**Original citation: **

### Goldberg, Leslie Ann, Jerrum, M. and Paterson, Michael S. (2001) The computational

### complexity of two-state spin systems. University of Warwick. Department of Computer

### Science. (Department of Computer Science Research Report). (Unpublished)

### CS-RR-386

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of two-state spin systems

Leslie AnnGoldberg

y

Dep't of Computer Siene

University of Warwik

MarkJerrum

z

Divisionof Informatis

University of Edinburgh

MikePaterson

x

Dep'tof Computer Siene

University of Warwik

November29, 2001

Abstrat

Thesubjetofthisartileisspin-systemsasstudiedin statistialphysis. Wefousonthe

aseoftwospins. Thisaseenompassesmodelsofphysialinterest,suhasthelassialIsing

model(ferromagnetiorantiferromagneti,withorwithoutanappliedmagneti eld)andthe

hard-oregasmodel. Therearethreedegreesoffreedom,orrespondingtoourparameters,

and . We wishto study theomplexity of(approximately) omputingthepartition funtion

in termsofthese parameters. Wepayspeialattentiontothesymmetriase=1forwhih

ourresultsaredepitedinFigure1. ExatomputationofthepartitionfuntionZ isNP-hard

exept in the trivial ase = 1, so weonentrate on the issue of whether Z anbe

om-puted withinsmallrelativeerrorin polynomialtime. Weshowthat thereisafullypolynomial

randomised approximationsheme (FPRAS)for thepartition funtion in the\ferromagneti"

region 1, but (unless RP = NP) there is no FPRAS in the \antiferromagneti" region

orrespondingto thesquare dened by 0< <1and 0< <1. Neitherofthese \natural"

regions|neitherthehyperbolanorthesquare |markstheboundarybetweentratableand

intratable. Inonediretion,weprovideanFPRAS forthepartition funtion within aregion

whihextendswellawayfrom thehyperbola. Intheotherdiretion,weexhibittwotiny,

sym-metri,intratableregionsextendingbeyondtheantiferromagnetiregion. Wealsoextendour

resultstotheasymmetriase6=1.

ResearhReport386, DepartmentofComputerSiene,Universityof Warwik, CoventryCV4 7AL,UK(Nov

2001). This workwas partially supportedby the EPSRC grant \SharperAnalysis of Randomised Algorithms: a

Computational Approah", the EPSRC grant GR/R44560/01 \Analysing Markov-hain based random sampling

algorithms" and the IST Programme of theEU underontrat numbers IST-1999-14186 (ALCOM-FT)and

IST-1999-14036 (RAND-APX).

y

leslieds.warwik.a.uk, http://www.ds.warwik.a.uk/les lie/, Department of Computer Siene,

UniversityofWarwik,Coventry,CV47AL,UnitedKingdom.

z

mrjds.ed.a.uk,http://www.ds.ed.a.uk/mrj/,DivisionofInformatis,UniversityofEdinburgh,JCMB,

TheKing'sBuildings,EdinburghEH93JZ,UnitedKingdom. x

mspds.warwik.a.uk, http://www.ds.warwik.a.uk/ msp/,DepartmentofComputerSiene,University

The subjet of this artile is \spin-systems" as studied in statistial physis. An instane of a

spin-system is an n-vertex graph G =(V;E). Let q 2 be an integer. A onguration of a spin

system onGis one oftheq

n

possibleassignments :V !f0;:::;q 1gof q spins to theverties

of G. (We shall usually refer to spins asolours.) Eah onguration has an energy H() whih

is the sum of individualontributionsfrom theedges and vertiesof G. The ontributionof eah

edge fi;jg 2 E is a speied funtion (here assumed symmetri) of the olours (i) and (j);

likewise, the ontribution of vertex k 2 V is a funtion of (k). Eah onguration has weight

w() = exp( H()=T), where T is a parameter of the system alled temperature. The partition

funtion of the system is the normalising fator Z =

P

exp( H()=T) that turns the weights

into probabilities. 1

Our goal in this paperis to study the omplexity of omputingthe partition funtion of spin

systems. We shall deal exlusively with two-spin (q = 2) systems, sine these already seem to

present enough of a hallenge. Moreover, thease q =2 enompasses modelsof physial interest,

suh as thelassial Ising model (ferromagneti orantiferromagneti, withor withoutan applied

magneti eld), or the hard-ore gas model. We refer to the two olours (spins) as \blue" and

\green". Sinew()=exp( H()=T)andH() isasum ofontributionsfromedgesandverties,

weanequivalentlytakeamultipliativeview,inwhihw()isdenedasaprodut ofontributions

fromtheindividualedgesandverties. (Allthiswillbesetupformallyinthenextsetion;however,

we hopethatthisinformalaount providesan adequatebasisforat leasta qualitativedisussion

of themainresultsof thepaper.)

Atrstsightitseemsasthoughtherearethreeparametersgoverningedgeontributions

(orre-spondingto blue-blue, blue-greenand green-green edges), and two governingvertexontributions

(orresponding to blueand green verties). But we maynormalisethe (multipliative)blue-green

edgeontributionto1,andthebluevertexontributionto 1also.

2

Thusthereareessentiallythree

degrees of freedom. We denote the (multipliative) blue-blue edge ontribution by , the

green-green by,and thegreen vertexontribution by. Infat |partly beause itiseasier to depit

a two-dimensionalparameterspae,andpartly beauseourunderstandingof thegeneral situation

is stillinomplete |we shallpay partiularattention to thespeial(symmetri) ase=1.

Figure 1 shows the regions in (;)-spae as lassied by our results when = 1. Exat

omputationofthepartitionfuntionZ isNP-hardexeptinthetrivialase=1sowe

onen-trate on the issue of whether Z an be omputed withinsmall relative error in polynomial time.

