<s>
In	O
computer	B-General_Concept
science	I-General_Concept
,	O
a	O
holographic	B-Algorithm
algorithm	I-Algorithm
is	O
an	O
algorithm	O
that	O
uses	O
a	O
holographic	B-Algorithm
reduction	B-Algorithm
.	O
</s>
<s>
A	O
holographic	B-Algorithm
reduction	B-Algorithm
is	O
a	O
constant-time	O
reduction	B-Algorithm
that	O
maps	O
solution	O
fragments	O
many-to-many	O
such	O
that	O
the	O
sum	O
of	O
the	O
solution	O
fragments	O
remains	O
unchanged	O
.	O
</s>
<s>
These	O
concepts	O
were	O
introduced	O
by	O
Leslie	O
Valiant	O
,	O
who	O
called	O
them	O
holographic	B-Algorithm
because	O
"	O
their	O
effect	O
can	O
be	O
viewed	O
as	O
that	O
of	O
producing	O
interference	O
patterns	O
among	O
the	O
solution	O
fragments	O
"	O
.	O
</s>
<s>
The	O
algorithms	O
are	O
unrelated	O
to	O
laser	O
holography	B-Algorithm
,	O
except	O
metaphorically	O
.	O
</s>
<s>
Their	O
power	O
comes	O
from	O
the	O
mutual	O
cancellation	O
of	O
many	O
contributions	O
to	O
a	O
sum	O
,	O
analogous	O
to	O
the	O
interference	O
patterns	O
in	O
a	O
hologram	B-Algorithm
.	O
</s>
<s>
Holographic	B-Algorithm
algorithms	I-Algorithm
have	O
been	O
used	O
to	O
find	O
polynomial-time	O
solutions	O
to	O
problems	O
without	O
such	O
previously	O
known	O
solutions	O
for	O
special	O
cases	O
of	O
satisfiability	B-Algorithm
,	O
vertex	O
cover	O
,	O
and	O
other	O
graph	O
problems	O
.	O
</s>
<s>
Holographic	B-Algorithm
algorithms	I-Algorithm
have	O
some	O
similarities	O
with	O
quantum	B-Architecture
computation	I-Architecture
,	O
but	O
are	O
completely	O
classical	O
.	O
</s>
<s>
Holographic	B-Algorithm
algorithms	I-Algorithm
exist	O
in	O
the	O
context	O
of	O
Holant	B-Algorithm
problems	I-Algorithm
,	O
which	O
generalize	O
counting	O
constraint	B-Application
satisfaction	I-Application
problems	I-Application
(	O
#CSP	O
)	O
.	O
</s>
<s>
A	O
#CSP	O
instance	O
is	O
a	O
hypergraph	O
G	O
=(	O
V	O
,	O
E	O
)	O
called	O
the	O
constraint	B-Application
graph	I-Application
.	O
</s>
<s>
A	O
Holant	B-Algorithm
problem	I-Algorithm
is	O
like	O
a	O
#CSP	O
except	O
the	O
input	O
must	O
be	O
a	O
graph	O
,	O
not	O
a	O
hypergraph	O
.	O
</s>
<s>
In	O
the	O
context	O
of	O
Holant	B-Algorithm
problems	I-Algorithm
,	O
the	O
expression	O
in	O
(	O
1	O
)	O
is	O
called	O
the	O
Holant	O
after	O
a	O
related	O
exponential	O
sum	O
introduced	O
by	O
Valiant	O
.	O
</s>
<s>
A	O
standard	O
technique	O
in	O
complexity	O
theory	O
is	O
a	O
many-one	B-Algorithm
reduction	I-Algorithm
,	O
where	O
an	O
instance	O
of	O
one	O
problem	O
is	O
reduced	O
to	O
an	O
instance	O
of	O
another	O
(	O
hopefully	O
