Original citation:
Goldberg, Leslie Ann (1999) Randomly sampling unlabelled structures. University of
Warwick. Department of Computer Science. (Department of Computer Science
Research Report). (Unpublished) CS-RR-356
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Leslie Ann Goldberg
April 20, 1999
Abstrat
Informally, an \unlabelled ombinatorial struture" is an objet suh as an
un-labelledgraph (inwhihthevertiesare indistinguishable)orastrutural isomerin
hemistry (in whih dierent atoms of the same type are indistinguishable).
Com-putationalexperimentssuhasthose desribedin thisvolume oftenrelyon random
sampling to generate inputs for the experiments. This paper surveys work on the
problem of eÆiently sampling unlabelled ombinatorial strutures from a uniform
distribution.
1 Introdution
Most of the experimental work desribed in this volume involves rst randomly sampling
ombinatorial strutures and seond using the randomly-hosen strutures as inputs to
omputationalexperiments. Inorderfortheexperimentstobevalid, thedistributionfrom
whih the ombinatorial strutures are drawn must be preisely speied. In order for
the experiments tobe omputationallyfeasible, the random-sampling algorithmsmust be
eÆient.
This survey is devoted to the problemof eÆiently samplingunlabelled ombinatorial
struturesfromauniformdistribution. ForinformationontherelatedproblemofeÆiently
listing unlabelled ombinatorialstrutures withoutdupliates, see [11℄ and [12℄.
Westartbygivinganexampleofanunlabelledombinatorialstruture|inpartiular,
an unlabelled graph
1
. Two graphs with vertex set V
n
= fv
1 ;:::;v
n
g are said to be
isomorphi if there is a permutation of V
n
whih transforms one graph into the other.
For example, the graphs G
1
and G
2
from Figure1 are isomorphi beause relabelling the
vertiesof G
1
aording tothe permutation(v
1 v
2 v
3 )(v
4 v
5 )(v
6
) transformsG
1
intoG
2 .
ThepermutationsofV
n
partitionthen-vertexgraphsintoequivalenelasses. Twographs
leslieds.warwik.a.uk,http://www.ds.warwik.a.uk/leslie/,DepartmentofComputer
Siene,UniversityofWarwik,Coventry,CV47AL,UnitedKingdom. Thisworkwaspartiallysupported
bytheESPRITProjetsRAND-II(Projet21726)andALCOM-IT(Projet20244)andbyEPSRCgrant
GR/L60982. 1
1
v 1
v 2
v 3
v
4
v 5
v 6
2
v 2
v 3
v 1
v
5
v 4
v 6
Figure1: G
1
andG
2
areisomorphi. Thepermutation(v
1 v
2 v
3 )(v
4 v
5 )(v
6
)transformsG
1
into G
2 .
areinthesameequivalenelass ifandonlyifthereissomepermutationwhihtransforms
one into the other. (That is, they are in the same equivalene lass if and only if they
are isomorphi.) Formally,an \unlabelled graph" is just an isomorphism lass of labelled
graphs. Informally,an\unlabelled graph"iswhat youget whenyoutake alabelled graph
and erase the labels on the verties. Forexample, the unlabelled graph orresponding to
the graphs inFigure 1is shown in Figure2.
We willfous onomputationalproblems suh asthe following:
Input: A positiveinteger n.
Output: An unlabelled graph hosen uniformly at random from the set of all
unlabelled graphs with n verties.
This omputational problem illustrates the dierene between unlabelled sampling and
labelled sampling: To hoose a labelled graph uniformly at random (u.a.r.) from the
set of all n-vertex graphs, one would just need to onsider eah unordered pair of the
n verties and hoose, independently with probability 1=2, whether or not to make it an
edge. However, thisapproahwouldnotworkforsamplingunlabelledgraphs. Forexample,
the probability of produing a simple path ontaining n verties (using this approah) is
(n!=2) times ashigh as the probability of produing the omplete graph on n verties. In
fat,itwasn'tuntil1987thatapolynomialexpeted-timealgorithmforuniformlysampling
unlabelledgraphswasdisovered. ThisalgorithmwasdisoveredbyWormald[31℄,building
t
t
t t
t
Figure 2: The unlabelled graphorrespondingto the graphs inFigure1.
