Approximate Counting via Correlation Decay in Spin Systems

Appro ximate Coun ting via Correlatio n Deca y in Spin Systems Liang Li ∗ P eking Unive r sit y liang.li@pk u.edu.cn Pin y an Lu Microsoft Researc h Asia pinyanl@mic rosoft.com Yitong Yin †∗ Nanjing Univ ersit y yinyt@nju.e du.cn Abstract W e give the first deterministic fully p o ly nomial-time appr oximation sc heme (FPT AS) for computing the par titio n function o f a tw o-state spin system on an arbitrar y graph, when the parameters of the system satisfy the unique ne s s co ndition on infinite regular trees. This condi- tion is of physical sig nificance a nd is b elieved to be the r ight bo undary be tw een a pproximable and inapproximable. The FPT AS is based o n the cor relation decay tec hnique in tro duced by Bandyopadh yay and Gamarnik [SOD A 06] and W eitz [STOC 06]. The cla s sic correla tion deca y is defined with resp ect to graph distance. Althoug h this definition has na tur al physical meanings , it do es no t directly supp ort a n FPT AS for systems on a rbitrary graphs, b eca use for gr aphs with un b o unded degrees, the lo cal co mputation that provides a desirable precis ion by corr elation decay ma y tak e sup e r-p olynomia l time. W e intro duce a notion of c omputational ly efficient c orr elation de c ay , in which the co rrelation decay is meas ured in a refined metric instea d o f gr aph distance. W e use a p otential metho d to analyze the a mo rtized b ehavior of this correlation de c ay and establish a co rrelatio n decay that guarantees a n inv e r se-p olyno mia l pr ecision by p olynomia l-time lo cal computation. This giv es us an FPT AS for spin systems on arbitrary graphs. This new notion of correla tion decay pro p erly reflects the algo rithmic asp ect of the spin sy s tems, and may b e used for des igning FP T AS for o ther co unt ing pro blems. 1 In tro duction Spin systems are we ll studied in S tatistical Physics. W e fo cus on t wo-stat e spin systems. An instance of a spin system is a graph G = ( V , E ). A confi guration σ : V → { 0 , 1 } assigns ev er y v ertex one of the tw o states. W e shall r efer th e t wo states as blue and green. T he contributions of lo cal inte r actions b et w een ad j acen t ve r tices are quantified by a matrix A =  A 0 , 0 A 0 , 1 A 1 , 0 A 1 , 1  =  β 1 1 γ  , where β , γ ≥ 0. The weig h t of an assignmen t is the pro duction of con tr ib utions of all lo cal in- teractions and the p artition function Z A ( G ) of a system is the summation of the w eigh ts o v er all p ossible assignment s . F ormally , Z A ( G ) = X σ ∈ 2 V Y ( u,v ) ∈ E A σ ( u ) ,σ ( v ) . ∗ This wo rk was done when these authors visited Microsoft Researc h Asia. † Supp orted by th e National Science F oundation of China und er Gran t No. 61003 023 and No. 6102 1062. 1 Although originated from Statistical Ph ysics, the sp in system is also accepted in C omputer Science as a framewo r k f or counting problems. Considering the t wo very w ell stud ied fr amew orks, the w eight ed Constrain t Satisfaction Problems (#C S P) [6, 7, 11, 12, 14, 19, 26] and Graph Homomor- phisms [8–10, 17, 28, 36], the tw o-state spin systems can b e view ed as the most basic setting in these framew orks: A Bo olean #CSP pr oblem with one s y m metric bin ary relation; or Graph Homomor- phisms to graph with tw o v ertices. Man y natur al combinatoria l problems can b e formulat ed as t wo -state spin sys tems. F or example, with β = 0 and γ = 1, Z A ( G ) is the num b er of indep enden t sets (or vertex co v ers ) of th e graph G . Giv en a matrix A , it is a computatio nal p roblem to compute Z A ( G ) wh er e graph G is given as inp ut. W e wan t to c h aracterize the computational complexit y of computing Z A ( G ) in terms of β and γ . F or exact computation of Z A ( G ), p olynomial time algorithms are kno w n only for the v ery restricted settings that β γ = 1 or ( β , γ ) = (0 , 0), and for all other setti n gs the p r oblem is pro ved to b e #P-Hard [8]. W e consider th e approximati on of Z A ( G ), with the fully p olynomial- time app ro ximation sc h emes (FPT AS ) and its r andomized relaxation the fully p olynomial-ti me randomized appro x im ation schemes (FPRAS). In a seminal pap er [48], Jerr um and Sinclair ga ve an FPRAS when β = γ > 1, whic h w as further extended to th e entire r egion β γ > 1 [4 1 ]. F or 0 ≤ β , γ ≤ 1 except that ( β , γ ) = (0 , 0) or (1 , 1), Goldb erg, Jerru m and Pat er s on p r o ve that the problem do not admit an FPRAS un less NP=RP [41]. F or the other v alues of the