simpler	O
)	O
problem	O
.	O
</s>
<s>
However	O
,	O
holographic	B-Algorithm
reductions	O
between	O
two	O
computational	O
problems	O
preserve	O
the	O
sum	O
of	O
solutions	O
without	O
necessarily	O
preserving	O
correspondences	O
between	O
solutions	O
.	O
</s>
<s>
It	O
is	O
convenient	O
to	O
consider	O
holographic	B-Algorithm
reductions	O
on	O
bipartite	O
graphs	O
.	O
</s>
<s>
Thus	O
,	O
and	O
have	O
exactly	O
the	O
same	O
Holant	O
value	O
for	O
every	O
constraint	B-Application
graph	I-Application
.	O
</s>
<s>
The	O
equivalence	O
of	O
these	O
two	O
counting	O
problems	O
can	O
also	O
be	O
proved	O
using	O
a	O
holographic	B-Algorithm
reduction	B-Algorithm
.	O
</s>
<s>
The	O
2-stretch	O
of	O
G	O
gives	O
a	O
bipartite	O
graph	O
H	O
=(	O
U	O
,	O
V	O
,	O
E	O
)	O
,	O
where	O
U	O
corresponds	O
to	O
the	O
edges	O
in	O
G	O
and	O
V	O
corresponds	O
to	O
the	O
vertices	O
in	O
G	O
.	O
The	O
Holant	B-Algorithm
problem	I-Algorithm
that	O
naturally	O
corresponds	O
to	O
counting	O
the	O
number	O
of	O
vertex	O
covers	O
in	O
G	O
is	O
The	O
truth	O
table	O
of	O
OR2	O
as	O
a	O
row	O
vector	O
is	O
(	O
0	O
,	O
1	O
,	O
1	O
,	O
1	O
)	O
.	O
</s>
<s>
which	O
is	O
the	O
Holant	B-Algorithm
problem	I-Algorithm
that	O
naturally	O
corresponds	O
to	O
counting	O
the	O
number	O
of	O
independent	O
sets	O
in	O
G	O
.	O
</s>
<s>
As	O
with	O
any	O
type	O
of	O
reduction	B-Algorithm
,	O
a	O
holographic	B-Algorithm
reduction	B-Algorithm
does	O
not	O
,	O
by	O
itself	O
,	O
yield	O
a	O
polynomial	O
time	O
algorithm	O
.	O
</s>
<s>
Valiant	O
's	O
original	O
application	O
of	O
holographic	B-Algorithm
algorithms	I-Algorithm
used	O
a	O
holographic	B-Algorithm
reduction	B-Algorithm
to	O
a	O
problem	O
where	O
every	O
constraint	O
is	O
realizable	O
by	O
matchgates	O
,	O
which	O
he	O
had	O
just	O
proved	O
is	O
tractable	O
by	O
a	O
further	O
reduction	B-Algorithm
to	O
counting	O
the	O
number	O
of	O
perfect	O
matchings	O
in	O
a	O
planar	O
graph	O
.	O
</s>
<s>
The	O
latter	O
problem	O
is	O
tractable	O
by	O
the	O
FKT	B-Algorithm
algorithm	I-Algorithm
,	O
which	O
dates	O
to	O
the	O
1960s	O
.	O
</s>
<s>
Soon	O
after	O
,	O
Valiant	O
found	O
holographic	B-Algorithm
algorithms	I-Algorithm
with	O
reductions	O
to	O
matchgates	O
for	O
#7Pl	O
-Rtw-Mon-3CNF	O
and	O
#7Pl	O
-3/2Bip	O
-VC	O
.	O
</s>
<s>
Both	O
problems	O
were	O
already	O
known	O
to	O
be	O
#P	O
-hard	O
when	O
ignoring	O
the	O
modulus	O
and	O
Valiant	O
supplied	O
proofs	O
of	O
#P	O
-hardness	O
modulo	O
2	O
,	O
which	O
also	O
used	O
holographic	B-Algorithm
reductions	O
.	O
</s>
<s>
Valiant	O
found	O
these	O