The general tehniques that have been used for sampling unlabelled ombinatorial
strutures are
1. indutive denitions/generating funtions,and
2. Burnside's Lemma(PolyaTheory).
These tehniques are desribed in Setion2 and Setion3 respetively.
2 Indutive Denitions
We willuse the following denitions to illustrate the \Indutive Denition" approah for
sampling unlabelled strutures. A tree is a onneted graph with no yles and a rooted
tree is a tree in whih a partiular vertex is distinguished as the root. An unlabelled
rooted tree is an isomorphism lass of rooted trees, where an isomorphism between two
rooted trees is required to map the root of one tree to the root of the other. Informally,
anunlabelled rootedtree iswhat youget when youtake arootedtree and erasethe labels
onthe verties,remembering whihvertex isthe root. Forexample, see Figure3. Let U
n
denote the set of unlabelled rooted trees with n verties and lett
n
denotethe size of U
n .
Nijenhuisand Wilf[25℄ showed that U
n
andt
n
an bedened indutivelyand thatthis
indutive denition an be used to sample u.a.r. from U
n
. Their main observation was
that every member of U
n
is onstruted exatly n 1 times by the following proedure:
For every d2f1;:::;n 1gand for every positive integer j suh that jd <n, letT
0
be a
memberofU
n jd
andletT
00
bea memberof U
d
. LetT
000
bethe unlabelledrooted n-vertex
graphformedby makingj opies ofT
00
t t t t
t t t
t R
t t t t
R
t t t t
R
Figure 3: The four unlabelled rooted trees with four verties.
to the root of T
0
. (The root of T
000
is taken to be the root of T
0
.) Outputd opies of T
000 .
Thus,
t n
(n 1)=
n 1 X d=1 X j:jd<n dt n jd t d ;
and so t
1 ;t
2
;::: an be omputed by dynami programming. Finally, the outline of the
samplingalgorithm(for n>2)is as follows (see also [27℄):
1. Choose the pair (j;d) with probability
dt n jd t d (n 1)tn .
2. Reursively hoose T
0
u.a.r. fromU
n jd
.
3. Reursively hoose T
00
u.a.r. from U
d .
4. Make j opies of T
00
and attah the rootof eah opy tothe rootof T
0 .
5. Letthe rootof T
0
be the root of the new n-vertex tree and outputthe new tree.
NijenhuisandWilf'sapproahwasextendedbyWilf[28℄,whoshowedhowtouniformly
samplefree(unrooted)trees. Wilf'salgorithmisalsobasedonanindutivedenition(i.e.,
a generatingfuntion) for the trees. This approahwill besystematised by Flajolet,
Zim-merman and VanCutsem in a forthomingpaper [9℄. The ompanion paper by the same
authorsonsamplinglabelled ombinatorialstrutures [8℄givesagoodidea ofthe
system-ati approah. The idea is to speify the unlabelled strutures using a formal grammar
involving set, sequene and yle onstrutions. Uniform sampling an then be done
au-tomatially using dynami programming. This systemati approah has also been used
by Goldberg and Jerrumto samplesome other tree-likeunlabelled strutures inSetion 4
of [15℄.
3 Burnside's Lemma
Burnside's Lemma is an important ingredient in the generating-funtion-based sampling
approahof[9℄. However, itturnsout thatthisingredientisusefulforsamplingeven when
beanitealphabetofardinalityk,andletGbeapermutationgroupon[m℄=f0;:::;m 1g.
For example,G ould bethe group inExample 1.
Example 1: An example of a group with degree
2
m = 4 is the group onsisting of the
followingpermutations.
identity,
(02),
(13), and
(02)(1 3).