(The preisenotionof eÆient approximation algorithmused isthe\fully polynomialrandomised

approximation sheme" orFPRAS, whihwillbedenedinx2.) Themainfeatures areasfollows:

1. To the North-East of the hyperbola is a \ferromagneti" region 1 within whih the

partitionfuntionmaybeapproximatedintheFPRASsense. Thisisdonebyredutiontoa

ferromagnetiIsingsystemwithexternaleld,whosepartitionfuntionmaybeapproximated

bya Markovhain Monte Carloalgorithm ofJerrum and Sinlair[7 ℄. Seex3.

2. Thesquaredenedby0< <1and0< <1isan\antiferromagneti"regionwithinwhih

the partitionfuntion ishard to approximate(unlessRP=NP). Essentially thisisbeause

\ground states" (i.e., most likely or most weighty ongurations) orrespond to maximum

uts inG. Thisat least istheintuition;the formalisationof itrequires some work. Seex4.

1

Readers who donot ndthe physial settingongenial, may thinkinstead of aweighted versionof the graph

homomorphismproblem. SeeSetion1.1 of[4 ℄. 2

### 0

### 0.5

### 1

### 1.5

### 2

### 2.5

### 3

### 0

### 0.5

### 1

### 1.5

### 2

### 2.5

### 3

PSfragreplaements

EASY

Thm.1

EASY

Thm.6.1 EASY

Thm.6.2

HARD

Thm.3

Figure1: Results forregions ofthe(;)-plane at =1.

3. Neither of these \natural" regions | neither the hyperbola nor the square | marks the

boundary between tratable and intratable. In one diretion, we provide an FPRAS for

the partition funtion within the light grey region, whih extends wellaway the hyperbola.

This FPRAS is based on the Markov hain Monte Carlo method, and its analysis uses the

\path-oupling"tehniqueof Bubleyand Dyer[2 ℄. Seex5.

4. In theother diretion,weexhibit two tiny,symmetri, intratableregions extendingbeyond

the \antiferromagneti" region, lose to the points (0;1) and (1;0). This is done by oding

up an inapproximable ombinatorial optimisation problem following Luby and Vigoda [9 ℄.

Seex6.

It willbe seen that our knowledge even of the =1 ase is inomplete: speially, we don't

know what happens in the remaining (medium intensity grey) regions. For example, we don't

knowwhether tratabilityismonotonein (or). Inthe remainingsetions,weprove theresults

depited inFigure 1and also extendthese resultsbeyond thesymmetri ase=1.

2 Denitions

Toformalise thelaimsmadeintheintrodutionweneedto denepreiselytheterms(two-state)

\spinsystem" and \FPRAS" (ournotionof eÆient approximateomputation).

In order to dene the partition funtion of a two-state spin spinsystem speied by weights

, and , it is onvenient to identify blue and green with the unit vetors (1;0)

0

and (0;1)

0 ,

respetively. (Primeswillbeused to denotetransposition,sospinsareolumn vetors.) Then the

partition funtionforagraph G=(V;E) maybeexpressed as

Z(G)=

X

Y

fi;jg2E (i)

0

A(j)

Y

k2V b

A=

1

1

and b=

1

;

and rangesover f(1;0)

0 ;(0;1)

0 g

V

. Toseethis, notethat eahof thefourpossibleassignmentsof

unit vetors to (i) and (j) piksouta distintelement of thematrixA, and similarlywithb.

The problemwhose omplexity we study is (;;)-Partition, dened as follows. Let ,

and be non-negative real numbers.

Name. (;;)-Partition.

Instane. A graphG.

Output. The quantityZ(G),whereZ is thepartitionfuntionwithparameters , and .

NotethatthegraphGaloneformstheprobleminstane,whihmeanswehaveaseparateproblem

for every triple (;;). (Our notation is intended to emphasise this.) Our goal is to map out

the tratableregionof theparameterspae. Toavoidthe issuesof speifyingand omputingwith

arbitrary real numbers,we assumethat , and arerational.

In thisarea, approximation algorithmsareusuallyviewed asomputingfuntions f :

!N,

where is a nitealphabetfor enoding probleminstanes. In the urrent appliation, however,

the outputmaybe anarbitrary rationalnumber. Rather thanredening awell-establishednotion

ofeÆientapproximateomputation,weshallstikwiththeusualdenition,andthenexplainhow

to view(;;)-Partition inthisframework.

A randomised approximation sheme fora ounting problem f :

! N (e.g., the numberof

mathings ina graph) is a randomised algorithm that takes asinputan instane x2

(e.g., an

enoding of a graph G) and an error tolerane " > 0, and outputs a number N 2 N (a random

variable of the\ointosses" made bythealgorithm)suh that, forevery instane x,

Pr

e "

f(x)N e

" f(x)

3

4

: (1)

We speak of a fully polynomial randomised approximation sheme, or FPRAS, if the algorithm

runs intime bounded by a polynomialin jxj and "

1

. It is a standard result that the number

3

4

appearingin (1)ouldbe replaedbyanynumberintheopeninterval (

1

2 ;1).

Tobringtheproblem(;;)-PartitionwithintheFPRASframework,wesuggestthe

follow-ing: Assume, andarerational,andletLbetheleastommonmultipleoftheirdenominators.

Then thedesired output Z(G) an be expressed as a rationalnumberz withdenominator L

n+m ,

where nis thenumberofverties inGand m thenumberof edges. Then ourgoal isto designan

FPRAS omputingL

n+m z.

3 The \ferromagneti region" is tratable

We arguethattheregion 1orrespondsfairlydiretly to theferromagnetiIsingmodel with

externaleld. It followsthatthere isan FPRASforthepartitionfuntionZ inthisregion. When

=1 thepartitionfuntionistriviallyomputablein polynomialtime.

To make thisorrespondene expliit,observe that

A=

s

1 0

0 =

1

1

1 0

0 =

where = ,and hene

(i) 0

A(j)=

s

(i)

0

1 0

0 =

1

1

1 0

0 =

(j)

= p

=

(1;=)(i)

(i) 0

b

A(j) (1;=)(j)

;

where

b

A=

1

1

;

and the nal equality uses the fat that spins are unit vetors. Thus we obtain the following

alternativeexpressionforthepartition funtion:

Z(G)=

m=2 X

Y

fi;jg2E (i)

0 b

A(j)

Y

k2V

1;(=)

d(k)

(k); (2)

where d(k) is the degree of vertex k. To verify (2), note that eah of thed(k) edgesinident at k

ontributes afator (1;=)(k) to Z(G),inaddition to the(1;)(k) already present.

