Approximate Capacities of Two-Dimensional Codes by Spatial Mixing

0.2 MB·

by  Yi-Kai Wang

Approximate Capacities of Two-Dimensional Codes
by Spatial Mixing
Yi-KaiWang∗† YitongYin∗‡ Sheng Zhong∗§
4
1
0 Abstract
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Weapplyseveraltechniquesdevelopedinrecentyearsforcountingalgorithmsandstatisticalphysics

b
to study the spatial mixing property of two-dimensional codes arising from local hard (independent
e
set) constraints, including: hard-square,hard-hexagon,read/write isolatedmemory(RWIM), and non-
F
attackingkings(NAK).Fortheseconstraints,theexistenceofstrongspatialmixingimpliestheexistence
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ofpolynomial-timeapproximationscheme(PTAS)forcomputingthecapacity. Theexistenceofstrong
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spatialmixingandPTASwaspreviouslyknownforthehard-squareconstraint.Weshowtheexistenceof

] strongspatialmixingforhard-hexagonandRWIMconstraintsbyestablishingthestrongspatialmixing
T
along self-avoiding walks, which implies PTAS for computingthe capacities of these codes. We also
I showthatfortheNAKconstraint,thestrongspatialmixingdoesnotholdalongself-avoidingwalks.
.
s
c
[
1 Introduction

2
v
1 We consider codes consisting of two-dimensional binary patterns of 0’s and 1’s arranged in rectangles,
3 satisfying constraints of forbidding certain local patterns. The capacity (or entropy) of a two-dimensional
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codemeasuresthemaximumratethattheinformation canbetransmittedthrough thisrepresentation. Com-
4
. putation of capacities of two-dimensional codes is highly nontrivial and has implications in information
1
theory [2, 4, 24, 9, 15, 5, 7, 20, 17, 14], probability [17, 14], statistical physics [6] and computation the-
0
4 ory [8]. A recent breakthrough [8] in symbolic dynamics shows that capacities of two-dimensional codes
1
characterize the class of Turing-computable reals. It is then a natural and fundamental problem to further
:
v classify the capacities of two-dimensional codes which can be computed efficiently, that is, in polynomial
i
X time.
r Inthispaper,weconsidertheclassoftwo-dimensionalcodeswhich: (1)canbedescribedbylocal(up-to
a
distance2)forbidden patterns,and(2)arisefromhard(independent set)constraints. Thisgivesusprecisely
the following well-studied constraints for two-dimensional codes: hard square (HS) [4, 17], hard hexagon
(HH) [2], read/write isolated memory (RWIM) [5, 7] and non-attacking kings (NAK) [24]. Among these
constraints, the capacities of hard-square and hard-hexagon are known to be efficiently computable [2, 17,
14]. Theefficiencyofcomputation ofthecapacityofRWIMorNAKisstillunknown.
It was first discovered in [6] an intrinsic connection between efficient computation of capacities of
two-dimensional codesandtheproperty ofbeingstrongspatialmixing(SSM),anotionoriginated fromthe
phasetransitionofcorrelationdecayinstatisticalphysics. Beingstrongspatialmixingmeansthecorrelation
∗StateKeyLaboratoryforNovelSoftwareTechnology,NanjingUniversity,China.
†SupportedbyNSFCgrants61272081and61321491. Email:[email protected].
‡SupportedbyNSFCgrants61272081and61321491. Email:[email protected].
§SupportedbyRPGE,NSFC-61321491,andNSFC-61300235.Email:[email protected].
1
betweenanybitsthatarefarawayfromeachotherdecaysrapidly, andhencethecapacity canbeefficiently
estimatedfromlocalstructure.
In a seminar work [25], the strong spatial mixing is introduced and is proved for independent sets
of graphs along self-avoiding walks, another essential object in statistical physics [13]. Specifically, for
independent sets, the maximum degree 5 is a phase transition threshold such that strong spatial mixing
holds for all graphs with maximum degree at most 5, but there are graphs of any maximum degree greater
than 5without spatial mixing. Adirect consequence isanefficient algorithm for approximately computing
hard-square (HS) entropy, because the hard-square constraint can be interpreted as independent sets of
two-dimensional grid, whose degree is less than 5. For the more complicated constraints of HH, RWIM,
and NAK, which correspond to independent sets of graphs of degree 6 or 8, we need stronger tools than
the generic ones used in [25] to verify the existence of strong spatial mixing and efficient algorithm for
computing thecapacity, orshowevidencesayingthattheymaynotexist.