Forg 2Gandi2[m℄,leti
g
denotetheimageofiunderg. Forexample,ifg =(02)wehave
0 g
=2, 1
g
=1, 2
g
=0 and 3
g
=3. Consider the set
m
of allwords of length m over the
alphabet . The group G has a natural ation on
m
whih is indued by permutations
of the m \bit positions". Under this indued ation, the permutation g 2 G maps the
word =a
0 a
1 :::a
m 1
to the word
g
=b
0 b
1 :::b
m 1
dened by b
j
=a
i
for all i;j 2[m℄
satisfyingi
g
=j. For example, if g =(02) and =0010 then
g
=1000 beause a
0
=0,
a 1
=0,a
2
=1,a
3
=0andb
2 =a 0 ,b 1 =a 1 ,b 0 =a 2 andb 3 =a 3
. TheationofGpartitions
m
intoanumberoforbits, whihare theequivalenelasses of
m
underthe equivalene
relationthatidentiesand wheneverthereexists g 2Gmappingto. Forexample,
ifk=2(so =f0;1g),m=4,andGisthegroupinExample1,thenthenineorbitsof
4
are(0000),(0001;0100),(0010;1000),(0101),(1010),(0011;0110;1001;1100),(1110;1011),
(1101;0111),and (1111). The \Orbit SamplingProblem"(for xed ) isas follows.
Input: A permutation group Gof degree m.
Output: A word in
m
so that eahorbit is equallylikely tobe output.
Mostofthe unlabelledsamplingproblems thatwe onsiderinthis surveyan beoded
up in terms of the orbit sampling problem. For example, we an represent an N-vertex
graph as itsadjaeny matrix, whih is a word in
( N
2 )
where =f0;1g. The unlabelled
N-vertex graphsare justtheorbitsof wordsin
( N
2 )
withrespet tothe \pairgroup"S
(2) N
.
(EahofthepermutationsinS
(2) N
orresponds toadierentpermutationof theN verties.
However the verties are permuted, the orresponding permutation in S
(2) N
performs the
orresponding ationon the adjaeny matrix.)
Now Burnside's Lemma
3
says that eah orbit omes up jGjtimes (asthe rst
ompo-nent)in the set of pairs
(;G)=f( ;g)j2
m
;g 2Gand
g
= g: (1)
2
Thedegree ofapermutation groupisthe numberof objetson whih thepermutations at. Not all
degree-4permutation groupshave4permutations.
3
Althoughthislemmaisommonlyreferredtoas\Burnside'sLemma",itisreallyduetoCauhyand
dividedintofourorbits(oneoneahrow)whihrepresentthe fourunlabelledgraphswith
3verties. Eahrowisrepresented 6timesin
3
beause anyof the 6permutationsofthe
verties maps the graph on the top row to itself. Similarly,the graph on the bottom row
has 6 automorphisms (permutations whih map it to itself). However, any graph on the
middle two rows has only two automorphisms: the identity, and the permutation whih
transposes the two degree-1 verties.
3.1 Unlabelled Graphs
TranslatingBurnside's Lemma to the speial ase of unlabelled graphs,we nd that eah
unlabelledN-vertex graph omes up N! times (asthe rst omponent)in the set of pairs
N
=f( ;g) j is a(labelled) N-vertex graph and g isa permutation
of the N verties whih maps toitself.g: (2)
The speial ase N =3 isillustrated in Figure4.
ThesignianeofBurnside'sLemma(Equation2)isthatinordertouniformlysample
N-vertex unlabelled graphs, we need only sample u.a.r. from
N
. This observation was
rst made by Dixon and Wilf [5℄. In order tosimplify our presentation of their idea (and
Wormald'srenement [31℄), for every permutation g of N verties, we willlet
g
denote
f( ;g)j is a (labelled)N-vertex graph and g maps toitself.g:
Thus,
N =
S g
g
, wherethe unionis over allpermutationsg of N verties.
WeannowstatetheoutlineofDixonandWilf'salgorithmforsamplingu.a.r.from
2. Choose g with probability jgj
j N
j
3. Choose ( ;g)u.a.r from
g .
Step 3 of the algorithm an be implemented in polynomial time: For eah yle of
(un-ordered)vertexpairsinduedbyg,deideindependently(withprobability1=2)whetherto
makethepairsedgesornon-edges. ItisnotknownhowtoimplementStep2inpolynomial
time. In partiular, itis not known howto ompute j
N
jin polynomial time
4 .