Suppose for the moment that = 1. When 1, i.e., when 1, equation (2) is, up to

an easily omputablefator, thepartition funtionfor aferromagneti Ising systemwith external

eld. Jerrum and Sinlair [7 ℄ have exhibited an FPRAS for omputingthe partition funtion of

suh systems,fromwhihit followsthat theregion 1 is tratable. Morepreisely:

Theorem 1 For any xed , satisfying 1 there is an FPRAS for (;;1)-Partition.

More generally, there is an FPRAS for (;;)-Partition provided, in addition, and

p

= (or and

p

=).

Proof. One we have provided a translation between the terminology of the urrent paper and

that of [7℄, it will be seen that the existene of an FPRAS is immediate from [7 , Thm. 5℄. (The

latter theoremsimplyassertstheexisteneof anFPRAS forestimatingthepartitionfuntionofa

ferromagneti Isingsystem.)

First, a brief desription of the Ising model. The Ising model is a two-spin model in whih

interations are symmetri under interhange of the two olours (spins): in our terminology =

= . In the ferromagneti Ising model, like spins are favoured over unlike, i.e., 1. There

may be an external (or applied) eld, that auses one olour to be favoured over the other: in

our terminology 6=1. The interations areallowed to vary from edgeto edge, provided they are

all ferromagneti. Thuswe mayhave aseparate matrixA

ij

assoiatedwith eah edge fi;jg 2E,

providedeahmatrix individuallysatises theonditionsstatedabove (diagonal entriesequaland

notlessthan1.) Theinterations withtheexternaleldmayalso vary,i.e., thevetor bmayvary

from vertex to vertex. However, one olour must be uniformly favoured over the other; in other

wordsthe parametermust beuniformlyat least1,oruniformlyat most1.

Inspetingequation(2), weseethattheaforementionedonditionsaremet,providedonlythat

(=) d(k)

= (=)

d(k)=2

is uniformlyat least 1 or at most 1. This willertainly be the ase if

=1. But itwillalso holdintheother situations identiedinthe statement ofthetheorem.

3

In order to give the detailsof the redutionfrom (;;)-Partition to Theorem5 of [7℄, we

need to show how to enode the input, that is, G, , and the quantities (=)

d(k)=2

as binary

3

approximation to isused,sine itself maybeirrational.)

One nal tehnial point onerning [7 , Thm. 5℄. In the proof of that theorem it is assumed

that the interation of the external eld with spins is uniform over all sites, whereas we require

here a non-uniform (though onsistently oriented) interation. The proof was organised in this

way for simpliity of presentation. The lean x is to routinely amend the proof by introduing

expliitindividualinterationstrengthsat thevarious sites. However, an alternative xthat does

not involve delvinginto theoriginal proof isto redue the aseof varyinginterationstrengthsto

thatofxed. Inpartiular,suppose"isourdesiredaurayparameterandonsideraninstaneG

with,foreah vertex v,an interationstrength

v

1. Let

Æ =

"

ndmax v

ln

v e

;

=1+Æ,and

z=

( 1)Æ

1++Æ

=

+1

+

1:

The Ising partition funtionfor thisinstane is losely approximatedbythe partition funtionof

a new instanein whihthe graph,G

0

,isformedfrom Gbyattahing

r v

: =

ln( v

=)

z

pendant edges to eah vertex v and giving eah vertex interation strength . To see that the

approximation is suÆientlylose, note that the relative weight of olouring v green rather than

blue inG

0 is

:

=

+1

+

r

v

:

Thusthedenitionsguarantee

e "=n

v

e

"=n

v :

as required.

Remark 1 When =1, expression (2) fatorises and the (exat) omputation of Z istrivial.

Remark 2 Anothersituationin whih(=)

d(k)

isassured tobeuniformlyatleast1 oratmost1

is when d(k) is onstant, i.e.,G is regular.

The parameter values not overed by Theorem 1, i.e., > and >

p

= (or < and

< p

=) present a onundrum. These orrespond to a situation, whih may be physially

unrealisti,inwhih some vertiesinlineto one olourand otherstothe other. Ontheonehand,

there is noobvious barrierto FPRASabilitywhen thisours. Onthe otherhand, theproof of[7,

Thm. 5℄ ertainly breaks down. The issue is that the quantity tanhB in [7 , eq. (2)℄ will be of

inonsistent sign, leading to negative weights w(X) in [7, eq. (3)℄. In thissituation, the so-alled

\subgraphs world"proessis no longerwelldened,asvarious\probabilities"beome negative.

4 The \antiferromagneti region" is intratable

Let

AP

be the approximation-preserving redution from [5 ℄. Let #Sat and #LargeCut be

Instane. A Boolean formula'inonjuntive normalform (CNF).

Output. The numberof satisfyingassignments to '.

Name. #LargeCut.

Instane. A positiveinteger k and a onnetedgraphG inwhih every ut

4

hassizeat mostk.

Output. The numberof size-k uts of G.

An AP-redution from #Sat to #LargeCut appears in [7 ℄.

5

For ertain , and (see

Lemma 2)wewillgive anAP-redution from#LargeCutto(;;)-Partition. The

ombina-tion of these redutions implies #Sat

AP

(;;)-Partition whih in turn impliesthat there

annot bean FPRAS for(;;)-Partitionunless NP=RP (seeSetion 3of [5℄).

Lemma 2 Let , and be xed parameters satisfying 0< <1, 0< <1 and >0. Then

#LargeCut

AP

(;;)-Partition.