1.1 Contributions
Previously it is known that strong spatial mixing holds for the hard-square constraint [25] and there ex-
ists a polynomial-time approximation scheme (PTAS) for computing its capacity [17, 14]. To analyze the
spatial mixing of two-dimensional codes arising from the aforementioned constraints, we apply several
techniques from the state of the art of counting algorithms and statistical physics, including: self-avoiding
walktree[25],sequential cavitymethod[6],branching matrix[18],thepotentialfunction proposed in[11],
connective-constant-based strong spatial mixing [22], and the necessary condition for correlation decay
in[23]. Wemakethefollowingdiscoveries:
1. Strong spatial mixing holds forthe hard-hexagon (HH)and theread/write isolated memory (RWIM)
constraints.
2. Consequently, thereexistPTASforcomputingthecapacities ofHHandRWIMconstraints.1
3. Forthenon-attacking-kings (NAK)constraint,strongspatialmixingdoesnotholdalongself-avoiding
walks.
This gives the first algorithm with provable efficiency for computing the capacity of RWIM constraint and
thefirststrongspatial mixingresultsforbothhard-hexagon andRWIMconstraints, andalsoshowsthatthe
NAKconstraint mightnotenjoysufficientspatialmixingtosupport efficientcomputation ofcapacity.
1.2 Related work
Computingthecapacitiesfordifferentconstrainedcodeshasbeenstudiedextensively. In[2],theexactsolu-
tionofhard-hexagon entropywasgiven. Themethodintroduced in[4]connectsthenumberofindependent
setstothecapacityandshowstheexistenceandboundsforcapacitiesofcertainimportanttwo-dimensional
(d,k)run-length constraints. In[9],the(d,k)run-length constraints withpositivecapacities arefullychar-
acterized. In [15], the bounds for the capacity of three-dimensional (0,1) run-length constrained channel
is given. In [24], numerical bounds on the capacity of NAK was given. In [5], the bounds for the RWIM
capacityisgiven. In[20],beliefpropagationisusedtoanalyzethecapacitiesoftwo-dimensionalandthree-
dimensional run-length limited constraint codes. In [6], sequential cavity method was used to show PTAS
1Althoughthehard-hexagon entropyisknowntobeexactsolvableduetoitsspecialstructure[2],weremarkthatthestrong
spatialmixingofthisimportantmodelisinterestingbyitself.
2
for computing the free energy and surface pressure for various statistical mechanics models on Zd, which
covers hard-square entropy and matching. In [17], a PTASfor computing the hard-square entropy is given
byergodictheoretictechniquesandmethodsfrompercolationtheory. In[14],itisprovedthatifanynearest
neighbortwo-dimensionalshiftoffinitetypesexhibitsSSM,thenthereisaPTASforcomputingtheentropy.
The strong spatial mixing was introduced in [25] for counting algorithms. The self-avoiding walk tree
wasintroduced in[25]todeal withBoolean-state pairwise constraints, andwasgeneralized in [3]and [16]
into its full-fledged power to deal with matching, and multi-state multi-wise constraints. These techniques
wereimprovedinaseriesofworks[18,11,22,21,23].
2 Preliminaries
2.1 Two-dimensional codes from hardconstraints
A two-dimensional binary codeword is an m n matrix of Boolean (0 and 1) entries. Let P be a k ℓ
× ×
Booleanmatrix,calledapattern. Atwo-dimensionalbinarycodewordW issaidtocontainpatternP ifP is
asubmatrixofW,respectingtherelativepositions. Formally,thereexistiandjsuchthatforany1 s k
≤ ≤
and1 t ℓ,itholdsthatW(i+s 1,j+t 1) = P(s,t). Weconsiderthefollowingtwo-dimensional
≤ ≤ − −
codesdefinedbyforbidding certainpatterns.
1. Hardsquare(HS)constraint: Acodeworddoesnotcontainpatterns
1
1 1 and .
1
(cid:18) (cid:19)
(cid:0) (cid:1)
Theconstraint forbidsanyhorizontally orvertically consecutive 1’s.
