<s>
In	O
computational	O
complexity	O
theory	O
and	O
game	O
complexity	O
,	O
a	O
parsimonious	B-Algorithm
reduction	I-Algorithm
is	O
a	O
transformation	O
from	O
one	O
problem	O
to	O
another	O
(	O
a	O
reduction	B-Algorithm
)	O
that	O
preserves	O
the	O
number	O
of	O
solutions	O
.	O
</s>
<s>
Informally	O
,	O
it	O
is	O
a	O
bijection	B-Algorithm
between	O
the	O
respective	O
sets	O
of	O
solutions	O
of	O
two	O
problems	O
.	O
</s>
<s>
A	O
general	O
reduction	B-Algorithm
from	O
problem	O
to	O
problem	O
is	O
a	O
transformation	O
that	O
guarantees	O
that	O
whenever	O
has	O
a	O
solution	O
also	O
has	O
at	O
least	O
one	O
solution	O
and	O
vice	O
versa	O
.	O
</s>
<s>
A	O
parsimonious	B-Algorithm
reduction	I-Algorithm
guarantees	O
that	O
for	O
every	O
solution	O
of	O
,	O
there	O
exists	O
a	O
unique	O
solution	O
of	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Parsimonious	B-Algorithm
reductions	I-Algorithm
are	O
commonly	O
used	O
in	O
computational	O
complexity	O
for	O
proving	O
the	O
hardness	O
of	O
counting	O
problems	O
,	O
for	O
counting	O
complexity	O
classes	O
such	O
as	O
#P	O
.	O
Additionally	O
,	O
they	O
are	O
used	O
in	O
game	O
complexity	O
,	O
as	O
a	O
way	O
to	O
design	O
hard	O
puzzles	O
that	O
have	O
a	O
unique	O
solution	O
,	O
as	O
many	O
types	O
of	O
puzzles	O
require	O
.	O
</s>
<s>
A	O
Parsimonious	B-Algorithm
reduction	I-Algorithm
from	O
problem	O
to	O
problem	O
is	O
a	O
reduction	B-Algorithm
such	O
that	O
the	O
number	O
of	O
solutions	O
to	O
is	O
equal	O
to	O
the	O
number	O
of	O
solutions	O
to	O
problem	O
.	O
</s>
<s>
If	O
such	O
a	O
reduction	B-Algorithm
exists	O
,	O
and	O
if	O
we	O
have	O
an	O
oracle	O
that	O
counts	O
the	O
number	O
of	O
solutions	O
to	O
which	O
is	O
an	O
instance	O
of	O
,	O
then	O
we	O
can	O
design	O
an	O
algorithm	O
that	O
counts	O
the	O
number	O
of	O
solutions	O
to	O
,	O
the	O
corresponding	O
instance	O
of	O
.	O
</s>
<s>
Just	O
as	O
many-one	B-Algorithm
reductions	I-Algorithm
are	O
important	O
for	O
proving	O
NP-completeness	O
,	O
parsimonious	B-Algorithm
reductions	I-Algorithm
are	O
important	O
for	O
proving	O
completeness	O
for	O
counting	O
complexity	O
classes	O
such	O
as	O
#P	O
.	O
Because	O
parsimonious	B-Algorithm
reductions	I-Algorithm
preserve	O
the	O
property	O
of	O
having	O
a	O
unique	O
solution	O
,	O
they	O
are	O
also	O
used	O
in	O
game	O
complexity	O
,	O
to	O
show	O
the	O
hardness	O
of	O
puzzles	O
such	O
as	O
sudoku	O
where	O
the	O
uniqueness	O
of	O
the	O
solution	O
is	O
an	O
important	O
part	O
of	O
the	O
definition	O
of	O
the	O
puzzle	O
.	O
</s>
<s>
Specific	O
types	O
of	O
parsimonious	B-Algorithm
reductions	I-Algorithm
may	O
be	O
defined	O
by	O
the	O
computational	O
complexity	O
or	O
other	O
properties	O
of	O
the	O
transformation	O
algorithm	O
.	O
</s>
<s>
For	O
instance	O
,	O
a	O
polynomial-time	O
parsimonious	B-Algorithm
reduction	I-Algorithm
is	O
one	O
in	O
which	O
the	O
transformation	O
algorithm	O
takes	O
polynomial	O
time	O
.	O
</s>
<s>
These	O
are	O
the	O
types	O
of	O
reduction	B-Algorithm
used	O
to	O
prove	O
#P	O
-Completeness	O
.	O
</s>
<s>
In	O
parameterized	B-General_Concept
complexity	I-General_Concept
,	O
FPT	B-General_Concept
parsimonious	B-Algorithm
reductions	I-Algorithm