Proof. Let k and G=(V;E) be an instane of #LargeCut and let n denote jVj and m denote

jEj. We wish to onstrut an instane G

0

=(V

0 ;E

0

) of (;;)-Partition. In order to make the

redution expliit, we will need to dene a quantity s whih depends upon , , and n. The

readershouldthinkof sassimply being asuÆientlylargepolynomialinn. Forompleteness,let

bea positive integer suh thatthequantity

: =

(min(;))

2

(max (;))

1

exeeds 1. It willthensuÆeto let sbe thesmallestintegersatisfyings

s=(2)

whih isat least

max 0

B B B

n+6

lg

1

;

ln

2 2n

(max(1;)) 2n

n2 5

() m

n

ln

1

max(;)

;

2ln max (;1=)n2

n+5

ln

; n

1

C C C A :

WenowgivetheonstrutionofG

0

. ForeveryvertexuofGletA u

andB

u

bedisjointsets ofsizes.

Let

V 0

= [

u2V A

u

[B

u

and

E 0

= [

u2V A

u

B

u !

[ 0

[

(u;v)2E [

i2f1;:::;sg f(A

u [i℄;A

v [i℄);(B

u [i℄;B

v [i℄)g

1

A :

Let (G

0

) denote the set of all two-spin ongurations on G

0

. For any subset W (G

0 ), let

Z W

(G 0

)denotetheontributionto Z(G

0

)orresponding toongurationsinW. Aonguration

4

Reallthata\ut"ofagraphisanunorderedpartitionofitsvertexsetintotwosubsetsandthatthesizeofthe

utisthenumberofedgesbetweenthetwosubsets. 5

Thedenitionof #LargeCutmayseemunnaturalbeause itisnoteasyingeneralto verifythepromisethat

noutsexeedingsize-kexistintheinputgraph. However,theredutionin[7 ℄anbeviewedasproduinganinput

u

B u

isoloured withthe otherspin. Every utof Gorrespondsto exatlytwo fullongurations:

If u andv areinthesame part oftheutthen A

u

and A

v

areoloured withthesame olour. If

is a fullongurationorresponding to asize-j utthen

Z fg (G 0 )= Y fi;jg2E 0 (i) 0 A(j) Y k2V 0 b 0

(k)=()

s(m j)

sn :

LetN bethenumberofsize-kutsofGandletCbethesetoffullongurationswhihorrespond

to size-k uts. Let

=2()

s(m k) sn ; so Z C (G 0

)=N :

We willnowshow

Z (G 0 ) C (G 0

)2

4

( +2

n

Z(G

0

)): (3)

Equation (3) implies

N Z(G 0 ) N + 1 4 : (4)

Toseethis,onsiderrsttheaseN =0. InthisaseZ(G

0

)=Z

(G 0

) C

(G 0

),soEquation(3)gives

Z(G 0

)(1 2

(n+4)

)2

4

and therefore 0Z(G

0

)= 1=4 soEquation (4)holds. If N >0 thenEquation (3)gives

Z(G 0

)=Z

C (G

0

)+Z

(G 0

) C

(G 0

)Z

C (G 0 ) 1+ 1 16N + 1 16N Z(G 0 ): Thus Z(G 0 ) 1+ 1 16N 1 1 16N Z C (G 0

)(1+

1 4N )Z C (G 0 );

whihimpliesEquation (4). FromEquation (4), we ndthat

N = Z(G 0 ) : (5)

Also, theoorfuntioninEquation(5)doesnotdistorttheauray overlymuh: An

approxima-tion to Z(G

0

) gives anapproximation to N. The detailsaboutthe auray of theapproximation

are thesame asthose intheproofof Theorem3of [5 ℄.

So,toonludetheproofweproveEquation(3). Wedo thisbysplitting(G

0

) C intoseveral

(potentially overlapping) sets and then summingthe partition funtionover these sets. Let F be

thesetoffullongurationsorrespondingtoutsofsizelessthank. Thensinethere areat most

2 n uts, Z F (G 0

)2

n () s 2 6 : (6)

The seond inequalityin Equation (6) follows from thefat that s is at least the rst term inits

denition.

Foru2V,leta u

bethesetofongurationsinwhihA

u

hasat leasts=greenvertiesandat

least s=blue verties. Let a=[

u a u . Z a u (G 0

)2

2ns

max (1;)

2ns

max(;)

ss=

2

6

at most2 2ns

. Eah ofthe2nsvertieshasweight at mostmax(1;). Alledge-weightsareat most

one, but eah of the s verties in B

u

has weight at most max(;)

s=

. The seond inequality in

Equation (7) followsfrom thefatthat sisat least theseond terminits denition.

Forw2[

u2V B

u ,let a

0 w

betheset of 2(G

0

) ain whihat least halfof theedges fromw

to [

u A

u

are monohromati. Let a

0 =[ w a 0 w

. We willshow

Z a 0 w (G 0

)2

(n+5) Z(G

0

)=(ns): (8)

Wewillusethefollowingnotationto establish(8). Foraonguration,and avertexwof G

0 ,let

nw bethe restrition of to V

0

fwg. Let R be the restritionsof ongurations in (G

0

) a

to V

0

fwg. In partiular,

R=fnwj 2(G

0

) ag:

Forevery 2R , let

Z R fg = Y fi;jg2E 0 i6=w;j6=w (i) 0 A(j) Y k2V 0 k6=w b 0 (k): Then Z a 0 w (G 0 ) X 2R Z R fg

max(;)

(1 1=)s

max (1;):

Also,

Z(G

0

)Z

(G 0

) a a

0 w (G 0 ) X 2R Z R fg

min(;)

(s=)+n min(1;): So Z a 0 w (G 0 )

max(;)

(1 1=)s

max(1;)

min(;)

(s=)+n min(1;) X 2R Z R fg min(;) (s=)+n min(1;)

max(;)

(1 1=)s

min(;)

(s=)+n

max(;1=)Z(G

0 )

max(;)

1

min(;)

2 ! s= max(;1=)Z(G 0 ) 2 (n+5) Z(G 0 )=(ns):

The seond-to-last inequality uses n s= and the nal inequality follows from the fat that

s

s=(2)

and the fat that s is at least the third term in its denition. Thus, Equation (8) is

established.