parameters, namely , 0 ≤ β < 1 < γ < 1 β or symmetrically 0 ≤ γ < 1 < β < 1 γ , the approxima bilit y of Z A ( G ) is n ot v er y well understo o d. It w as sho wn in [41] that by coupling a simple heat-bath random walk, there exists an additional r egion of β and γ whic h adm it some FPRAS . Th e tru e charact erization of approxima bilit y is still left op en. Within this un kno wn region, there lies a critical curv e with physica l significance, called the uniqueness threshold. The phase tr an s ition of Gibb s measure o ccurs at this threshold curv e. Such statistica l physics phase tr an s itions are b eliev ed to coincide w ith the transitions of computational complexit y . Ho wev er, th ere are only very few examples w here the connection is rigorously prov ed. One example is the hardcore (counting ind ep end ent set) mo del. It was conjectured in [56] by Mossel, W eitz and W ormald, and settled in a line of works by Dy er, F rieze and Jerru m [23], W eitz [61], Sly [58], and ve ry r ecen tly Galanis, Ge, ˇ Stefank ovi ˇ c, Vigod a and Y ang [31] that in the hard core mo del the un iqueness threshold essentia lly c h aracterizes the app ro ximability of the partition function. It will b e very interesting to observ e th e similar transition in spin systems. 1.1 Main results W e ext en d the appro ximable r egion (in terms of β and γ ) of Z A ( G ) to the u niqueness threshold in tw o-state spin sy s tems, whic h is b eliev ed to b e the righ t b oun dary b etw een appro ximable and inapproxi mable. Sp ecifically , we form ulate a criterion for β and γ su c h that there is a unique Gibbs measure on all infin ite regular trees 1 , and pro v e that there is an FPT AS for computing Z A ( G ) when this u niqueness condition is satisfied. This impro ves the app r o ximable b oun d ary (dashed lines in Figure 1) pro v id ed by the heat -b ath rand om w alk in [41]. Moreo ver, th e algorithm is deterministic. The FPT AS is based on th e correlation deca y tec hnique first used in [1, 61] for appr o ximate coun ting. W e elab orate a bit on the ideas. A spin system induces a natural probabilit y distribution o v er all configurations called the Gibbs measure w here the pr ob ab ility of a configuration is p rop or- 1 T echnically , there is a small integralit y gap caused b y the contin uous generali zation of the condition. The formal statement is given in the fol lowing section. 2 0 0 .5 1 1 . 5 2 2 . 5 3 0 0 .5 1 1 .5 2 2 .5 3 β γ 0 < β , γ < 1 βγ = 1 uniqueness thr eshold threshold achieved by heatbath random walk Figure 1: O u r FPT AS works for the region b etw een the critical cur v e of th e uniqueness threshold and the curve β γ = 1. The heat-bath random walk in [41] w orks for the r egion b et we en the dashed line and β γ = 1. tional to its weigh t. Due to a standard self-redu ction pro cedu re, computin g Z A ( G ) is reduced to computing the marginal d istribution of the state of one v ertex, whic h is made plaus ib le b y W eitz in [61] with the self-a voi ding w alk (SA W) tree construction. F or efficiency of computation, the marginal distribution of a v ertex is estimated using only a lo cal neighborho o d around th e vertex. T o justify the precision of the estimation, w e show that f ar-a wa y v ertices ha v e little influence on the marginal distrib ution. This is done by analyzing the rate w ith whic h the correlation b et we en t wo vertice s deca ys as they are far a wa y fr om eac h other. The correlation deca y by itself is a p h enomenon of physical significance. One of our main disco ve r ies is that t w o-state spin systems on any graphs ha ve exp onen tial correlation deca y wh en the ab ov e u niqueness condition is satisfied. 1.2 T ec hnical contributions The tec hn ique of u sing correlation deca y to design FPT AS for partition fu nctions is deve lop ed in the hard core mo d el. W e in tro duce seve r al n ew ideas to adapt the chall enges arising f rom spin systems. W e b eliev e these c hallenges are t ypical in counting problems, and the new ideas will make the correlation deca y tec hnique m ore applicable for approxi mate count in g. 1. The correlation d eca y tec hniqu e used in [61] relies on a monotonicit y pr op erty sp ecific to the hardcore m o del. Correlation deca ys in graphs are reduced via this monotonicit y to the deca ys in in finite regular trees, w h ile the later ha v e solv ab le phase transition thresholds. It w as already observed in [61] that suc h monotonicit y may not generally hold f or other mo d els. Indeed, it do es n ot h old for spin systems. W e devel op a more general metho d which do es not 3 rely on monotonicit y: W e dir ectly compute the correlation deca y in arbitrary trees (and as a result in arbitrary graph s via the SA W tree reduction), and use the p oten tial method to analyze the amortized b eha vior of correlation deca y . 