Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice, for the first time computationally accessing three-dimensional lattices of physical relevance. The method offers straightforward generalizations to several other counting problems.
Deep Dive into Counting Complex Disordered States by Efficient Pattern Matching: Chromatic Polynomials and Potts Partition Functions.
Counting problems, determining the number of possible states of a large system under certain constraints, play an important role in many areas of science. They naturally arise for complex disordered systems in physics and chemistry, in mathematical graph theory, and in computer science. Counting problems, however, are among the hardest problems to access computationally. Here, we suggest a novel method to access a benchmark counting problem, finding chromatic polynomials of graphs. We develop a vertex-oriented symbolic pattern matching algorithm that exploits the equivalence between the chromatic polynomial and the zero-temperature partition function of the Potts antiferromagnet on the same graph. Implementing this bottom-up algorithm using appropriate computer algebra, the new method outperforms standard top-down methods by several orders of magnitude, already for moderately sized graphs. As a first application, we compute chromatic polynomials of samples of the simple cubic lattice,
arXiv:0904.4010v1 [cond-mat.stat-mech] 27 Apr 2009
Coun
ting
Complex
Disordered
States
b
y
E ien
t
P
attern
Mat
hing:
Chromati
P
olynomials
and
P
otts
P
artition
F
un tions
Mar
Timme1
,
F
rank
v
an
Bussel1
,
Denn
y
Fliegner1
,
and
Sebastian
Stolzen
b
erg2
1
Max
Plan k
Institute
for
Dynami s
&
Self-Or
ganization,
Bunsenstr.
10,
37073
Göttingen,
Germany
and
2
Dep
artment
of
Physi s,
Cornel
l
University,
109
Clark
Hal
l,
Itha
a,
New
Y
ork
14853-2501,
USA
(Dated:
25
No
v
em
b
er
2008)
Coun
ting
problems,
determining
the
n
um
b
er
of
p
ossible
states
of
a
large
system
under
ertain
onstrain
ts,
pla
y
an
imp
ortan
t
role
in
man
y
areas
of
s ien e.
They
naturally
arise
for
omplex
disordered
systems
in
ph
ysi s
and
hemistry
,
in
mathemati al
graph
theory
,
and
in
omputer
s ien e.
Coun
ting
problems,
ho
w
ev
er,
are
among
the
hardest
problems
to
a ess
omputationally
.
Here
w
e
suggest
a
no
v
el
metho
d
to
a ess
a
b
en
hmark
oun
ting
problem,
nding
hromati
p
olynomials
of
graphs.
W
e
dev
elop
a
v
ertex-orien
ted
sym
b
oli
pattern
mat
hing
algorithm
that
exploits
the
equiv
alen e
b
et
w
een
the
hromati
p
olynomial
and
the
zero-temp
erature
partition
fun tion
of
the
P
otts
an
tiferromagnet
on
the
same
graph.
Implemen
ting
this
b
ottom-up
algorithm
using
appropriate
omputer
algebra,
the
new
metho
d
outp
erforms
standard
top-do
wn
metho
ds
b
y
sev
eral
orders
of
magnitude,
already
for
mo
derately
sized
graphs.
As
a
rst
appli ation
w
e
ompute
hromati
p
olynomials
of
samples
of
the
simple
ubi
latti e,
for
the
rst
time
omputationally
a essing
three-dimensional
latti es
of
ph
ysi al
relev
an e.
The
metho
d
oers
straigh
tforw
ard
generalizations
to
sev
eral
other
oun
ting
problems.
Giv
en
a
set
of
dieren
t
olors,
in
ho
w
man
y
w
a
ys
an
one
olor
the
v
erti es
of
a
graph
su
h
that
no
t
w
o
adja en
t
v
erti es
ha
v
e
the
same
olor?
The
answ
er
to
this
question
is
pro
vided
b
y
the
hromati
p
olynomial
of
a
graph
[1,
2
℄,
whi
h
giv
es
the
n
um
b
er
of
p
ossible
olorings
as
a
fun tion
of
the
n
um
b
er q
of
olors
a
v
ailable.