Foru2V,letb u

be theset ofongurationsinwhihB

u

hasatleasts=green vertiesandat

least s=blue verties. Let b=[

u b

u

. By analogyto Equation (7), we get

Z b

u (G

0

)2

6

=n: (9)

For w2[

u2V A

u ,let b

0 w

bethe setof 2(G

0

) bin whih at leasthalfof the edgesfrom w

to [

u B

u

are monohromati. Letb

0 =[ w b 0 w

. By analogyto Equation (8), we get

Z b 0 w (G 0

)2

(n+5)

Z(G

0

Z (G

0

) C

(G 0

)Z

F (G

0

)+Z

a (G

0

)+Z

a 0

(G 0

)+Z

b (G

0

)+Z

b 0

(G 0

): (11)

To see that (11) holds, onsiderany onguration whih is not in a[a

0

[b[b

0

. Consider any

vertexu ofG. Sine 62a

u

,more than1 1=of thenodesinA

u

have a ertainolour. So,sine

62a

0 w

forany w2B

u

, all of B u

is oloured with the other olour. Finally,sine 62b

0 w

for any

w2A

u

,all of A u

isoloured withthesame olour. We onludethat is full,soit iseither inF

or inC.

Lemma2 hasthefollowingonsequene

Theorem 3 Let , and bexed parameters satisfying 0<<1, 0< <1 and >0. Then

thereis no FPRAS for (;;)-Partition unless NP=RP.

5 An additional tratable region

Theorem3 showed thatthere isunlikelyto bean FPRASfor(;;)-Partitionwhen0< <1

and 0 < < 1. Theorem 1 showed that in theregion 1 there is an FPRAS. In thissetion

we willassumethat<1and either >1or >1. Our aim isto identifyan additionalregion

where there isstillanFPRAS.The FPRASis basedon thesimulationof thesingle-siteheat-bath

Markovhain,whih isstudiedin Setion5.1.

5.1 Rapid mixing within the region

The single-site heat-bath hain for the two-state partition funtion works as follows. Given a

(onneted) n-vertex input graph G = (V;E), (G) is the state spae (the set of ongurations,

i.e.,thesetofall2-olouringsofG,inludingimproperolourings). Fromaonguration2(G),

thehainrsthoosesavertexx2V u.a.r. Let(x!g)denotetheongurationobtainedfrom

byolouringx green,and (x!b) theongurationorrespondingto olouringx blue. Let

p(x;g)()=

Z

f(x!g)g (G)

Z

f(x!g)g

(G)+Z

f(x!b)g (G)

and p(x;b)()=1 p(x;g)(). The new state istaken to be(x!g) withprobabilityp(x;g)()

and (x!b) otherwise.

Wewillusepathoupling[2℄toprovethatsingle-siteheatbathisrapidlymixing. Weadoptthe

notationfrom[3 ℄. LetS (G)

2

bethesetofpairsofongurationswithHamming-distane1. If

and

0

areongurationswhihdisagreeonlyatvertexvthen (;

0

)(theproximityofand

0 )is

denedtobethedegreeofvinG,whihwedenote[v℄. Thedistanefuntionisgivenintheusual

way: Foreah pair(;

0

)2(G)

2

,P(;

0

) isthesetof allsequenes =

1 ;

2 ;:::;

r 1

; r

=

0

with (

i ;

i+1

)2S fori2f1;:::;r 1g. Thedistane funtionis denedby

Æ(;

0

)= min

P(; 0

)

r 1

X

i=1 (

i ;

i+1

); (12)

whihan be writtenas

Æ(;

0

)=

X

v2V I

v

(;

0

where I v

(;) is the indiator for the event that and dier at vertex v, i.e., the event

(v) 6=

0

(v). Note that if Z

fg

(G) and Z

f 0

g

(G) are both positive, then there is a hain =

1 ; 2 ;:::; r 1 ; r = 0

whih minimises the right-hand-side of (12), and for whih eah

i has Z f i g

(G) >0. For example, if >0 then the hain is onstruted by rst olouring some green

vertiesblueand thenolouringsome blue vertiesgreen.

We will now dene a oupling whih, for every (X

0 ;Y

0

) 2 S and every (X

1 ;Y

1

) 2 (G)

2 ,

givestheprobabilityof ajoint transition from(X

0 ;Y

0

) to (X

1 ;Y

1

). SupposethatX

0

and Y

0 dier

on v. The oupling will be the optimal one, subjet to the assumption that the same vertex x

is seleted in X

0

and in Y

0

. First, a vertex x is hosen u.a.r. If x is not a neighbour of v then

the same olour is hosen for x in X

1

and in Y

1

. If x is a neighbour of v, then with

probabil-ity min(p(x;g)(X

0

);p(x;g)(Y 0

)), X

1

= X

0

(x ! g) and Y

1

= Y

0

(x ! g), and with probability

min(p(x;b)(X

0

);p(x;b)(Y 0

)), X

1

= X

0

(x ! b) and Y

1

= Y

0

(x ! b). The rest of the oupling is

fored bytherequirementthat themarginalsbe orret.

Thepath ouplinglemma in[2 , 3℄guarantees that thehainisrapidlymixingas longasthere

is an " n

> 1=poly (n) suh that for every pair (X

0 ;Y

0

) 2 S, E(Æ(X

1 ;Y

1

)) (1 "

n )Æ(X 0 ;Y 0 ).

In partiular, the total variation distane between the t-step distribution of the hain and the

stationary distribution isat most"after onlyln(n" 1

)=" n

steps.

So, suppose that X

0

and Y

0

dier at vertex v. For onreteness, suppose that X

0

(v) is blue.

Forevery neighbourwof v, letb

w

denote thenumberofneighboursof w,other thanv,whih are

oloured blue in X

0

(or equivalently, in Y 0

). Letg

w

denote the numberof neighboursof w,other

than v whihare olouredgreen inX

0

. Thus[w℄=b

w

+g

w

+11. Let

f w (i)= [w℄ i [w℄ i + i = 1

1+()

i

[w℄ :

Note thatp(w;b)(X

0

)=f

w (g

w

)and p(w;b)(Y

0

)=f

w (g

w

+1).