2. T o h a ve an FPT AS, the marginal distribution of a single v ertex should b e approximable up to certain precision from a lo cal neighborh o o d of p olynomial size. The classic correlation deca y is measured with resp ect to graph d istance. The lo cal n eigh b orho o ds in this sense are balls in the graph metric. A SA W tree enumerates all paths originating from a vertex. F or graph s of unboun ded degree s, the SA W tree transform ation ma y hav e the balls offering desirable precisions explo de to sup er-p olynomial sizes. W e intro d uce the notion of c omputational ly efficie nt c orr elation de c ay . Corr elation deca y is now measured in a refined metric, whic h h as the adv ant age that a d esirable precision is ac hiev able b y a ball (in the new metric) of p olynomial size eve n after the SA W tree tran s for- mation. W e p ro ve an exp onentia l correlation deca y in th is n ew m etric wh en the u n iqueness is satisfied. As a result, w e ha v e an FPT AS for arbitr ary graphs as long as the uniqueness condition holds. 1.3 Related w orks The approximati on for partition f unction h as b een extensiv ely studied with b oth p ositiv e [18, 2 9 , 39, 47, 48, 50, 60] an d n egativ e r esults [3, 5, 13, 32, 33, 37, 38, 56]. Some sp ecial p roblems in these framew ork are w ell studied combinato r ial p roblems, e.g. count in g indep enden t sets [23, 29, 53] and graph coloring [4, 20–22, 30 , 40, 43 – 47, 54, 55, 60 ]. S ome dichoto m ies (or trichoto mies) of complexit y for appro ximate counting CSP w ere also obtained [24, 27, 31, 58 ]. Almost all k n o wn approximat ion coun ting algorithms are based on random sampling [25, 51], usually thr ough the famous Marko v Chain Mont e Carlo (MCMC) metho d [16, 49]. There are v er y few d eterministic approximati on algorithms for an y counting p roblems. Some notable examples in clud e [1 , 2, 34, 42, 59]. In a ve r y r ecen t w ork [57], Sinclair, S riv asta v a, and Thurley giv e an FPT AS using correlation deca y for the t wo -state spin systems on bou n ded degree graphs. They allo w the t wo -state spin systems to ha ve an external field, and the uniqu eness thresholds th ey used are defined with resp ect to sp ecific maxim u m degrees. 2 Definitions and Stateme n ts of Results A spin system is describ ed b y a graph G = ( V , E ). A c onfigur ation of the system is one of the 2 | V | p ossible assignment s σ : V → { 0 , 1 } of states to vertice s . W e also u se tw o colors blue and gr e e n to denote these tw o states. Let A =  A 0 , 0 A 0 , 1 A 1 , 0 A 1 , 1  =  β 1 1 γ  , where β , γ ≥ 0. The Gibbs me asur e is a distribution o ver all configurations defined b y µ ( σ ) = 1 Z A ( G ) Y ( u,v ) ∈ E A σ ( u ) ,σ ( v ) . The normalization factor Z A ( G ) = P σ ∈ 2 V Q ( u,v ) ∈ E A σ ( u ) ,σ ( v ) is called the p artition function . F rom this d istribution, w e can define the marginal pr obabilit y p v of v to b e colored b lu e. Let σ Λ b e a confi gu r ation defined on v ertices in Λ ⊂ V . W e call v ertices v ∈ Λ fixe d ve r tices, and v 6∈ Λ 4 fr e e ve r tices. W e use p σ Λ v to den ote the marginal p robabilit y of v to b e colored blue conditioned on the configuration of Λ b eing fixed as σ Λ . Definition 1 A spin system on a family of g r aphs is said to have exp onential c orr elation de c ay if for any gr aph G = ( V , E ) in the family, any v ∈ V , Λ ⊂ V and σ Λ , τ Λ ∈ { 0 , 1 } Λ , | p σ Λ v − p τ Λ v | ≤ exp( − Ω(dist( v, ∆))) . wher e ∆ ⊂ Λ is the sub set on which σ Λ and τ Λ differ, and d ist( v , ∆) i s the shor test distanc e fr om v to any v e rtex in ∆ . This definition is equiv alen t to the “strong sp atial mixing” in [61] with an exp onen tial r ate. It is stronger than the standard notion of exp onen tial correlation deca y in Statistical Ph ysics [15], where the d eca y is measur ed with resp ect to dist( v , Λ) instead of dist( v , ∆). The marginal probability p σ Λ v in a tree can b e compu ted by the follo wing recursion. Let T b e a tree ro oted by v . W e den ote R σ Λ T as the r atio of the probabilities that ro ot v is blu e and green, resp ectiv ely , wh en imp osing the condition σ Λ . F ormally , R σ Λ T = p σ Λ v 1 − p σ Λ v (when p σ Λ v = 1, let R σ Λ v = ∞ b y con ven tion). Su pp ose that the