It
is
a
p
olynomial
in q
of
degree N
,
the
n
um
b
er
of
v
erti es
of
the
graph.
The
hromati
p
olynomial
is
losely
related
to
other
graph
in
v
arian
ts
e.g.
to
the
reliabilit
y
and
o
w
p
olynomials
of
a
net
w
ork
or
graph
(fun tions
that
hara terize
its
omm
uni ation
apabilities)
and
to
the
T
utte
p
olynomial.
These
are
of
widespread
in
terest
in
graph
theory
and
omputer
s ien e
and
p
ose
similar
hard
oun
ting
problems.
The
hromati
p
olynomial
is
also
of
dire t
relev
an e
to
statisti al
ph
ysi s
as
it
is
equiv
alen
t
to
the
zero-temp
erature
partition
fun tion
of
the
P
otts
an
tiferromagnet
[3
,
4℄:
The
P
otts
mo
del
[3℄
onstitutes
a
paradigmati
hara terization
of
systems
of
in
tera ting
ele tromagneti
momen
ts
or
spins,
where
ea
h
spin
an
b
e
in
one
out
of q ≥2
states;
it
th
us
generalizes
the
Ising
mo
del
where q = 2 .
F
or
an
tiferromagneti
in
tera tions,
neigh
b
oring
spins
tend
to
disalign
su
h
that
at
zero
temp
erature,
the
partition
fun tion
of
the
P
otts
an
tiferromagnet
oun
ts
the
n
um
b
er
of
ground
states
of
a
spin
system
just
as
the
hromati
p
olynomial
oun
ts
the
n
um
b
er
of
prop
er
olorings
of
the
same
graph.
F
or
su ien
tly
large q
there
are
many
system
on gurations
in
whi
h
al
l
pairwise
in
tera tion
energies
are
minimized
at
zero
temp
erature.
Indeed,
these
systems
exhibit
a
large
n
um
b
er
of
disordered
ground
states
that
is
exp
onen
tially
in reasing
with
system
size.
Th
us
the
P
otts
mo
del
exhibits
p
ositiv
e
ground
state
en
trop
y
,
an
ex eption
to
the
third
la
w
of
thermo
dynami s.
Exp
erimen
tally
,
omplex
disordered
ground
states
and
related
residual
en
trop
y
at
lo
w
temp
eratures
ha
v
e
b
een
observ
ed
in
v
arious
systems
[510
℄.
Although
there
are
sev
eral
analyti al
approa
hes
to
nd
hromati
p
olynomials
for
families
of
graphs
and
to
b
ound
their
v
alues
[14
,
11
17
℄,
there
is
no
losed
form
solution
to
this
oun
ting
problem
for
general
graphs.
Algorithmi ally
it
is
hard
to
ompute
the
hromati
p
olynomial,
b
e ause
the
omputation
time
in
general
in reases
exp
onen
tially
with
the
n
um
b
er
of
edges
in
the
graph
[18
℄.
It
also
strongly
dep
ends
on
the
stru ture
of
the
graph
and
rapidly
in reases
with
the
graph’s
size,
and
the
degrees
of
its
v
erti es,
f.
[19
21
℄.
Therefore,
most
studies
on
hromati
p
olynomials
up
to
date
ha
v
e
fo
used
on
small
graphs
and
families
of
graphs
of
simple
stru ture
and
lo
w
v
ertex
degrees,
e.g.
t
w
o-dimensional
latti e
graphs
[15
17
℄
(an
in
teresting
re en
t
attempt
to
analyti ally
study
simple
ubi
latti es
onsidered
strips
with
redu ed
degrees
[11
℄).
In
fa t,
it
is
not
at
all
straigh
tforw
ard
to
omputationally
a ess
larger
graphs
with
more
in
v
olv
ed
stru ture,
in luding
ph
ysi ally
relev
an
t
three-dimensional
latti e
graphs.
Finding
the
hromati
p
olynomial
of
a
graph
th
us
onstitutes
a
hallenging,
omputationally
hard
problem
of
sta
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