
the	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
method	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
branch	B-Algorithm
and	I-Algorithm
cut	I-Algorithm
method	O
that	O
combines	O
both	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
and	O
cutting	B-Algorithm
plane	I-Algorithm
methods	I-Algorithm
.	O
</s>
<s>
Branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
algorithms	I-Algorithm
have	O
a	O
number	O
of	O
advantages	O
over	O
algorithms	O
that	O
only	O
use	O
cutting	B-Algorithm
planes	I-Algorithm
.	O
</s>
<s>
Further	O
,	O
the	O
solutions	O
of	O
the	O
LP	B-Algorithm
relaxations	I-Algorithm
can	O
be	O
used	O
to	O
provide	O
a	O
worst-case	O
estimate	O
of	O
how	O
far	O
from	O
optimality	O
the	O
returned	O
solution	O
is	O
.	O
</s>
<s>
Finally	O
,	O
branch	B-Algorithm
and	I-Algorithm
bound	I-Algorithm
methods	O
can	O
be	O
used	O
to	O
return	O
multiple	O
optimal	O
solutions	O
.	O
</s>
<s>
We	O
focus	O
on	O
the	O
feasibility	B-Application
problem	O
,	O
which	O
is	O
to	O
decide	O
whether	O
there	O
exists	O
an	O
n-by-1	O
vector	O
satisfying	O
.	O
</s>
<s>
If	O
n	O
(	O
the	O
number	O
of	O
variables	O
)	O
is	O
a	O
fixed	O
constant	O
,	O
then	O
the	O
feasibility	B-Application
problem	O
can	O
be	O
solved	O
in	O
time	O
polynomial	O
in	O
m	O
and	O
log	O
V	O
.	O
This	O
is	O
trivial	O
for	O
the	O
case	O
n	O
=	O
1	O
.	O
</s>
<s>
In	O
the	O
special	O
case	O
of	O
0-1	O
ILP	O
,	O
Lenstra	O
's	O
algorithm	O
is	O
equivalent	O
to	O
complete	O
enumeration	O
:	O
the	O
number	O
of	O
all	O
possible	O
solutions	O
is	O
fixed	O
(	O
2n	O
)	O
,	O
and	O
checking	O
the	O
feasibility	B-Application
of	O
each	O
solution	O
can	O
be	O
done	O
in	O
time	O
poly(m, log V )	O
.	O
</s>
<s>
Since	O
integer	B-Algorithm
linear	I-Algorithm
programming	I-Algorithm
is	O
NP-hard	O
,	O
many	O
problem	O
instances	O
are	O
intractable	O
and	O
so	O
heuristic	O
methods	O
must	O
be	O
used	O
instead	O
.	O
</s>
<s>
For	O
example	O
,	O
tabu	B-Algorithm
search	I-Algorithm
can	O
be	O
used	O
to	O
search	O
for	O
solutions	O
to	O
ILPs	O
.	O
</s>
<s>
To	O
use	O
tabu	B-Algorithm
search	I-Algorithm
to	O
solve	O
ILPs	O
,	O
moves	O
can	O
be	O
defined	O
as	O
incrementing	O
or	O
decrementing	O
an	O
integer	O
constrained	O
variable	O
of	O
a	O
feasible	O
solution	O
while	O
keeping	O
all	O
other	O
integer-constrained	O
variables	O
constant	O
.	O
</s>
<s>
It	O
is	O
often	O
the	O
case	O
that	O
the	O
matrix	O
which	O
defines	O
the	O
integer	B-Algorithm
program	I-Algorithm
is	O
sparse	O
.	O
</s>
<s>
Let	O
be	O
the	O
number	O
of	O
variables	O
of	O
the	O
integer	B-Algorithm
program	I-Algorithm
.	O
</s>
<s>
Then	O
it	O
was	O
shown	O
in	O
2018	O
that	O
integer	B-Algorithm
programming	I-Algorithm
can	O
be	O
solved	O
in	O
strongly	O
polynomial	O
and	O
fixed-parameter	O
tractable	O
time	O
parameterized	O
by	O
and	O
.	O
</s>
<s>
That	O
is	O
,	O
for	O
some	O
computable	O
function	O
and	O
some	O
constant	O
,	O
integer	B-Algorithm
programming	I-Algorithm
can	O
be	O
solved	O
in	O
time	O
.	O
</s>