ro ot of T has d c h ild ren. Let T i b e the subtree ro oted b y the i -th c hild of the ro ot. T h e d istributions on distinct subtrees are indep end ent. A calculation then giv es that R σ Λ T = d Y i =1 β R σ Λ T i + 1 R σ Λ T i + γ . (1) It is of p hysic al significance to study the Gibbs measures on infinite ( d + 1)-regular trees b T d [35]. In b T d , the recursion is of a symmetric form f ( x ) =  β x +1 x + γ  d . T here m a y b e m ore than one Gibb s measures on infinite graphs. W e sa y that the s ystem has the uniqu e ness if there is exact one Gibb s measure. Let ˆ x = f ( ˆ x ) b e the fixed p oin t of f ( x ). I t is k n o wn [52, 54] that the spin system on b T d undergo es a phase transition at | f ′ ( ˆ x ) | = 1 with uniqu eness when | f ′ ( ˆ x ) | = d (1 − β γ )( β ˆ x +1) d − 1 ( ˆ x + γ ) d +1 ≤ 1. This motiv ates the follo w ing d efinition Γ( β ) = inf  γ ≥ 1    ∀ d ≥ 1 , d (1 − β γ )( β ˆ x + 1) d − 1 ( ˆ x + γ ) d +1 ≤ 1  . F or a fixed 0 ≤ β < 1, the Γ( β ) giv es the b oun d ary th at all infinite r egular trees b T d exhibit uniqueness when Γ( β ) ≤ γ ≤ 1 β . W e call Γ ( β ) the uniq ueness thr eshold . Indeed, for an y d ≥ 1, there is a critical Γ d ( β ) suc h th at b T d exhibits uniqueness when Γ d ( β ) < γ < 1 β . F u r thermore, there is a finite cru cial D > 1 such that Γ D ( β ) = Γ( β ). That is, b T D has the highest uniqueness th r eshold Γ( β ) among all b T d . W e remark that for tec hn ical reasons, we treat d as r eal num b ers thus Γ( β ) is slightly greater than the one defined b y intege r d s. An in teger version of Γ( β ) is giv en in Section 6, where a sligh tly impro ved and tigh t analysis is giv en for the s p ecially case β = 0. Definition 2 A f u l ly p olynom i al-time appr oximation scheme (FPT A S) for Z A ( G ) is an algorithm that given as input an instanc e G and an ǫ > 0 , outputs a numb er Z in time p oly ( | G | , 1 ǫ ) such that (1 − ǫ ) Z A ( G ) ≤ Z ≤ (1 + ǫ ) Z A ( G ) . 5 In Definition 1, the correlat ion deca y is measur ed in graph distance. I n order to supp ort an FPT AS f or graph s with unboun ded degrees, we need to define the follo wing refined metric. Definition 3 L e t T b e a r o ote d tr e e and M ≥ 2 b e a c onstant. We define the M -b ased depth L M ( v ) of a vertex v in T r e cursively as fol lows: L M ( v ) = 0 if v is the r o ot of T ; and for ev e ry child u of v , if v has d ≥ 1 childr en, L M ( u ) = L M ( v ) + ⌈ log M ( d + 1) ⌉ . If every v ertex in T has d < M c hildren, L M ( v ) is p recisely the d epth of v . I f there are vertice s ha ving d ≥ M children, we actually replace eve r y su c h vertex and its d c hildren with an M -ary tree of depth ⌈ log M ( d + 1) ⌉ , and L M ( v ) is the d epth of v in this new tree. Definition 4 L e t T b e a r o ote d tr e e and M ≥ 2 b e a c onstan t. L et B M ( L ) = { v ∈ T | L M ( v ) ≤ L } , c al le d an M -based L -ball , b e the set of ve rtic es in T whose M -b ase d depth s ar e no gr e ater than L ; and let B ∗ M ( L ) , c al le d an M -based L -closed-ball , b e the set of v e rtic es in B M ( L ) and al l their childr en in T . The main tec hn ical result of the paper is the follo wing theorem whic h establishes an exp onen tial correlation deca y in the refined metric when the uniqueness condition h olds. Theorem 5 (C omputationally E fficien t Correlation Deca y) L et 0 ≤ β < 1 , β γ < 1 , and γ > Γ( β ) . Ther e exists a sufficiently lar ge c onstant M which dep ends only on β and γ , such that on an arbitr ary tr e e T , f or any two c onfigur ations σ Λ and τ Λ which differ on ∆ ⊂ Λ , if B ∗ M ( L ) ∩ ∆ = ∅ then | R σ Λ T − R τ Λ T | ≤ exp( − Ω( L )) . The n ame computationally efficie n t correlation d eca y is due to the fact th at | B M ( L ) | ≤ M L in any tree, th us an exp onentia l deca y w ould imply a p olynomial-size B M ( L ) giving an inv erse- p olynomial precision. Theorem 5 has the follo wing implications via W eitz’s self-a voiding tree construction [61]. Theorem 6 L e t 0 ≤ β < 1 , β γ < 1 , γ > Γ( β ) . It is of exp onential c orr elation de c ay for the Gibbs me asur e on any gr aph. Theorem 7 L e t 0 ≤ β < 1 , β γ < 1 , γ > Γ( β ) . Ther e is an FPT AS for c omputing the p artition function Z A ( G ) for arbitr ary gr aph G . By symmetry , in Theorem 5, 6, and 7, the roles of β and γ can b e switc hed. In the Section 3, we will sh o w the FPT AS implied by Theorem 5, follo wed b y a formal treatmen t of the uniqueness threshold in Section 4, and fin ally the form al pr o of of Theorem 5 in S ection 5. 