2. Hardhexagon(HH)constraint: Acodeworddoesnotcontain patterns
1 0 1
1 1 , and .
1 1 0
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
Theconstraint forbidsanyhorizontally, vertically, oranti-diagonally consecutive 1’s.
3. Read/writeisolatedmemory(RWIM)constraint: Acodeworddoesnotcontainpatterns
1 0 0 1
1 1 , and .
0 1 1 0
(cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
Theconstraint forbidsanyhorizontally, diagonally oranti-diagonally consecutive 1’s. Thehorizontal
patterncorrespondstothereadrestriction: notwoconsecutivepositionsmaystore1’ssimultaneously.
The two diagonal patterns correspond to the write restriction: no two consecutive positions in the
memorycanbechanged duringonerewritingphase.
4. Non-attacking kings(NAK)constraint: Acodeworddoesnotcontain patterns
1 0 1 1 0
1 1 , , and .
1 1 0 0 1
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
Theconstraint forbidsanyhorizontally, vertically, diagonally oranti-diagonally consecutive 1’s.
3
Figure1: Localstructures ofLHH,LRWIM andLNAK
Throughoutthepaper,weassumeCons HS,HH,RWIM,NAK tobeoneoftheconstraintsdefined
asabove. LetNCons bethenumberofm n∈B{ooleanmatricessatisfyin}gconstraintCons. Weobservethat
m,n ×
the above two-dimensional codes can be equivalently defined as independent sets of certain lattice graphs.
Infact,restricting tothetwo-dimensional codesdefinedbylocalforbidden patterns (ofdimension upto2),
theyaretheallfourcaseswhichcanbedescribed asindependent sets.2
Let Z be the integer field. Let LCons = (Z2,ECons) be the infinite lattice graph with vertex set Z2
Cons
whoseedgesetE isdefinedasfollows,respectively:
EHS = ((i,j),(i+1,j)),((i,j),(i,j +1)) : i,j Z2 ;
{ ∀ ∈ }
EHH = ((i,j),(i+1,j)),((i,j),(i,j +1)),((i,j),(i+1,j 1)) : i,j Z2 ;
{ − ∀ ∈ }
ERWIM = ((i,j),(i,j +1)),((i,j),(i +1,j +1)),((i,j),(i+1,j 1)) : i,j Z2 ;
{ − ∀ ∈ }
ENAK = ((i,j),(i+1,j)),((i,j),(i,j +1)),((i,j),(i+1,j +1)),((i,j),(i +1,j 1)) : i,j Z2 .
{ − ∀ ∈ }
The lattices LHS and LHH are just two-dimensional grid lattice and hexagonal lattice, respectively. The
latticesLHH,LRWIM,andLNAK areshowninFigure1.
WedenotebyLCons = LCons[[m] [n]]theinducedsubgraphofLCons onthefinitevertexset[m] [n].
m,n × ×
Itiseasytoverifythatfortheconsidered constraints Cons HS,HH,RWIM,NAK ,wehave
∈ { }
NCons = #IS(LCons),
m,n m,n
where#IS(G)denotesthenumberofindependentsetsofgraphG. Moreover,eachcodewordofsizem n
satisfying constraint Cons corresponds toaunique independent setoflattice LCons such that the1’sin×the
m,n
codewordindicates theverticesintheindependent set.
Definition2.1. LetCons HS,HH,RWIM,NAK . ThecapacityofconstraintCons,denotedCCons,is
∈ { }
definedby
log NCons log #IS(LCons)
2 m,n 2 m,n
CCons = lim = lim .
m,n→∞ mn m,n→∞ mn
0 1
2Otherforbiddenpatternsofdimensionupto2mayalsodefineindependent sets,suchastheonedescribedby and
(cid:18)1 0(cid:19)
1 0
,however,thiscaseisjustaunionoftwodisjointinstancesofhardsquare.
(cid:18)0 1(cid:19)
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2.2 Spatial Mixing
Weadoptnotionsandterminologiesfromstatisticalphysicstodescribetheprobabilityspaceofindependent
sets. LetG(V,E)beagraph, whereeachvertexisinoneofthetwostates 0,1 , suchthatstate1iscalled
{ }
occupied and state 0 is called unoccupied. Each configuration σ 0,1 V indicates a subset I V
σ
∈ { } ⊆
of vertices such that a vertex is occupied means it is in I . Given a finite graph G(V,E), the uniform
σ
probability measureµ overallindependent setsofGcanbedefinedas
G
1 I isanindepdent setinG,
σ 0,1 V, µ (σ) = ZG σ
G
∀ ∈ { } (0 otherwise,
whereZ = #IS(G)isthenumberofindependent setsofG. HereZ isalsocalledthepartitionfunction,
G G
andµ iscalledtheGibbsmeasureofindependent sets.