Wormald's algorithm [31℄ uses the rejetion sampling method to get around
omput-ingj
N
j. Thebasiideaofrejetionsamplingisasfollows. ItmaybetoodiÆulttosample
fromthe desireddistribution. Sowhatthe user does insteadistosample fromsome other
(more tratable) distribution. Imagine the desired distribution asbeing \saled down" so
thatitts underneaththe moretratable distribution. Todrawasamplefromthe desired
distribution, the user rst draws a samplefrom the more tratable distribution. The user
then uses thesample todeterminethe probabilitywith whihthe moretratable
distribu-tionover-representsthis sample(relativetothe\saleddown" desireddistribution). With
this probability, the sample is rejeted and the samplingproess is re-started. Otherwise,
the sample is output. The method is useful when it is easy to determine the
probabil-ity with whih a given sample should be rejeted and, furthermore, the overall rejetion
probability islow(so the expeted runningtime until asample is outputis small).
The outline of Wormald's algorithm is as follows, where i
N
represents the identity
permutation onN verties and p
g
represents the probabilitywith whih permutation g is
hosen (so
P g
p g
=1). Appropriate hoies forp
g
will bedisussed below.
1. Input N
2. Choose g with probability p
g .
3. Choose ( ;g)u.a.r. from
g .
4. Output( ;g) with probability
p i
N j
i N
j j
g j
pg
. Otherwise rejet.
It is straightforward to hek that the probability that any given pair ( ;g) from
N is
outputis
p i
N j
i N
j
,sothe algorithmdoessampleu.a.r.from
N
aslongasCriterion 1(below)
is met (soStep 4 an be implemented).
4
Dixon andWilf showhowtoimplementStep 2in polynomialtime on average providedthevalueof
j N
j isknown. (Theirmethod involveshoosing aonjugaylasswith theappropriate probabilityand
g
p i
N j
i N
j j
g j
p g
1:
Furthermore, the rejetion probability is 0 whenever g =i
N
. Thus, the expeted running
time of the sampling algorithmis polynomial aslong as the other riteria beloware met.
Criterion 2: The probabilities p
g
must be hosen so that Step 2 an be
imple-mented inpolynomialtime
5 .
Criterion 3: The probabilities p
g
must be hosen so that Step 4 an be
imple-mented inpolynomialtime.
Criterion 4: The probabilities p
g
must be hosen so that, for some positive
onstant and every N, we have p
i N
N
. (This ensures that the expeted
number of trials beforean outputis produed (withno rejetion)is atmost N
.)
Wormald[31℄ showed howtohoose the probabilitiesp
g
sothat these riteriaare met.
Thus, he gave an expeted polynomial-timealgorithmfor sampling unlabelled graphs.
3.2 Extending Wormald's Method
Letusnowonsider howtoextendWormald'smethodtothegeneral\Burnside'sLemma"
framework. Let bea xed alphabet. Forany permutation g of [m℄,let
(;g)=f( ;g)j2
m
and
g
= g:
Then(;G)fromEquation1equals
S g2G
(;g). WeanwritetheoutlineofWormald's
algorithmusing this notation, where i
m
denotes the identity permutationof degree m:
1. Input a permutationgroup G
2. Choose g 2G with probabilityp
g .
3. Choose ( ;g) u.a.r. from(;g).
4. Output( ;g)with probability
p im
j(;im)j
j(;g)j
pg
. Otherwise rejet.