3 An FPT AS for the P artition F unction Assuming that Th eorem 5 is tru e, w e sho w that w hen 0 ≤ β < 1 and Γ( β ) < γ < 1 β , there is an FPT AS for the partition function Z A ( G ) for arbitrary graph G . The FPT AS is based on appro ximation of R σ Λ G,v = p σ Λ v / (1 − p σ Λ v ), the r atio b et w een the probabilities that v is blue and green, resp ectiv ely , when imp osing the condition σ Λ . 6 The self-a v oiding wa lk tree is in tr o duced by W eitz in [61] for calculating R σ Λ G,v . Giv en a graph G = ( V , E ), w e fix an arbitrary order < of v ertices. Originating from an y vertex v ∈ V , a self- a v oiding wa lk tr ee, denoted T SA W ( G, v ), is constr u cted as follo ws. Every ve r tex in T SA W ( G, v ) corresp onds to one of the wa lks v 1 → v 2 → · · · → v k in G suc h that v 1 = v , all edges are d istinct and v 1 , . . . , v k − 1 are distinct, i.e. the self-av oiding w alks originating from v and those app ended with a v ertex closing a cycle. The ro ot of T SA W ( G, v ) corresp ond s to the trivial wa lk v . The v ertex v 1 paren ts v 2 in T SA W ( G, v ), if and only if their resp ectiv e w alks w 1 and w 2 satisfy that w 2 = w 1 → u for some u . F or a leaf of T SA W ( G, v ) wh ose walk closes a cycle, supp osed that the cycle is u → v 1 → · · · v k → u , fix the leaf to b e blue if v 1 > v k and green otherw ise. When a configuration σ Λ is imp osed on Λ ⊂ V of the original graph G , f or an y ve r tex of T SA W ( G, v ) whose corresp ondin g w alk ends at a u ∈ Λ, the col or of the vertex is fixed to b e σ Λ ( u ). W e abuse the notation and d enote the resulting configuration on T SA W ( G, v ) by σ Λ as w ell. This no v el tree constru ction has the adv an tage that the pr obabilities are exactl y the same in b oth the original spin system and the constructed tr ee. Theorem 8 (W eitz [61]) L e t T = T SA W ( G, v ) . It holds that R σ Λ G,v = R σ Λ T Due to (1), in a tree T , the f ollo wing r ecursion holds for R σ Λ T : R σ Λ T = d Y i =1 β R σ Λ T i + 1 R σ Λ T i + γ . The base case is either when the curr en t ro ot v ∈ Λ, i.e. v ’s color is fi xed, in which case R σ Λ T = ∞ or R σ Λ T = 0 (dep ending on whether v is fi xed to b e blue or green), or when v is f ree and has no c hildren , in whic h case R σ Λ T = 1 ( this is co n sisten t with the recursion since the ou tcome of an empt y pro du ct is 1 b y con ven tion). F or β γ < 1, the recursion is monotonically decreasing with resp ect to ev ery R σ Λ T i . An up p er (lo w er) b oun d of R σ Λ T can b e compu ted by replacing R σ Λ T i in the recursion b y their r esp ectiv e lo wer (upp er) boun ds. Alg orithm 1 computes the lo wer or upp er boun d of R σ Λ T up to v ertices in M -based L -closed ball B ∗ M ( L ). F or the ve r tices outside B ∗ M ( L ), it uses the trivial b ounds 0 ≤ R ≤ ∞ . Due to the m onotonicit y of the recursion, it holds that R ( T , σ Λ , L, 0 , true ) ≤ R σ Λ T ≤ R ( T , σ Λ , L, 0 , false ) . Note that the naiv e low er b ound 0 (or the upp er b oun d ∞ ) of R for a ve r tex outside B ∗ M ( L ) can b e ac hiev ed by fixin g the v ertex to b e green (or b lue). D en ote b y τ 0 and τ 1 the configurations ac hieving th e lo wer and upp er b oun ds resp ectiv ely . It is easy to see that τ 0 = τ 1 = σ Λ in B ∗ M ( L ). Then d ue to Theorem 5, there is a constan t α < 1 su c h that | R ( T , σ Λ , L, 0 , false ) − R ( T , σ Λ , L, 0 , true ) | = | R τ 1 T − R τ 0 T | = O ( α L ) . T o compute R σ Λ G,v for an arbitrary graph G , w e first constru ct the B ∗ M ( L ) of T = T SA W ( G, v ), and run Algorithm 1. Due to Theorem 8, R σ Λ G,v = R σ Λ T , th u s it return s R 0 and R 1 suc h that R 0 ≤ R σ Λ G,v ≤ R 1 and R 1 − R 0 = O ( α L ). S ince p σ Λ v = R σ Λ G,v / (1 + R σ Λ G,v ), we can output p 0 = R 0 R 0 +1 and p 1 = R 1 R 1 +1 so that p 0 ≤ p σ Λ v ≤ p 1 and p 1 − p 0 = R 1 R 1 +1 − R 0 R 0 +1 ≤ R 1 − R 0 = O ( α L ). 7 Algorithm 1: Estimate R σ Λ T based on B ∗ M ( L ) R ( T v , σ Λ , L, d paren t , l b ): Input : Ro oted tree T v ; configuration σ Λ ; M -based dep th L ; p arent degree d paren t ; Bo olean indicator l b of lo w er b ound. Output : Low er (or upp er) b oun d of R σ Λ T computed from vertic es in B ∗ M ( L ). b egin Supp ose r o ot v h as d c h ildren and let T i b e the sub tree ro oted by the i -th child; if v ∈ Λ then if σ Λ ( v ) = blue t hen ret urn ∞ ; else return 0; else if L < 0 then if l b = true then return 0; else return ∞ ; else L ′ ← − L − ⌈ log M ( d paren t + 1) ⌉ ; return Q d i =1 β R ( T i ,σ Λ ,L ′ ,d, ¬ lb )+1 R ( T i ,σ Λ ,L ′ ,d, ¬ lb )+ γ ; The runnin g time