G
GivenG(V,E),letv V beavertexandρ 0,1 Λ anindependent setofvertexsetΛ V. Letpρ
∈ ∈ { } ⊂ G,v
denotethemarginalprobability ofv beingunoccupied conditioned onconfiguration ρ,thatis
ρ
p = Pr [σ = 0 σ = ρ].
G,v σ∼µG v | Λ
Forthisconditional probability space, wesaythattheverticesinΛarefixed tobeρ.
For infinite graph G, the uniform probability measure µ cannot be defined since uniform distribution
G
cannotbedefinedoncountably infiniteset. Sowedealwithfinitesubgraphs.
Spatialmixing,alsocalledthecorrelation decay,isthepropertythattheinfluenceofanarbitrarily fixed
boundary on the marginal distribution on a vertex decreases exponentially as the distance between them
grows. Itisoneoftheessential conceptsinStatisticalPhysics.
Definition 2.2 (Weak Spatial Mixing). The independent sets of an infinite graph G(V,E) exhibits weak
spatial mixing (WSM) if there exist constants β,γ > 0, such that for every finite vertex set U V, every
⊂
v U,andanytwoindependent setsσ,τ ofthevertexboundary ∂U = w U uw E,u U ,itholds
∈ { 6∈ | ∈ ∈ }
that
pσ pτ β exp( γ dist(v,∂U)),
G,v − G,v ≤ · − ·
wheredist(v,∂U)istheshortest(cid:12)distance bet(cid:12)weenv andanyvertexin∂U.
(cid:12) (cid:12)
In order for algorithmic applications, we need a stronger version of spatial mixing, called the strong
spatialmixing,whichisintroduced in[25].
Definition 2.3 (Strong Spatial Mixing). The independent sets of an infinite graph G(V,E) exhibits strong
spatial mixing (SSM) if there exist constants β,γ > 0, such that for every finite vertex set U V, every
⊂
v U,andanytwoindependent setsσ,τ ofthevertexboundary ∂U = w U uw E,u U ,itholds
∈ { 6∈ | ∈ ∈ }
that
pσ pτ β exp( γ dist(v,∆)),
G,v − G,v ≤ · − ·
where∆ ∂U isthesetofverti(cid:12)cesonwhich(cid:12)σ andτ differanddist(v,∆)istheshortestdistancebetween
⊆ (cid:12) (cid:12)
v andanyvertexin∆.
Thedifference between WSMandSSMisthat SSMrequires that the correlation decay still holds even
withtheconfiguration ofasubset∂U ∆ofnearbyverticestov tobearbitrarilyfixed. Itiseasytoseethat
\
SSMimpliesWSM.Moreover,wehavethefollowingeasybutusefulproposition forindependent sets.
Proposition 2.4. IfaninfinitetreeT exhibitsSSM,thenallsubtreesofT exhibitSSM.
The proposition is implied by the simple observation that fixing vertices to be unoccupied effectively
prunesthetreetoanarbitrary subtree, whiletheSSMstillholds.
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2.3 Self-avoiding walktree
Ontrees,thereisaneasyrecursionformarginalprobabilities. LetT beatreerootedbyv,andv ,v ,...,v
1 2 d
thechildren ofv. Foreach 1 i d, letT denote the subtree ofT rooted byv . Foranyindependent set
i i
≤ ≤
σ ofasubsetΛofverticesinT,consider theratioRσ = (1 pσ )/pσ betweenmarginalprobabilities of
T − T,v T,v
beingoccupied andunoccupied atvertexv. Thefollowingrecursion iswellknown:
d
1
Rσ = , (1)
T 1+Rσi
i=1 Ti
Y
whereσ istherestriction ofσ onT .