An appropriate measure of \inputsize" for the input G is the degree of G, beause every
degree-m permutation group an be speied by a set of O(m) permutations [18℄. Thus,
the analogue of Criterion 4 states that there is a positive onstant suh that for every
5
Notethathoosingp
g =
jgj j
N j
wouldmakeWormald'salgorithmequivalenttoDixonandWilf's. Thus,
possible input group G of degree m, we must have p im
m . On the other hand, the
analogueof Criterion 1 implies
p g
p i
m
j(;g)j
j(;i
m )j
; (3)
whihimplies
p im
j(;i
m )j
j(;G)j
= jj
m
j(;G)j
:
Thus, weannotsimultaneouslysatisfythetworiteriaunlessthereisapositiveonstant
suh that forevery possible input Gof degree m we have
j(;G)jm
jj
m
: (4)
That is, the size of (;G) must be at most a polynomial times the size of the part
of (;G) whih orresponds to the identity permutation. We will say that a family
of permutation groups has \too many non-trivial symmetries" (with respet to the xed
alphabet)ifthere isnot apositiveonstant suhthat everygroup Ginthe family
sat-isesEquation4. Sinesomefamiliesofunlabelledombinatorialstrutures doorrespond
tofamiliesof groupswith toomany non-trivialsymmetries, Wormald's methodannotbe
used in general for unlabelled sampling. Wormald has shown [31℄ how to use the method
to eÆiently sampleunlabelled r-regular graphsfor r 3. (Note that for xed r 3,the
unlabelled r-regular graphs do not have too many non-trivial symmetries. In fat, most
unlabelledr-regulargraphs have only the identity as asymmetry.)
3.3 Strutures with Too Many Non-Trivial Symmetries
A (labelled) multigraph is given by a set of verties and a multiset
6
of unordered pairs
of verties, whih are alled edges. An unlabelled multigraph is an isomorphism lass of
labelledmultigraphs. Thedegree sequene ofmultigraphGis asequene tellinghowmany
vertiesofeahdegreeGhas. Considerthefollowingunlabelledsamplingproblem: Givena
degreesequeneinwhiheahdegreeisboundedfromabovebyaonstant,selet,uniformly
at random, an unlabelled onneted multigraph with the given degree sequene. This
samplingproblemarises intheontextof samplingstruturalisomers inhemistry[7,15℄.
Itisanexampleof asamplingproblemtowhihWormald'smethoddoesnotapply: Ifthe
degree sequene has many verties of degree 1 or 2 then the unlabelled multigraphs with
the degree sequene have too many non-trivialsymmetries.
Goldberg and Jerrum [15℄ have given a polynomial expeted-time algorithm for this
samplingproblembasedonthefollowingidea. EveryunlabelledmultigraphGisassoiated
with a unique \ore"
7
whih has no verties of degree 1 or 2. To randomly generate a
multigraph G, the algorithm rst generates the ore of G and then extends the ore by
adding trees and hains of trees to obtainG.
6
A multiset is likeasetexept that itanontainmorethan oneopyof anobjet,so amultigraph
anontainseveral(indistinguishable) edgesbetweenanygivenpairofverties.
7
Bender and Caneld [1℄, Bollobas [3℄ and Wormald [29℄. A onguration (for a given
degreesequene) isalabelledombinatorialstruturewhihanbeviewed asarenement
of a multigraph with the degree sequene. For any given degree sequene, the orbits of
all ongurations (with respet to the appropriate permutation group) orrespond to the
unlabelled multigraphs with the degree sequene. Sine the degree sequene of the ore
has no verties of degree 1 or 2, the ores do not have too many non-trivial symmetries.
(This follows from an extension of Bollobas's analysis of unlabelled regular graphs [2℄.)
Thus, the algorithm uses Wormald's method to generate the ore. (If the ore is not
onneted,itisrejeted. Thefatthatthisdoesnothappentoooftenfollowsfromaresult
of Wormald[30℄.)
After generatingthe oreof the randommultigraph, the algorithmextends theore by
adding trees and hains of trees. This part of the algorithm is based on the
generating-funtion approahmentioned inSetion 2.
Itisanopenproblemtosampleunlabelledmultigraphsgivenageneral degreesequene
(inwhihdegrees neednotbeboundedfromaboveby aonstant). Goldbergand Jerrum's
method is not appliable when the degrees are unbounded. In fat, the problem with
unbounded degrees seems to bediÆult even inthe labelled ase (see [21, 23,6℄).
3.4 A General Approah Based on Burnside's Lemma
Jerrum [19℄ proposed a general way to use Burnside's Lemma for unlabelled sampling.