We requirethat, forevery pair(X

0 ;Y

0

)2S whih disagree onvertex v,

1 1 n [v℄+ 1 n X wv jf w (g w

+1) f

w (g

w

)j[w℄(1 "

n

)[v℄: (14)

Equation (14) follows from (13). The probabilitythat X

1

and Y

1

dier on v is equal to 1 1=n,

whih is the probability that v is not hosen. If w is hosen then the probability that X

1 and

Y 1

dier on w is jp(w;b)(Y

0

) p(w;b)(X

0

)j. The [v℄ on the right-hand-side of (14) represents

Æ(X 0

;Y 0

). In order to establish(14), it suÆesto showthatforevery neighbourw ofv,

jf w

(g w

+1) f

w (g

w

)j[w℄; (15)

for some <1, dependingonlyon , and . Then we an take "

n

=(1 )=n. We willidentify

regions where(15) holds. We startbyfousingon theasewhere >1>. (Thease >1>

is symmetri to thisase, and willbe handledbelow.)

Sine<1,f

w

(i)is aninreasingfuntionof i,and sojf

w

(i+1) f

w

(i)j=f w

(i+1) f

w (i)

foralli. Tosatisfy(15)foralln, G,v andw,itissuÆientto showthat,forallintegersi0and

all real 1,

1

1+y

i+1 x

1

1+y

i x

: (16)

where y= and x

=

. Ofourse, we onlyreallyrequire thisinequalityforinteger values

Foranyxed i0,let i

=y x

. So

i

isa dereasingfuntionof , withderivative

i

ln.

The derivativewithrespet to ofthelogarithm of

1

1+y

i 1

1+

i

=

(1 y)

(1+y

i

)(1+

1 i

)

is

1

ln

1

1+y

i

1

1+

1

i

;

adereasingfuntionof. Thisontinuousfuntiontendsto+1as!0,andtendsto ln <0

as!1. Thereforethereisaunique(nite)positivevalue

i

forwhihthisderivativevanishes,

and at whih the maximum value of the left-hand sideof inequality(16) with respet to the real

variable isattained.

To map the boundary of the region of (;;)-spae for whih (16) holds, we an therefore

solve the following simultaneous pair of relations, expressing the onditions that the maximising

value of yieldsa valueat most :

1

i

ln

1

1+y

i

1

1+

1

i

= 0; and (17)

1

1+y

i 1

1+

i

i

: (18)

Eliminating theexpliitourrenes of

i

,we ndthefollowingquadratiinequalityfor

i :

2 i

y+

i

(1 y)

ln

10: (19)

Solvingthisquadratifor1=

i

impliestheinequality

i

2ln

1 y+

p

(1 y)

2

+y(2ln)

2

: (20)

Sine

i

is dereasing in i, (20) is satised for all i 0 if and only if it is satisedfor i = 0.

Sine we anhoose<1arbitrarilyand theright-hand sideof(20)isan inreasingfuntionof,

we an replae by1 in(20) butmakethe inequalitystrit, i.e.,

i

0

<C(ln) (21)

where

C(z)=

2z

1 +

p

(1 )

2

+4z

2

: (22)

Equation (17)yields

ln

0

=D(

0

); (23)

where theright-handsideis aninreasing funtionof

0 ,

D(z)=

1

1+z

1

1+z

1

1

: (24)

We an use (21) and (23) to derive an upper bound on as a funtion of and . Sine

=

0

0

0

as required. Our nalboundisgiven by

<C(ln)e

D(C(ln))

: (25)

Notethattheleft-hand-sideof(18)isadereasingfuntionof fortheritialase,i=0. Thus

we an get a simpler (but worse) boundbyonsidering theextreme ase, = 0. Here, C(z) =z

and D(z)=1+z,so(25)givesthe bound: <eln.

The region denedby <1, >1 is symmetri to the regionthat we have just onsidered.

In partiular,theblue{greensymmetry yieldsthefollowingrelationships:

p(w;g)(X

0

) = 1 p(w;b)(X

0

)=1 f

w (g

w

)=1=(

1

gw

[w℄ gw

+1)

= 1=(1+

1

b w

+1

b w

+1 [w℄

)=

^ f

w (b

w

+1);

and

p(w;g)(Y

0

)=1 f

w (g

w

+1)=1=(1+

1

b w

b

w [w℄

)=

^ f

w (b

w );

where ^ f

w

isderivedfrom f

w

byreplaing;; by

1

;; respetively. Our requirement is

j ^ f

w (b

w +1)

^ f

w (b

w

)j[w℄

n ;

whihissymmetritoEquation(15)and thisisthereforemetwithintheregiondenedby <1,

>1and

1

<C(ln)e

D(C(ln))

(fromEquation (25)).

Thus,we have shownthe followingresult.

Lemma 4 The single-site heat-bath Markov hain for the two-state partition funtion is rapidly

mixing within the regions dened by: <1 and either

1. >1 and <C(ln)e

D(C(ln))

, or

2. >1 and 1=<C(ln)e

D(C(ln))

,

where funtions C andD are dened in Equations (22) and(24).

Corollary 5 Thesingle-siteheat-bath Markov hain for the two-statepartition funtionisrapidly

mixing within the regions dened by: <1 and either

1. >1 and <eln, or

2. >1 and 1=<eln.

Figure 2 shows a portion of the bounding surfae of the region desribed in Lemma 4. The

rapid-mixingregionliesbelowthissurfae,whihrepresentsthelogarithmofthemaximalwithin

the region1<2and 0 <1.