i i
Theself-avoidingwalk(SAW)treeisintroducedin[25]totransformagraphintoatreewhilepreserving
the marginal probability. Let G(V,E) be a graph, finite or infinite. For each vertex u V, we fix an
∈
arbitrary order > for the neighbors of u. Let v V be an arbitrary vertex. A tree T rooted by v can be
u
∈
naturally constructed from all self-avoiding walks v = v v ... v starting from vertex v V,
0 1 ℓ
→ → → ∈
after whichforanywalkv ... w v ... v such that v ,w E and v > v ,wedelete
k ℓ ℓ ℓ w k
→ → → → → { } ∈
the corresponding node and the subtree from T. The resulting tree is denoted as T = T (G,v). This
SAW
construction identifies each vertex in T (many-to-one) to a vertex in G. Thus for any independent set σ of
Λ V,wehaveacorresponding configuration inT,whichisstilldenoted asσ byabusing thenotation.
⊂
Theorem2.5(Weitz[25]). ForanyfinitegraphG(V,E), v V,Λ V andanyindependent setσ ofΛ,it
∈ ⊂
holdsthat
pσ = pσ ,
G,v T,v
whereT = T (G,v).
SAW
In[3]and[16],theself-avoiding walktreeisgeneralized todealwithgeneral constraints.
3 Computing the capacity by SSM
We apply the sequential cavity method of Gamarnik and Katz [6], which gives efficient approximation al-
gorithmforcomputingtheentropyoflatticemodelsinstatisticalmechanicsusingthestrongspatialmixing.
In[14],thiswasusedforapproximately computingtheentropy ofthe2-dimensional Markovrandomfields
exhibiting strong spatial mixing. These approximation algorithms, along withthe one given in [17]for the
hard-square entropy, are all in the framework of polynomial-time approximation scheme (PTAS), which is
definedasfollows.
Definition 3.1. We say that there exists a polynomial-time approximation scheme (PTAS) for computing a
realnumberC [0,1]ifforanyǫ > 0anumberCˆ canbereturnedintimepoly(1)suchthat Cˆ C ǫ.
∈ ǫ | − |≤
Here we restate the proof in [6] for the implication from SSM to the existence of PTASin our context
forself-containness.
Theorem 3.2. For Cons HS,HH,RWIM,NAK , if the independent sets of LCons exhibit SSM,then
∈ { }
wehaveaPTASforcomputing CCons.
We define some notations. Consider the vertex set [2t +1] [2t + 1] of the finite lattice LCons .
× 2t+1,2t+1
Let Q = (k,ℓ) : k [t 1],ℓ [2t +1] (k,ℓ) : k = t,ℓ [t 1] denote the first t 1 rows
{ ∈ − ∈ } ∪ { ∈ − } −
6
and the first t 1 entries of the t-th row in [2t+1] [2t+1], and let 0 0,1 Q be the configuration
Q
fixing all verti−ces in Q to be unoccupied. Consider u×niformly distributed in∈de{pend}ent set σ of LCons .
2t+1,2t+1
Let p = Pr[σ = 00 ] be the marginal probability of central point (t,t) being unoccupied conditioned
t t,t Q
|
onthatallverticesinQbeingfixedtobeunoccupied.
Lemma 3.3. If the independent sets of LCons exhibit SSM, then for any constant 0 < ǫ < 1, there is a
t = O(log 1)suchthat
ǫ
1
log2 p −CCons ≤ ǫ.
(cid:12) t (cid:12)
(cid:12) (cid:12)
Proof. Let n t be sufficiently large. W(cid:12)e use the finite l(cid:12)attice LCons on vertex set [n] [n] to connect
≫ (cid:12) (cid:12) n,n ×
the marginal probability pt in aconstant size (t = O(log 1ǫ))instance to the capacity CCons defined on the
infinitelatticeLCons.
Foreachi,j [n],letS = (k,ℓ) [n] [n]: k i t, l j t denotethe(2t+1) (2t+1)
i,j
∈ { ∈ × | − |≤ | − |≤ } ×
squarecenteredat(i,j)(truncatedifitgoesbeyondtheboundaryof[n] [n]. LetS′ denotethesetofthose
vertices (i,j) [n] [n] whose S are not truncated by the boundar×y of LCons, that is, S′ = (i,j)
∈ × i,j n,n { ∈
[n] [n] :t < i,j n t .