of this algorithm relies on the s ize of B ∗ M ( L ) in T SA W ( G, v ). T h e maximum degree of T SA W ( G, v ) is b ounded b y the maxim u m degree of G , whic h is trivially b ounded b y n , th us | B ∗ M ( L ) | ≤ n | B M ( L ) | ≤ nM L . Th e runn ing time of the algorithm is O ( | B ∗ M ( L ) | ) = O ( nM L ). By setting L = log α ǫ , w e can app ro ximate 1 − p σ Λ v within absolute error O ( ǫ ) in time O ( n · p oly( 1 ǫ )). F or β < 1 < γ , it h olds that 0 < R σ Λ G,v < 1 for free v thus 1 − p σ Λ v > 1 2 , th erefore the ab ov e pro cedur e approxima tes (1 − p σ Λ v ) within factor (1 ± O ( ǫ )). W e h a ve an FPT AS for (1 − p σ Λ v ). The partition function Z A ( G ) can b e co mp uted from p σ Λ v b y the follo wing standard routine. Let v 1 , . . . , v n en um erate the v ertices in G , and let σ i , i = 0 , 1 , . . . , n , b e the configurations fixing the first i vertic es v 1 , . . . , v i to b e green, where σ 0 means all vertices are free. Th e probabilit y measure of σ n (all green) can b e computed as µ ( σ n ) = n Y i =1 Pr[ v i is green | v 1 , . . . , v i − 1 are green] = n Y i =1 (1 − p σ i − 1 v i ) . On the other hand, it is easy to see that µ ( σ n ) = γ | E | Z A ( G ) b y definition of µ . Thus Z A ( G ) = γ | E | µ ( σ n ) = γ | E | Q n i =1 (1 − p σ i − 1 v i ) . Notice th at γ | E | > 1. Th er efore, an FPT AS for (1 − p σ Λ v ) implies an FPT AS for Z A ( G ). 8 4 The uniqueness threshold In th is section, w e formally define th e u niqueness thr eshold Γ( β ) and the cr itical D . W e also p r o ve sev eral p rop ositions regardin g th ese quantiti es whic h are useful for the analysis of the correlation deca y . Definition 9 L e t 0 ≤ β < 1 b e a fixe d p ar ameter. Supp ose that 1 ≤ γ < 1 β and d ≥ 1 . L et x ( γ , d ) b e the p ositive solution of x =  β x + 1 x + γ  d . (2) Define that f ( x ) =  β x +1 x + γ  d . Then x ( γ , d ) is th e p ositiv e fixed p oint of f ( x ). F or γ < 1 β , f ( x ) =  β + 1 − β γ x + γ  d is con tinuous and strictly decreasing ov er x ∈ [0 , ∞ ), and it holds th at f (0) = 1 γ d > 0 and f (1) =  1+ β 1+ γ  d < 1 γ d ≤ 1, thus f ( x ) has a uniqu e fixed p oint ov er x ∈ (0 , 1). Therefore, for 1 ≤ γ < 1 β and d ≥ 1, x ( γ , d ) is well defin ed and x ( γ , d ) ∈ (0 , 1). Definition 10 L et Γ( β ) = inf  γ ≥ 1    ∀ d ≥ 1 , d (1 − β γ )( β x ( γ , d ) + 1) d − 1 ( x ( γ , d ) + γ ) d +1 ≤ 1  . We write Γ = Γ( β ) for short if no ambiguity is c ause d. Note that Γ can b e equiv alen tly d efined as Γ = inf  γ ≥ 1    ∀ d ≥ 1 , d (1 − β γ ) x ( γ , d ) ( β x ( γ , d ) + 1) ( x ( γ , d ) + γ ) ≤ 1  , b ecause x ( γ , d ) satisfies (2). The follo wing lemma states th at for 0 ≤ β < 1, Γ( β ) is we ll-defin ed and nont r ivial. Lemma 11 F or 0 ≤ β < 1 , i t hol ds that 1 < Γ( β ) < 1 β . Pro of: W e first show that Γ > 1. It is su fficien t to sho w that if γ ≤ 1 then there exists a d such that d (1 − β γ ) x ( β x +1)( x + γ ) > 1, wh ere x satisfies that x =  β x +1 x + γ  d . By con tradiction, supp ose that γ ≤ 1 and for all d ≥ 1, d (1 − β γ ) x ( β x +1)( x + γ ) ≤ 1 wher e x satisfies that x =  β x +1 x + γ  d . Th en , 1 ≥ d (1 − β γ ) x ( β x + 1)( x + γ ) = d (1 − β γ ) β x + γ x + (1 + β γ ) ≥ d (1 − β γ ) β x + γ x + 2 . Sp ecifically , sup p ose that d is sufficiently large so the follo w ings h old β d exp  d (1 − β γ ) d − 3  < d (1 − β γ ) − 3 β , and exp  − γ d d (1 − β γ ) − 3  > γ d (1 − β γ ) − 3 . 9 Case.1: x ≥ γ . Then γ x ≤ 1. Thus, 1 ≥ d (1 − β γ ) β x + γ x + 2 ≥ d (1 − β γ ) β x + 3 , whic h imp lies that x ≥ d (1 − β γ ) − 3 β . On the other hand, x =  β x + 1 x + γ  d ≤  β x + 1 x  d ≤  β + β d (1 − β γ ) − 3  d ≤ β d exp  d (1 − β γ ) d − 3  < d (1 − β γ ) − 3 β , a con tradiction. Case.2: x < γ . Then β x ≤ β γ < 1. T h u s, 1 ≥ d (1 − β γ ) β x + γ x + 2 ≥ d (1 − β γ ) γ x + 3 , whic h imp lies that x ≤ γ d (1 − β γ ) − 3 . On the other hand, x =  β x + 1 x + γ  d ≥ 1 ( x + 1) d ≥  1 + γ d (1 − β γ ) − 3  − d ≥ exp  − γ d d (1 − β γ ) − 3  > γ d (1 − β γ ) − 3 , a con tradiction. W e p ro ceed to sh o w that Γ < 1 β . It is sufficient to show that there exists a 1 < γ < 1 β suc h that for all d ≥ 1, d (1 − β γ ) x ( β x +1)( x + γ ) ≤ 1, wh ere x satisfies that x =  β x +1 x + γ  d . If β = 0, then x =  1 x + γ  d ≤ 1 γ d . Thus, d (1 − β γ ) x ( β x + 1)( x + γ ) = dx x + γ ≤ dx ≤ d γ d ≤ 1 e ln γ , where the last inequalit y can b e ve rified by taking the maximum of d γ d o v er d . Th erefore, setting γ = e 1 e , it holds that d (1 − β γ ) x ( β x +1)( x + γ ) ≤ 1 e ln γ = 1. On the other hand, if 0 < β < 1, c h o osing an arb itrary constan t α ∈ (exp( − 1 − β e ) , 1) wh ic h also satisfies that α ∈ ( β , 1), and assuming γ ∈ h 1 − ( α − β ) β , 1 β  ⊆ (1 , 1 β ), w e h a ve x =  β x + 1 x + γ  d =  β + 1 − β γ x + γ  d ≤ ( β + 1 − β γ ) d ≤ α d . Th us, d (1 − β γ ) x ( β x + 1)( x + γ ) ≤ d (1 − β γ ) x ≤ (1 − β γ ) dα d ≤ (1 − β γ ) − e ln α , 10 where the last inequ alit y is also p ro ved by taking the maximum of dα d . T herefore, we can c h o ose γ = max n 1 − ( α − β ) β , 1 β − e ln(1 /α ) β o , which in deed satisifes γ ∈ (1 , 1 β ), to guarante e that d (1 − β γ ) x ( β x +1)( x + γ ) ≤ (1 − β γ ) − e ln α ≤ 1. Therefore, for 0 ≤ β < 1, there alwa ys exists a 1 < γ < 1 β suc h that for all d ≥ 1, it holds that d (1 − β γ ) x ( β x +1)( x + γ ) ≤ 1, wh ere x satisfies that x =  β x +1 x + γ  d . This implies Γ < 1 β . Definitions 12 L et γ ( d ) b e the solution γ of d (1 − β γ ) x ( γ , d ) ( β x ( γ , d ) + 1)( x ( γ , d ) + γ ) = 1 (3) over γ ∈ (1 , 1 β ) , and define γ ( d ) = 1 b y c onvention if such solution do es not exist. The follo wing lemma states th at γ ( d ) is well- d efined and captures the uniqueness thresh old for differen t instances of d . Lemma 13 The f ol lowings hold f or γ ( d ) : 1. γ ( d ) is a wel l-define d fu nction for d ≥ 1 . 2. Γ = sup d ≥ 1 γ ( d ) . 3. Ther e exists a finite c onstant D > 1 such that Γ = γ ( D ) , and D is a stationary p oint of γ ( d ) , i.e. d γ d d    d = D = 0 . Pro of: 1. W e fi rst show that for an y d ≥ 1, there exists at most one γ ∈ (1 , 1 β ) satisfying (3), whic h will imply that γ ( d ) is we ll-defined. Observe that f or an y fixed d ≥ 1, x ( γ , d ) is strictly decrea sing with resp ect to γ o v er γ ∈ (1 , 1 β ). By con tradiction, assume that for some d ≥ 1, x is non-decreasing o ver γ . Th en x =  β x +1 x + γ  d =  β + 1 − β γ x + γ  d is str ictly d ecreasing o ver γ , a con tradiction. Therefore, 1 − β γ x ( γ ,d )+ γ m us t b e strictly decreasing with resp ect to γ , or otherwise x =  β + 1 − β γ x + γ  d w ould h a ve b een n on-decreasing, con tradicting th at x ( γ , d ) is strictly decreasing. Com bin ing th ese together, w e ha ve d (1 − β γ ) x ( γ , d ) ( β x ( γ , d ) + 1)( x ( γ , d ) + γ ) = d (1 − β γ ) x ( γ , d ) + γ · 1 β + 1 x ( γ ,d ) is str ictly decreasing o ve r γ ∈ (1 , 1 β ). Th u s, there exists at most one γ ∈ (1 , 1 β ) satisfying (3). Therefore, γ ( d ) is well- defined. 11 2. W e then sho w th at Γ = sup d ≥ 1 γ ( d ). F or any d ≥ 1, let Γ d ( β ) = inf  γ ≥ 1    d (1 − β γ ) x ( γ , d ) ( β x ( γ , d ) + 1)( x ( γ , d ) + γ ) ≤ 1  . Note th at for any d ≥ 1, when γ → 1 β , d (1 − β γ ) x ( γ ,d ) ( β x ( γ ,d )+1)( x ( γ, d )+ γ ) → 0 < 1, th u s Γ d ( β ) < 1 β . In addition to that, since d (1 − β γ ) x ( γ ,d ) ( β x ( γ ,d )+1)( x ( γ, d )+ γ ) is strictly decreasing o v er γ ∈ (1 , 1 β ), Γ d ( β ) is ei ther equal to the uniqu e solution γ of d (1 − β γ ) x ( γ ,d ) ( β x ( γ ,d )+1)( x ( γ, d )+ γ ) = 1 o ver γ ∈ (0 , 1 β ) or equal to 1 if suc h solution do es not exist. T herefore, Γ d ( β ) = γ ( d ) . Since d (1 − β γ ) x ( γ ,d ) ( β x ( γ ,d )+1)( x ( γ, d )+ γ ) is strictly d ecreasing o ver γ ∈ (1 , 1 β ), for an y γ ∈ (1 , 1 β ) that γ ≥ Γ d ( β ) for all d ≥ 1, it holds that d (1 − β γ ) x ( γ ,d ) ( β x ( γ ,d )+1)( x ( γ, d )+ γ ) ≤ 1 for all d ≥ 1, i.e. γ ≥ Γ( β ). Th u s , Γ( β ) ≤ sup d ≥ 1 Γ d ( β ). Th e other direction Γ( β ) ≥ sup d ≥ 1 Γ d ( β ) is universal. Therefore, Γ( β ) = sup d ≥ 1 Γ d ( β ) = sup d ≥ 1 γ ( d ) . 3. W e sho w th at th ere is a finite D > 1 that γ ( D ) = sup d ≥ 1 γ ( d ). First notice that D > 1. By con tradiction assume that D = 1. Su b stituting x in (1 − β γ ) x = ( β x + 1)( x + γ ) with the p ositiv e solution of x =  β x +1 x + γ  giv es us a γ < 1. T hen b y con v entional definition, γ ( D ) = 1. F rom the p revious analysis, w e kno w that γ ( D ) = sup d ≥ 1 γ ( d ) = Γ and due to Lemma 11, Γ > 1. A con tradiction. W e treat x = x ( γ ( d ) , d ) as a single-v ariate function of d . W e claim that x → 0 as d → ∞ . By con tradiction, if x is b ounded a w ay fr om 0 b y a constan t as d → ∞ , then x =  β + 1 − β γ x + γ  d ≤  β + 1 − β x +1  d → 0 as d → ∞ , a contradicti on. Therefore, when d → ∞ , it must hold that γ ( d ) > 1, b ecause if otherwise γ ≤ 1, since x → 0 as d → ∞ , it holds that x =  β x +1 x + γ  d → 1 γ d , whic h app r oac hes either 1 or ∞ as d → ∞ , whic h contradicts th at x → 0 as d → ∞ . W e just sho w that γ ( d ) > 1 for sufficient ly large d , whic h means that f or these d s, γ ( d ) is defined by (3 ) instead of defined by the conv en tion γ ( d ) = 1. Thus, for sufficien tly large d , x = x ( γ ( d ) , d ) and γ = γ ( d ) can b e treated as single-v ariate fu nctions of d s atisfying b oth (2) and (3). F or β γ < 1, it h olds that x =  β x +1 x + γ  d ≤ 1 γ d , th u s d γ d ≥ d (1 − β γ ) γ d ≥ d (1 − β γ ) x = ( β x + 1)( x + γ ) ≥ γ , where the equalit y holds by (3). Thus, γ ( d ) ≤ d 1 d +1 . 