are	O
used	O
;	O
these	O
are	O
parsimonious	B-Algorithm
reductions	I-Algorithm
whose	O
transformation	O
is	O
a	O
fixed-parameter	O
tractable	O
algorithm	O
and	O
that	O
map	O
bounded	O
parameter	O
values	O
to	O
bounded	O
parameter	O
values	O
by	O
a	O
computable	O
function	O
.	O
</s>
<s>
Polynomial-time	O
parsimonious	B-Algorithm
reductions	I-Algorithm
are	O
a	O
special	O
case	O
of	O
a	O
more	O
general	O
class	O
of	O
reductions	O
for	O
counting	O
problems	O
,	O
the	O
polynomial-time	B-Algorithm
counting	I-Algorithm
reductions	I-Algorithm
.	O
</s>
<s>
One	O
common	O
technique	O
used	O
in	O
proving	O
that	O
a	O
reduction	B-Algorithm
is	O
parsimonious	O
is	O
to	O
show	O
that	O
there	O
is	O
a	O
bijection	B-Algorithm
between	O
the	O
set	O
of	O
solutions	O
to	O
and	O
the	O
set	O
of	O
solutions	O
to	O
which	O
guarantees	O
that	O
the	O
number	O
of	O
solutions	O
to	O
both	O
problems	O
is	O
the	O
same	O
.	O
</s>
<s>
One	O
can	O
show	O
that	O
any	O
boolean	O
formula	O
can	O
be	O
rewritten	O
as	O
a	O
formula	O
in	O
3-CNF	B-Application
form	O
.	O
</s>
<s>
Any	O
valid	O
assignment	O
of	O
a	O
boolean	O
formula	O
is	O
a	O
valid	O
assignment	O
of	O
the	O
corresponding	O
3-CNF	B-Application
formula	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Hence	O
,	O
this	O
reduction	B-Algorithm
preserves	O
the	O
number	O
of	O
satisfying	O
assignments	O
,	O
and	O
is	O
a	O
parsimonious	B-Algorithm
reduction	I-Algorithm
.	O
</s>
<s>
The	O
hardness	O
reduction	B-Algorithm
from	O
3SAT	O
to	O
Planar	O
3SAT	O
given	O
by	O
Lichtenstein	O
has	O
the	O
additional	O
property	O
that	O
for	O
every	O
valid	O
assignment	O
of	O
an	O
instance	O
of	O
3SAT	O
,	O
there	O
is	O
a	O
unique	O
valid	O
assignment	O
of	O
the	O
corresponding	O
instance	O
of	O
Planar	O
3SAT	O
,	O
and	O
vice	O
versa	O
.	O
</s>
<s>
Hence	O
the	O
reduction	B-Algorithm
is	O
parsimonious	O
,	O
and	O
consequently	O
Planar	O
#3SAT	O
is	O
#P	O
-complete	O
.	O
</s>
<s>
Seta	O
Takahiro	O
provided	O
a	O
reduction	B-Algorithm
from	O
3SAT	O
to	O
this	O
problem	O
when	O
restricted	O
to	O
planar	O
directed	O
max	O
degree-3	O
graphs	O
.	O
</s>
<s>
The	O
reduction	B-Algorithm
provides	O
a	O
bijection	B-Algorithm
between	O
the	O
solutions	O
to	O
an	O
instance	O
of	O
3SAT	O
and	O
the	O
solutions	O
to	O
an	O
instance	O
of	O
Hamiltonian	O
Cycle	O
in	O
planar	O
directed	O
max	O
degree-3	O
graphs	O
.	O
</s>
<s>
Hence	O
the	O
reduction	B-Algorithm
is	O
parsimonious	O
and	O
Hamiltonian	O
Cycle	O
in	O
planar	O
directed	O
max	O
degree-3	O
graphs	O
is	O
#P	O
-complete	O
.	O
</s>
<s>
Shakashaka	O
is	O
an	O
example	O
of	O
how	O
parsimonious	B-Algorithm
reduction	I-Algorithm
could	O
be	O
used	O
in	O
showing	O
hardness	O
of	O
logic	O
puzzles	O
.	O
</s>
<s>
The	O
reduction	B-Algorithm
from	O
Planar	O
3SAT	O
given	O
by	O
Demaine	O
,	O
Okamoto	O
,	O
Uehara	O
and	O
Uno	O
also	O
provides	O
a	O
bijection	B-Algorithm
between	O
the	O
set	O
of	O
solutions	O
to	O
an	O
instance	O
of	O
Planar	O
3SAT	O
and	O
the	O
set	O
of	O
solutions	O
to	O
the	O
corresponding	O
instance	O
of	O
Shakashaka	O
.	O
</s>
<s>
Hence	O
the	O
reduction	B-Algorithm
is	O
parsimonious	O
,	O
and	O
the	O
counting	O
version	O
of	O
Shakashaka	O
is	O
#P	O
-complete	O
.	O
</s>