× ≤ − }
Foreachi,j [n],letQ = (k,ℓ) [n] [n]: k [i 1],ℓ [n] (k,ℓ) [n] [n] :k = i,ℓ
i,j
∈ { ∈ × ∈ − ∈ }∪{ ∈ × ∈
[j 1] denotethefirsti 1rowsandthefirstj 1entriesofthei-throwin[n] [n],andlet0 0,1 Qi,j
i,j
− } − − × ∈{ }
betheconfigurationfixingallverticesinQ tobeunoccupied. Consideruniformlydistributedindependent
i,j
set σ of LCons. Foreach i,j [n], let p = Pr[σ = 00 ] be the marginal probability of vertex (i,j)
n,n ∈ i,j i,j | i,j
beingunoccupied conditioned onthatallverticesinQ beingfixedtounoccupied.
i,j
Let0betheconfigurationfixingallverticesin[n] [n]tobeunoccupied. Forauniformlydistributedin-
dependentsetσofLCons,wehavePr[σ = 0] = 1/NC×ons whereNCons denotesthenumberofindependent
n,n n,n n,n
setsofLCons. Moreover, itholdsthat
n,n
Pr[σ = 0] = Pr[σ = 0 0 ]= p .
i,j i,j i,j
|
i,j∈[n] i,j∈[n]
Y Y
Therefore, wehave
Cons
log N 1 1
2 n,n
= log .
n2 n2 2 p
i,j
i,j∈[n]
X
Letp′ = Pr[σ = 00 0 ]where0 istheconfigurationfixingallboundaryverticesofS inLCons to
i,j i,j | i,j∧ B B i,j n,n
beunoccupied. Notethatfor(i,j) S′andanyconsideredconstraintCons,thedistanceinLconsisatleast
∈
thegriddistance distorted byaconstant factor, thustheshortest distance betweenvertex(i,j) S′ andthe
boundary isΩ(t). Suppose that theindependent sets ofLCons exhibits SSM.Then forsomet∈= O(log 1),
n,n ǫ
wehave p p′ ǫforall(i,j) S′.
| i,j − i,j| ≤ ∈
Furthermore, for those (i,j) S′, it is easy to see that p′ = p . And for (i,j) LCons S′, itholds
∈ i,j t ∈ n,n \
thatp [1,1]. Therefore,wehave
i,j ∈ 2
(n−2t)2log2 pt1+ǫ 1 log 1 (n−2t)2log2 pt1−ǫ +(4nt−t2).
n2 ≤ n2 2 p ≤ n2
i,j
i,j∈[n]
X
7
whenn→ ∞,wehaveCCons = log2nN2nC,onns = n12 i,j∈[n]log2 pi1,j and
P
1 1
log2 p +ǫ ≤ CCons ≤ log2 p ǫ.
t t
−
Therefore,
1 1 1 2ǫ 2ǫ
log2 p −CCons ≤ log2 p ǫ −log2 p +ǫ = log2 1+ p ǫ ≤ log2 1+ 1/2 ǫ = O(ǫ).
(cid:12) t (cid:12) t− t (cid:18) t− (cid:19) (cid:18) − (cid:19)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
The exact value of p can be relatively efficiently computed because the graph on which p is defined
t t
has bounded treewidth. Precisely, for all considered constraints Cons, the treewidth of the finite graph
LCons is O(t). And the independent set is covered by the framework considered in [26]. The value
2t+1,2t+1
of p can thus be computed exactly by the dynamic programming algorithm introduced in [26] with time
t
complexity2O(t) poly(t)= poly(1). CombinedwithLemma3.3,Theorem3.2isproved.
· ǫ
4 SSM of hard-hexagon and RWIM
It is well known that the independent sets of two-dimensional grid LHS exhibits SSM [25]. Wenow prove
thefollowingtheorem.
Theorem4.1. Theindependent setsofLHH andLRWIM exhibits SSM.
It is well known that C is exact solvable [2]. Applying Theorem 3.2, we have the following new
HH
algorithmic resultforC .
RWIM
Corollary 4.2. ThereexistsPTASforcomputing C andC .
HH RWIM
4.1 Branching matrix
TheSSMisproved onasupertree oftheself-avoiding walktree fortherespective lattice. Thissupertree is
amulti-type Galton-Watson treegenerated byabranching matrixwhosedefinitionisintroduced in[18].