Theideaistoonsiderthe followingbipartitegraph: The vertiesontheleft-handsideare
allwords in
m
. Thevertiesontheright-handsideare allpermutationsinG. Thereisan
edge from word to permutationg if and only if
g
= . For example, when =f0;1g
and m=4andG isthe groupdesribed inExample 1,weget the graphinFigure5. This
graph essentially implements Burnside's Lemma: The edges in the graph are the pairs
in (;G). Thus, Burnside's Lemma (Equation 1) says that the set of left-endpoints of
edges ontains jGjrepresentatives of eah orbit.
Now onsider a random walk on this graph: When the walk is at a given node, it
hooses aneighbourof the nodeu.a.r. andmovestotheneighbour. Ifweviewthe random
walk from the perspetive of the right-hand verties we an see that it has a stationary
distribution,andthat,inthestationarydistribution,theprobabilityofbeingatanynodeis
proportionaltoitsdegree. Thus,thestationarydistributionallowsustosampleedges(and
thus, left end-points) u.a.r. By Burnside's Lemma, the stationary distribution therefore
allows us to sample orbits u.a.r. Three issues arise when we onsider how to use this
method tosample orbits:
1. How long doesit take to simulate aright-to-left step of the random walk?
2. How long doesit take to simulate aleft-to-rightstep of the random walk?
3. How many steps of the randomwalk do we needto simulate inorder to get lose to
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1011
1010
1100
1101
identity
(0 2)
(0 2) (1 3)
(1 3)
1110
1111
Figure5: ThebipartitegraphdepitingBurnside's Lemmafor =f0;1g,m=4,andthe
expliitly. We are looking for sampling algorithms with run-times that are polynomial
in m, the degree of G, but the number of left-hand verties is jj
m
and the number of
right-hand verties ould be as large as m!. Fortunately, we an simulate the random
walk without expliitly onstruting the graph. Taking a step from right to left an be
done in polynomial time: For eah yle of the permutation orresponding to the urrent
right-hand vertex, a letter in is hosen u.a.r. (This is just a generalisation of Step 3 of
Dixon and Wilf's algorithm.) Taking a step from left toright is not so easy. In fat, it is
equivalentunderrandomisedpolynomial-timeredutionstotheSetwiseStabiliser problem,
whih inludes Graph Isomorphism as a speial ase. Nevertheless, there are signiant
lasses of groups G forwhih this step an be implemented inpolynomial time. Luks has
shown that the lass of p-groups|groups in whih every element has order a power of p
for some prime p|is anexample of suh a lass [22℄.
Before movingontothe thirdissue,wemention anothersituationinwhihthe
simula-tion of the left-to-right step seems towork suÆiently well. Peter Cameron has observed
that the random walk desribed in this setion ould be dened for any group ation,
not just for the speial ase of a permutation group G ating on
m
by permuting
posi-tions. Thegeneralsettingisasfollows: Givenapoint ,seletu.a.r.agroupelementgthat
xes ,andthenseletapointthatisxedbyg. Thisrandomwalkhasbeenimplemented
in ertainalgorithmsfor determiningthe onjugaylasses of a nite group [26℄.
Clearly the eetiveness of the random walk as a basis for general-purpose sampling
proedures for unlabelled strutures depends upon the third issue: \How many steps of
the random walk do we need to simulate in order to get lose to the stationary
distribu-tion?" Inordertoanswerthis question,weneedsome denitions. Forany two probability
distributions and
0
ona niteset ,wedene thevariation distane between and
0
tobe
D(;
0
)=max
A
j(A)
0
(A)j=
1
2 X
x2
j(x)
0 (x)j:
We say that the \tolerane-Æ mixing time" of the random walk is the minimum t suh
that for all start verties v
0
and for all t
0
t, D(
t 0
(v 0
);) Æ, where denotes the
stationarydistributionofthe walk,and
t 0
(v 0
)denotesthedistributionaftert
0
steps, given
that the walk starts at vertex v
0
. We say that the random walk is\rapidly mixing" if its
\tolerane-Æ mixing time" is at most a polynomial in m (the input size | the degree of
the group G) and log(1=Æ).