two	O
problems	O
by	O
a	O
computer	O
search	O
that	O
looked	O
for	O
problems	O
with	O
holographic	B-Algorithm
reductions	O
to	O
matchgates	O
.	O
</s>
<s>
He	O
called	O
their	O
algorithms	O
accidental	B-Algorithm
algorithms	I-Algorithm
,	O
saying	O
"	O
when	O
applying	O
the	O
term	O
accidental	O
to	O
an	O
algorithm	O
we	O
intend	O
to	O
point	O
out	O
that	O
the	O
algorithm	O
arises	O
from	O
satisfying	O
an	O
apparently	O
onerous	O
set	O
of	O
constraints.	O
"	O
</s>
<s>
The	O
"	O
onerous	O
"	O
set	O
of	O
constraints	O
in	O
question	O
are	O
polynomial	O
equations	O
that	O
,	O
if	O
satisfied	O
,	O
imply	O
the	O
existence	O
of	O
a	O
holographic	B-Algorithm
reduction	B-Algorithm
to	O
matchgate	O
realizable	O
constraints	O
.	O
</s>
<s>
After	O
several	O
years	O
of	O
developing	O
(	O
what	O
is	O
known	O
as	O
)	O
matchgate	O
signature	O
theory	O
,	O
Jin-Yi	O
Cai	O
and	O
Pinyan	O
Lu	O
were	O
able	O
to	O
explain	O
the	O
existence	O
of	O
Valiant	O
's	O
two	O
accidental	B-Algorithm
algorithms	I-Algorithm
.	O
</s>
<s>
These	O
two	O
problems	O
are	O
just	O
special	O
cases	O
of	O
two	O
much	O
larger	O
families	O
of	O
problems	O
:	O
#2k	O
-1Pl-Rtw-Mon-kCNF	O
and	O
#2k	O
-1Pl-k/2Bip	O
-VC	O
for	O
any	O
positive	O
integer	O
k	O
.	O
The	O
modulus	O
7	O
is	O
just	O
the	O
third	O
Mersenne	O
number	O
and	O
Cai	O
and	O
Lu	O
showed	O
that	O
these	O
types	O
of	O
problems	O
with	O
parameter	O
k	O
can	O
be	O
solved	O
in	O
polynomial	O
time	O
exactly	O
when	O
the	O
modulus	O
is	O
the	O
kth	O
Mersenne	O
number	O
by	O
using	O
holographic	B-Algorithm
reductions	O
to	O
matchgates	O
and	O
the	O
Chinese	O
remainder	O
theorem	O
.	O
</s>
<s>
Around	O
the	O
same	O
time	O
,	O
Jin-Yi	O
Cai	O
,	O
Pinyan	O
Lu	O
and	O
Mingji	O
Xia	O
gave	O
the	O
first	O
holographic	B-Algorithm
algorithm	I-Algorithm
that	O
did	O
not	O
reduce	O
to	O
a	O
problem	O
that	O
is	O
tractable	O
by	O
matchgates	O
.	O
</s>
<s>
Instead	O
,	O
they	O
reduced	O
to	O
a	O
problem	O
that	O
is	O
tractable	O
by	O
Fibonacci	O
gates	O
,	O
which	O
are	O
symmetric	O
constraints	O
whose	O
truth	O
tables	O
satisfy	O
a	O
recurrence	O
relation	O
similar	O
to	O
one	O
that	O
defines	O
the	O
Fibonacci	B-Algorithm
numbers	I-Algorithm
.	O
</s>
<s>
They	O
also	O
used	O
holographic	B-Algorithm
reductions	O
to	O
prove	O
that	O
certain	O
counting	O
problems	O
are	O
#P	O
-hard	O
.	O
</s>
<s>
Since	O
then	O
,	O
holographic	B-Algorithm
reductions	O
have	O
been	O
used	O
extensively	O
as	O
ingredients	O
in	O
both	O
polynomial	O
time	O
algorithms	O
and	O
proofs	O
of	O
#P	O
-hardness	O
.	O
</s>