Definition 4.3 (Restrepo et al. [18]). A branching matrix M is an m m matrix of nonnegative integral
×
entries. Eachbranching matrixrepresents arootedtreeT generated bythefollowingrules:
M
eachnodeofT isinoneofthemtypes 1,...,m ,withtherootbeingtype1;
M
• { }
everytype-inodehasexactlyM manychildren oftypej.
ij
•
Note that if M is irreducible, then T is an infinite tree. The maximum arity of the tree is d =
M
max m M . The following lemma gives a relation between SSM and the maximum eigenvalue
1≤i≤m j=1 ij
ofbranching matrix.
P
TheSSMisprovedbythefollowingmainlemma,whichcanbeimpliedbyTheorem1.3of[22]through
thenotionofconnectiveconstant. Herewerestatetheprooftothelemmawithoutusingconnectiveconstant.
ThefollowinglemmarelatestheSSMtothemaximumeigenvalue ofbranching matrix.
8
Lemma 4.4 (implicit in [22]). Let G(V,E) be an infinite graph with maximum degree d + 1. Let γ =
inf [1+(1+x)d](1+x). Ifforeveryv V,thereexistsabranching matrixM satisfying:
x∈[0,+∞) dx ∈
1. thetreeT generated byM isasupertree ofT (G,v), and
M SAW
2. thelargest eigenvalue λ∗ ofM islessthanγ,
thentheindependent setsofGexhibitsSSM.
4.2 Potential analysisofcorrelation decay
By Theorem 2.5, the SSM on graph G can be implied by the SSM on its SAW tree, which can be implied
bytheSSMonanysupertreeoftheSAWtree,accordingtoProposition2.4. WethenverifytheSSMforthe
supertree T oftheSAWtreegenerated byabranching matrix M whoseeigenvalues satisfy the condition
M
ofLemma4.4. Thiscanbedonebyshowingthatthesystem(1)onthetreeT converges atanexponential
M
ratewhiletheboundary conditions arearbitrarily fixed. Twokeyideasfortheanalysisare:
Instead of analyzing theconvergence of theratios R = 1−p ofthe marginal probabilities asstraight-
• p
forwardlyintherecursion(1),weanalyzetheconvergenceofthe“potentials”φ = sinh−1(√R). This
potential functionwasintroduced in[11]andlaterusedin[22],seemingtocapturetheverynatureof
hard(independent set)constraint.
Weusethel2-norm(sumofsquares)tomeasuretheerrorsofpotentialsforverticesofdifferenttypes.
• 2
Thesameschemewasproposed in[22].
Without loss of generality, we consider m m branching matrix M of Boolean (0 and 1) entries. For
×
branchingmatriceswithentriesgreaterthan1,wecanrefineeachoftheinvolvedtypestoanumberoftypes
so that the resulting branching matrix has Boolean entries, generates the same tree as before, and has the
same largest eigenvalue. LetT be the infinite tree generated by M. A set S of vertices in T is a cutset
M M
if any infinite path in T intersects with S. Let σ denote an independent set of cutset S. For any vertex
M
v in T , let T be the subtree rooted by v and x the ratio between probabilities of v being occupied and
M v v
unoccupied inT conditioned onvertices inS beingfixedasσ. Suppose v isoftypei. By(1),wehavethe
v
followingrecursion:
1 ifv / S,
Qmj=1(1+xj)Mij ∈
xv = 0 ifv S andσv = 0, (2)
 ∈

 ifv S andσ = 1.
v
∞ ∈
where xj is the corresponding ratioof probabilities at the child of type j in its subtree. Note that Mij ∈
0,1 andthefunctiondependsonlyonthosex withM = 0,thustherecursion iswell-defined. Letη be
j ij
{ } 6
anotherindependent setofcutsetS. Wecandefineanewsequence ofquantities x′ inthesameway.
v
Consider anyfinitegraph Gandavertexv,andanytwoindependent setsρ,η ofvertexsubsetΛwhere
ρ,η disagree only at vertices at least ℓ far away from v. Let T = T (G,v) and T a supertree of T.
SAW M
Then by the same argument as in Proposition 2.4, there must exist two independent sets σ,τ of a cutset S
of T , disagreeing only at vertices at least ℓ far away from the root v of T , such that pρ = pσ and
M M T,v TM,v
pη = pτ . Therefore, byTheorem2.5anddefinitionsofthequantities x andx′ asabove,wehave
T,v TM,v v v
1 1
pρ pη = pρ pη = pσ pτ = x x′ .