Jerrum [19℄ showed that the random walk is rapidly mixingif the group G is yli or
symmetri. If G isyli (i.e., it is generated by a single permutation)then the orbits do
not have too many non-trivial symmetries, so a sampling algorithm based on Wormald's
method might also work. If G is the symmetri group then the orbits have too many
non-trivialsymmetriestosatisfyEquation4soWormald'smethodwillnotwork. However
it is interesting to note that an analogous situation ours: A polynomially-largefration
of the pairs in (;G)have word 000 in the rst omponent.
Goldberg and Jerrum [14℄showed that the randomwalk is not always rapidly mixing:
tolerane-3
mixingtimeisexponentialinthe degreeofG. Thus, there isaninnitefamily
of groups for whih the variation distane is at least 1=3 after exponentially-many steps.
The groups in this family are unompliated from a group-theoreti point of view: The
permutationsinthe groupsommuteand haveorderthree. Thus,the groupsarep-groups,
and eah step of the random walk an be simulated in polynomial time. However, the
groups are ompliated from a ombinatorial point of view
8
. It would be interesting to
know for whih families of groups the random walk is rapidly mixing. As far as I know,
[19℄ and [14℄ are the only known results about this.
GoldbergandJerrum'snegativeresultleavesopenthepossibilitythatforaxed
alpha-bet there might be some other eÆient sampling algorithm(one whih does not simulate
the random walk) whih takes input G and outputs a right-hand vertex (or a left-hand
vertex) drawn fromthe stationarydistribution ofthe randomwalk (orfroma distribution
losetothestationarydistribution). Aswehaveseenbefore,samplingfromthe stationary
distribution of the left-hand verties is easy provided we an sample from the stationary
distribution of the right-handverties. A more generalresult of this form isgiven in[19℄.
Thus, an interesting open question is whether there exists an eÆient sampling
algo-rithm whih takes input G and outputs a right-hand vertex (i.e., a group element) with
(approximately) the orret distribution. We onlude this setion by desribing weak
evidene indiating that suh an algorithmmay not exist.
First, we willbe more spei about the notion of an \approximately orret"
distri-bution. A \good"samplingalgorithmwill
1. run in poly(m;
1
) time, where m is the degree of the input group, and is an
auray parameter,and
2. produe a random right-hand vertex v suh that for any vertex v, the probability
withwhihv isreturnedis intherange [(1 )p;(1+)p℄, where pisthe probability
of being at vertex v inthe stationary distribution of the randomwalk.
Seond, we willdene the \yle index polynomial" of a permutationgroup G. When
this polynomial is evaluatedat k, the value is
1
jGj X
g2G k
(g) ;
where (g) denotes the numberof yles inpermutation g.
Finally, we will state a urious onsequene whih would our if a good sampling
algorithmdid exist.
8
The groups are onstruted to mimi the \Swendsen-Wang Proess," whih is a dynamis for the
q-state Potts model. The slow mixing is onneted to a rst-order phase transition in the number of
1. Foranyxed k>1,evaluatingthe yleindex polynomialofapermutationgroupG
atk is #P-hard.
2. Foranyxedpositivenon-integer k,approximately evaluatingtheyleindex
polyno-mialofapermutationgroupGatk ishard. (Speially,thereisnofullypolynomial
randomised approximation sheme (fpras) unless RP=NP).
3. Foranyxed positiveinteger k,approximately evaluatingtheyleindex polynomial
of apermutationgroup Gat k is easy. (Speially,there is anfpras.)
Items 1 and 2 are true (whether or not there is a good sampling algorithm). Item 1
was shown to be true by Goldberg [13℄ and item 2 was shown to be true by Goldberg
and Jerrum [13, 20℄. Jerrum [19℄ showed that Item 3 would be true if there were a good
sampling algorithm for every xed alphabet size k. It may be that items 1{3 are all
true (though this would be somewhat surprising). Alternatively, it may be that there is
no eÆient algorithm for sampling permutations (right-hand verties) in the \Burnside's
Lemma"framework. Resolvingthis open question would bevery interesting.
Aknowledgements
I thank Mark Jerrum for useful omments on an earlier draft of this survey. Most of my
work inthis area isjointwith Mark and Ihave used materialfromour joint papers inthe
survey.
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