<s>
In	O
computational	O
complexity	O
theory	O
,	O
a	O
gap	B-Algorithm
reduction	I-Algorithm
is	O
a	O
reduction	B-Algorithm
to	O
a	O
particular	O
type	O
of	O
decision	O
problem	O
,	O
known	O
as	O
a	O
c-gap	O
problem	O
.	O
</s>
<s>
Such	O
reductions	O
provide	O
information	O
about	O
the	O
hardness	O
of	O
approximating	B-Algorithm
solutions	O
to	O
optimization	O
problems	O
.	O
</s>
<s>
Gap	B-Algorithm
reductions	I-Algorithm
can	O
be	O
used	O
to	O
demonstrate	O
inapproximability	O
results	O
,	O
as	O
if	O
a	O
problem	O
may	O
be	O
approximated	O
to	O
a	O
better	O
factor	O
than	O
the	O
size	O
of	O
gap	O
,	O
then	O
the	O
approximation	B-Algorithm
algorithm	I-Algorithm
can	O
be	O
used	O
to	O
solve	O
the	O
corresponding	O
gap	O
problem	O
.	O
</s>
<s>
Note	O
that	O
-approximating	O
the	O
solution	O
to	O
an	O
instance	O
of	O
is	O
at	O
least	O
as	O
hard	O
as	O
solving	O
the	O
-gap	O
version	O
of	O
.	O
</s>
<s>
A	O
gap-producing	O
reduction	B-Algorithm
is	O
a	O
reduction	B-Algorithm
from	O
an	O
optimization	O
problem	O
to	O
a	O
c-gap	O
problem	O
,	O
so	O
that	O
solving	O
the	O
c-gap	O
problem	O
quickly	O
would	O
enable	O
solving	O
the	O
optimization	O
problem	O
quickly	O
.	O
</s>
<s>
The	O
term	O
gap-producing	O
arises	O
from	O
the	O
nature	O
of	O
the	O
reduction	B-Algorithm
:	O
the	O
optimal	O
solution	O
in	O
the	O
optimization	O
problem	O
maps	O
to	O
the	O
opposite	O
side	O
of	O
the	O
gap	O
from	O
every	O
other	O
solution	O
via	O
reduction	B-Algorithm
.	O
</s>
<s>
A	O
simple	O
example	O
of	O
a	O
gap-producing	O
reduction	B-Algorithm
is	O
the	O
nonmetric	O
Traveling	B-Algorithm
Salesman	I-Algorithm
problem	I-Algorithm
(	O
i.e.	O
</s>
<s>
We	O
can	O
reduce	O
from	O
the	O
Hamiltonian	O
path	O
problem	O
on	O
a	O
given	O
graph	O
G	O
=	O
(	O
V	O
,	O
E	O
)	O
to	O
this	O
problem	O
as	O
follows	O
:	O
we	O
construct	O
a	O
complete	O
graph	O
G	O
 '	O
=	O
(	O
V	O
,	O
E	O
 '	O
)	O
,	O
for	O
the	O
traveling	B-Algorithm
salesman	I-Algorithm
problem	I-Algorithm
.	O
</s>
<s>
A	O
gap-preserving	O
reduction	B-Algorithm
is	O
a	O
reduction	B-Algorithm
from	O
a	O
c-gap	O
problem	O
to	O
a	O
c'-gap	O
problem	O
.	O
</s>
<s>
If	O
c	O
 '	O
>	O
c	O
,	O
then	O
this	O
is	O
a	O
gap-amplifying	O
reduction	B-Algorithm
.	O
</s>
<s>
This	O
problem	O
is	O
a	O
form	O
of	O
the	O
Boolean	B-Algorithm
satisfiability	I-Algorithm
problem	I-Algorithm
(	O
SAT	O
)	O
,	O
where	O
each	O
clause	O
contains	O
three	O
distinct	O
literals	O
and	O
we	O
want	O
to	O
maximize	O
the	O
number	O
of	O
clauses	O
satisfied	O
.	O
</s>