G,v − G,v T,v − T,v TM,v − TM,v 1+x − 1+x′ ≤ | v − v|
(cid:12) v v(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12)
9
To prove the SSM, it is then sufficient to show it always holds that x x′ exp( Ω(ℓ)) for any
| v − v| ≤ −
independent setsσ,τ incutsetS ofT whichdisagreeonlyatverticesℓfarawayfromtherootv.
M
ProofofLemma4.4. Foreachi 1,2,...,m ,wedefine
∈ { }
1
f (x)= ,
i mj=1(1+xj)Mij
wherex = (x ,x ,...,x ). By(2),ifthecurrenQtrootvisunfixedandisoftypei,thenwehavex = f (x)
1 2 m v i
andx′ = f (x′). Wedefinethepotentials asy = φ(x )andy = φ(x )forallj = 1,2,...,m,wherethe
v i v v j j
potential function φ()isgivenby
·
φ = sinh−1(√x).
Thisisthepotential function usedin[11,22,12]. Supposedv isoftypei,themappingfromy toy canbe
v
deduced as
y = g (y)= φ(f (φ−1(y ),φ−1(y ),...,φ−1(y ))).
v i i 1 2 m
Sinceφisstrictlymonotone, theinverseφ−1(y)iswell-defined.
Suppose y = φ(x ) and y′ = φ(x′) are the respective potentials at every vertex v defined by the
v v v v
boundary conditions σ,τ oncutset S. Forv S, bythe mean value theorem, there exist y˜ [y ,y′], and
6∈ j ∈ j j
accordingly x˜ = φ−1(y˜ ), j = 1,2,...,m,suchthat
j j
∀
m m
∂g f (x˜) M x˜
y y′ = g (y) g (y′) i y y′ i ij j M y y′ .
| v − v| | i − i | ≤ j=1(cid:12)∂y˜j(cid:12)| j − j| ≤ s1+fi(x˜) j=1s1+x˜j ij| j − j|
X(cid:12) (cid:12) X p
(cid:12) (cid:12)
DuetoCauchy-Schwarz, wehave (cid:12) (cid:12)
m m
f (x˜) M x˜
y y′ 2 i ij j M y y′ 2.
| v − v| ≤ 1+f (x˜) 1+x˜  ij| j − j|
i j
j=1 j=1
X X
 
Letd = m M . Notethatd dwhered+1isthemaximum degreeoftheoriginal graph G. Recall
i j=1 ij i ≤
that γ = inf [1+(1+x)d](1+x). It can be verified that by such definition γ is nondecreasing in d.
P x∈[0,+∞) dx
Let x¯ = m (1+x˜ )Mij 1/di 1. Wehave fi(x˜) = 1 , and by Jensen’s inequality it can be
j=1 j − 1+fi(x˜) 1+(1+x¯)di
verifiedth(cid:16)aQt m Mijx˜j (cid:17)dix¯. Therefore,itholdsthat
j=1 1+x˜j ≤ 1+x¯
P m m
d x¯ 1
y y′ 2 i M y y′ 2 M y y′ 2. (3)
| v − v| ≤ (1+(1+x¯)di)(1+x¯) ij| j − j| ≤ γ ij| j − j|
j=1 j=1
X X
FixacutsetS anda∆ S suchthattheshortestdistancefromanyvertexin∆totherootofthetreeis
⊆
ℓ. Foreach1 t ℓandi 1,2,...,m ,letǫ(t) beanm-vectorsuchthatǫ(t) isthemaximumpotential
≤ ≤ ∈ { } i
differencesquare y y′ 2 foranyvertexv oftypeiatdepthℓ t(assumingtheroothasdepth0)defined
| v− v| −
byanytwoboundaryconditionsσ,τ onS disagreeingonlyon∆. Supposethatǫ(t) = y y′ 2foravertex
i | v− v|
oftypeiatlevelℓ tforaparticular pairofboundary conditions σ,τ onS. Wehave
−
m m
1 1
ǫ(t) = y y′ 2 M y y′ 2 M ǫ(t−1).
i | v − v| ≤ γ ij| j − j| ≤ γ ij j
j=1 j=1
X X
10