5.2 Reduing approximate ounting to sampling

Lemma4showedthatthesingle-siteheat-bathMarkovhainisrapidlymixingwithinthespeied

region. Thus, thisMarkovhain provides a fully-polynomialapproximatesampler(FPAS) forthe

### 1.2

### 1.4

### 1.6

### 1.8

### 2

### 0.2

### 0.4

### 0.6

### 0.8

### 0

### 5

### 10

PSfragreplaements

log

Figure 2: Logarithmof themaximalintheregion 1< 2 and 0 <1.

given a onnetedgraphG=(V;E) and anauray parameter"2(0;1℄, outputsa onguration

2(G)aording to a measure

G

whihsatises d

TV (

G ;

G

)" where

G

()=

Z fg

(G)

Z(G)

and d

TV

denotes total variation distane. Usingthe method of Jerrum,Valiant and Vazirani [8 ℄,

itisstraightforwardto showthattheFPAS anbeturnedintoan FPRASfor(;;)-Partition

withinthe region. Thus,weobtainthefollowingtheorem.

Theorem 6 There is an FPRAS for (;;)-Partition when the xed parameters , and

arein the regions dened by <1 and either

1. >1 and <C(ln)e

D(C(ln))

, or

2. >1 and 1=<C(ln)e

D(C(ln))

,

where funtions C andD are dened in Equations (22) and(24).

Corollary 7 There is an FPRAS for (;;)-Partition when the xed parameters , and

arein the regions dened by <1 and either

1. >1 and <eln, or

2. >1 and 1=<eln.

The speial ase of =1 gives a numerial lower boundof about 1.3211 for and respetively.

Toaidthereader,weprovidesomedetailsshowinghowtoturntheFPASintoanFPRASwithin

Region 1. Region 2 an be handledbya similarargument, exploitingthe blue{greensymmetry as

1 m i 1 i

Thus, ourjobis to approximate

Z(G)=Z(G

m )= Z(G m ) Z(G m 1 ) Z(G m 1 ) Z(G m 2 ) Z(G 1 ) Z(G 0 ) Z(G 0 ):

The quantity Z(G

0

) is easy to ompute,so themaintaskisto estimate thequantity

% i = Z(G i ) Z(G i 1 ) :

Supposethat e

i

is theedge (x

i ;y

i

). For spins, let

s i 1

denote theset of all ongurations in

(G i 1

) in whih x

i

and y

i

are assigned spin s. Let

i 1

denote the set of all ongurations in

(G i 1

) inwhih x

i

andy

i

areassigned dierent spins. Then

% i = P 2 b i 1 Z fg (G i 1 )+ P 2 g i 1 Z fg (G i 1 )+ P 2 i 1 Z fg (G i 1 ) P 2(G i 1 ) Z fg (G i 1 ) : (26)

We need a method for estimating %

i

. Consider the followingexperiment (whih makes sense,

sine ;1 ): Sample a onguration 2 (G

i 1

) with weight

G i 1

(). If 2

b i 1

, output

\yes". If 2 g

i 1

, output \yes" with probability = and \no" otherwise. If 2

i 1

, output

\yes"withprobability1= and\no"otherwise. From(26),wededuethattheprobabilitythatthe

algorithm outputs \yes" is% i

=. Thus,we an aurately estimate %

i

by applyingtheexperiment

to several outputs of the FPAS. (We need several outputs beause the FPAS has measure

G i 1

not

Gi 1

.) Also, we an onlude that %

i

. It is known [8℄ that as long as %

i

1=poly (n),

where n=jVj,thentherequirednumberof samplesisonlypolynomialinnand"

1

,sowegetan

FPRAS. Detailsan befoundintheproofof Proposition3.4of [6 ℄.

We onlude this setion by showing that, in the region of interest, %

i

1=(1+). To start

with,we observe that

Z g i 1 (G i 1

)Z

i 1 (G i 1 ): (27)

To see (27), onsiderthe injetion whih maps every 2

g i 1 to 0 2 i 1

byolouring y

i blue.

Sine 1 and 1,Z

f 0

g (G

i 1

)Z

fg (G

i 1

). Thus, from (26), sine1,

% i Z b i 1 (G i 1

)+Z

i 1 (G i 1 ) Z g i 1 (G i 1

)+Z

b i 1

(G i 1

)+Z

i 1 (G i 1 ) Z b i 1 (G i 1

)+Z

i 1 (G i 1 ) Z b i 1 (G i 1

)+(1+)Z

i 1 (G i 1 ) 1 1+ :

6 An additional intratable region

In the previous setion, we saw that the tratable region extends beyond that dened by the

hyperbola 1. The main result of thissetion is that the intratable region extends beyond

the squaredened by <1 and <1. Speially,we show:

Theorem 8 Let bea suÆiently small onstant ( =10

7

will do), and suppose that 1

1 +, 0 and

1

2

2. Then there is no FPRAS for (;;)-Partition unless

roles of and reversed.

The region overed by Theorem 8 is admittedly small. The estimates in the proof ould

un-doubtedly be tightened with a view to expanding the range of parameter values overed by the

theorem. However, sine our main aim is to unover some intratable region of positive volume

lying outsidethe square,we shallinsteadaim to keep thetehnialompliationsto a minimum.

Ourstartingpointisaninapproximabilityresultonerningindependentsetsinboundeddegree

graphs. It iswellknownthat thatit isNP-hard to determinethesizeof a maximumindependent

set in a graph of maximum degree 4. A result of Berman and Karpinski [1 , Thm 1(iv)℄ tells us

more:

Proposition 9 For any ">0, it isNP-hard to determinethe size of a maximumindependent set

in a graph G to withinratio

73

74

+",even when G isrestrited tohave maximum degree4.

(By \determiningthesize:::withinratio"wemeanomputinganumber

^

k suhthatk

^

k k,

wherekisthesizeofamaximumindependentsetinG.) Thepossibilityofestablishingresultssuh

as Proposition9hasbeenopenedup bythe theoryof \polynomially hekable proofs"(PCPs).

Proof of Theorem 8. Our proof strategy is to design a redution that takes a graph G = (V;E)

of maximum degree 4 and forms a graph G

0

with the followinginformal property: The partition

funtion Z(G

0

) of thenew graphG

0

determinesthesizeofthelargest independentset inGwithin

ratio 0:99. Sine suh a tight performane guarantee is preluded by Proposition 9, this will be

enough to establishthe result.

We now desribe the onstrution of G

0

from G. For every vertex u of G let A

u

be a distint

set of sizer,where r is aonstant to be determinedlater. Thendene

V 0

= [

u2V A

u

and

E 0

= [

fu;vg2E A

u

A

v :

Presently,we shallargue thatthepartition funtionof G

0

isboundedbelow andabove asfollows:

Z(G 0

)(1+)

rk

(28)

and

Z(G 0

)

k X

i=0

n

i

(1+)

ri r(n i)

X

j=0

r(n i)

j

j

(1+)

r 2

m j

j

; (29)

where n=jVj, m =jEj and k is thesize of a maximum independent set inG. It transpires that

when the parameters , and satisfy the onditions of the theorem, these inequalities loate

lnZ(G

0

) rather aurately: see inequality (32). Thus a good estimate for Z(G

0

) provides a good

estimate fork.

The lower bound(28) isthe easier of thetwo to justify. LetI be any independent setin Gof

sizek. Thelowerbound(28)omesfromonsideringjusttheongurationswhihassignbluetoall

vertiesin

S

u2VnI A

u

. Sine 1 and thereare nogreen-greenedges, every suh onguration

ontributesatleast

j

tothepartitionfuntion,wherejisthenumberofgreenvertiesin. Sine

the greenvertiesarefreelyseleted from a setof sizerk,inequality(28)is nowimmediate.

The upper bound (29) is not muh more diÆult, if viewed in the right way. A base for a

u

forevery blokA

u

ontainingagreenvertex, eitheruisinI oruisadjaentto avertexinI.

Everyongurationhasat leastone base,sinewemaytakeI to beanymaximalindependentset

withinthe subgraphofG induedbythevertexset

fu2V :A

u

ontainsat least one greenvertex g:

Itisonvenienttothinkoftheterm\base"asapplyingbothtothevertexsetI inGandthevertex

set S

u2I A

u

inG

0 .

For eah base, we shall estimate the total weight of ongurations with that base, and then

sumoverallpossiblebases. Thiswillleadtooverounting,sineeahongurationhasmanybases

in general. This is ne, as we are shooting for an upper bound. The key observation is that, in

any ongurationwith base I, eah green vertexlying outside the base isadjaent to some green

vertex lying inside. Thus the number of green-green edges is at least as large as the number of

green non-baseverties.

Withtheseonsiderationsinmind,theformulain(29)maybereadleft-to-rightasfollows: (i)i

ranges over thepossiblesizes ofa base, k beingan upperboundsineanybase isan independent

set inG;(ii) n

i

isaboundonthenumberofbases ofsizei;(iii)(1+)

ri

ountsolourings ofthe

base-verties;(iv)jisthenumberofgreenvertiesamongthenon-baseverties,rangingfromj=0

(no green verties) to j =r(n i) (allgreen); (v)

j

omes from thej green verties; and nally

(vi) (1+)

r 2

m j

j

isan upperboundon edgeweights, sine there mustbe at leastj green-green

edges.

Next,we simplifytheupperbound(29) byapproximatingthe two sums:

Z(G

0

)

k X

i=0

n

i

(1+)

ri

(1+)

r 2

m rn X

j=0

rn

j

j

j

=(1+)

r 2

m k X

i=0

n

i

(1+)

ri

(1+)

rn

(1+)

r 2

m

(1+)

rn 2

n k X

i=0

(1+)

ri

(1+)

r 2

m

(1+)

rn 2

n+1

(1+)

rk

; (30)

where thenalinequalityassumes(as willertainlybe thease) that(1+)

r

2.

Taking logarithmsof(28) and (30)we may sandwihlnZ(G

0

) asfollows:

rkln(1+)lnZ(G

0

)(1+

1

+

2

+

3

)rkln(1+); (31)

where

1

= r

2

mln(1+)

rkln(1+)

;

2 =

rnln(1+)

rkln(1+)

; and

3 =

(n+1)ln2

rkln(1+)

:

Now m 2n sine G has maximum degree 4. Furthermore, k

1

4

n sine G is 4-olourable

2

2 and0 10 ,wehave thefollowingboundson

1 ,

2

and

3 :

1

2rn

kln(1+)

8r

ln(1+)

0:002

2

n

kln(1+)

4

ln(1+)

0:001

3

(1+o(1))4ln2

rln(1+)

0:007;

for suÆientlylargen.

Thusfrom (31),

rkln(1+)lnZ(G

0

)1:01rkln(1+);

and hene

0:99k

0:99lnZ(G 0

)

rln(1+)

k: (32)

Finally, suppose , and are as stated in the theorem, and that there is an FPRAS for

(;;)-Partition. Then we would be able to ompute an approximation L to lnZ(G

0

) within

additive error 1 (say), inpolynomial time, with high probability. But then 0:99L=1000ln(1+)

(rounded to the nearest integer) would approximatethe sizeof a maximum independent set in G

to withinratio uniformlybetter than

73

74

. By Proposition9,thisentailsRP=NP.

Referenes

[1℄ P. Berman and M. Karpinski, On some tighter inapproximabilityresults (extended abstrat),

Proeedings of the 26th EATCS International Colloquium on Automata, Languages and

Pro-gramming (ICALP),(Springer-Verlag, 1999)200{209.

[2℄ R.BubleyandM.Dyer,Pathoupling: AtehniqueforprovingrapidmixinginMarkovhains,

Proeedings ofthe 38th IEEEAnnualSymposium onFoundations of ComputerSiene,(IEEE,

LosAlamitos,1997) 223{231.

[3℄ M. Dyer and C. Greenhill, Random walks on ombinatorial objets, in J. D. Lamb and D. A.

Preee, eds.,Surveys in Combinatoris 1999,vol. 267 ofLondon Mathematial Soiety Leture

Note Series, (CambridgeUniversityPress,Cambridge, 1999) 101{136.

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