12 Recall th at γ ( d ) > 1 for all sufficiently large d , th us there is a fi nite d suc h that γ ( d ) is b ound ed from b elo w by a constant greater than 1. On th e other hand , γ ( d ) ≤ d 1 d +1 = 1 as d → ∞ . Therefore, there is a fin ite D su c h that γ ( D ) = su p d ≥ 1 γ ( d ). Due to Lemma 13, this implies γ ( D ) = Γ. Since γ ( D ) = sup d ≥ 1 γ ( d ) and D is neither infinite nor equ al to 1, D m ust b e a stationary p oint of γ ( d ), i.e. d γ d d    d = D = 0. W e can th en d efine the crucial D wh ich generates the highest u n iqueness th r eshold Γ. Definition 14 L et D b e the value satisfying γ ( D ) = Γ . L et X = x (Γ , D ) . It is ob vious that b oth (2) and (3) hold for γ = Γ, d = D , and x = X . Two less obvious but v ery useful ident ities are giv en in the f ollo wing lemma. Lemma 15 The f ol lowings hold f or Γ , D and X . 1 . β Γ ≤ p β Γ < D − 1 D + 1 ; 2 . ln  β X + 1 X + Γ  = 2( β X + 1) ( D + 1)( β X + 1) − 2 D = 2 D (1 − β Γ) X ( β X + 1)(2 DX − ( D + 1)( X + Γ)) . Pro of: 1. Since γ = Γ, d = D , and x = X s atisfies (3), it holds that D (1 − β Γ) = ( β X + 1)( X + Γ) X =  β X + Γ X  + β Γ + 1 ≥ 2 p β Γ + β Γ + 1 =  1 + p β Γ  2 , where the inequalit y is due to the in equalit y of arithmetic an d geometric means. Th us, D ≥ 1+ √ β Γ 1 − √ β Γ . Th er efore, D − 1 D + 1 = 1 − 2 D + 1 ≥ p β Γ ≥ β Γ , where the last inequalit y is implied trivially by that 0 ≤ β < 1 and Γ > 1. 2. Recall that X = x (Γ , D ) and Γ = γ ( D ), where x ( γ , d ) is defined by (2 ), and γ ( d ) is defined b y (3). Th u s, x = x ( γ ( d ) , d ) and γ = γ ( d ) can b e treated as single-v ariate f unctions of d satisfying b oth (2) and (3). The follo wing iden tity is imp lied by (3): d (1 − β γ ) x = ( β x + 1)( x + γ ) . (4) T aking the deriv ativ es with resp ect to d at d = D for b oth s id es of (4), w e h a ve  ( β ( x + γ ) + ( β x + 1) − d (1 − β γ )) d x d d + ( β x ( d + 1) + 1) d γ d d      d = D = (1 − β γ ) x | d = D . 13 Due to Lemma 13, it holds that d γ d d    d = D = 0. Then d x d d     d = D = (1 − β Γ) X β ( X + Γ) + ( β X + 1) − D (1 − β Γ) = (1 − β Γ) X 2 β ( X + Γ) X + ( β X + 1) X − ( β X + 1)( X + Γ) (applying (4)) = (1 − β Γ) X 2 β X 2 − Γ . (5) Recall that x ( γ , d ) is defined by (2 ). Applyin g logarithm to b oth side of (2), we h a ve ln x = d ln  β x + 1 x + γ  . T aking the partial deriv ativ es with resp ect to d for b oth sides, 1 x ∂ x ∂ d = ln  β x + 1 x + γ  + β d ( β x + 1) · ∂ x ∂ d − d ( x + γ ) · ∂ x ∂ d . whic h imp lies that ∂ x ∂ d = x ( β x + 1)( x + γ ) ln  β x +1 x + γ  ( β x + 1)( x + γ ) + d (1 − β γ ) x = x ( β x + 1)( x + γ ) ln  β x +1 x + γ  ( β x + 1)( x + γ ) + ( β x + 1)( x + γ ) (applying (4)) = x 2 ln  β x + 1 x + γ  . Due to the total deriv ativ e form u la, and that d γ d d    d = D = 0, d x d d     d = D = ∂ x ( γ , d ) ∂ γ · d γ ( d ) d d     d = D + ∂ x ( γ , d ) ∂ d     d = D = 0 + ∂ x ( γ , d ) ∂ d     d = D = X 2 ln  β X + 1 X + Γ  . Com bin ing w ith (5 ), w e ha ve ln  β X + 1 X + Γ  = 2(1 − β Γ ) X β X 2 − Γ The equations in the lemma are consequences of the ab o v e equation. S p ecifically , 2(1 − β Γ) X β X 2 − Γ = 2 D (1 − β Γ) X D ( β X − 1)( X + Γ) + D (1 − β Γ) X = 2( β X + 1)( X + Γ ) D ( β X − 1)( X + Γ) + ( β X + 1)( X + Γ) (applying (4)) = 2( β X + 1) ( D + 1)( β X + 1) − 2 D ; 14 and 2(1 − β Γ) X β X 2 − Γ = 2 D (1 − β Γ) X D ( β X + 1)( X − Γ) − D (1 − β Γ) X = 2 D (1 − β Γ) X D ( β X + 1)( X − Γ) − ( β X + 1)( X + Γ) (applying (4)) = 2 D (1 − β Γ) X ( β X + 1)(2 DX − ( D + 1)( X + Γ)) . 5 Computationally Efficien t Correlation Deca y W e pro v e Th eorem 5, ju stifying the compu tationally efficien t correlation deca y . W e use R v and R v + δ v to r esp ectiv ely denote the lo wer and u p p er b oun ds of R σ Λ T where T is ro oted by v . F or fixed v ertices v ∈ B ∗ M ( L ), set R v = ∞ if v is b lue (and R v = 0 if v is green) and δ = 0. F or all free v ertices v ∈ B ∗ M ( L ), supp osed that v has d 1 c hildren fixed to b e blue, d 0 c hildren fixed to b e green, and d free c hildr en v 1 , . . . , v d , the recursion (1) giv es that R v + δ v = β d 1 γ d 0 d Y i =1 β R v i + 1 R v i + γ and R v = β d 1 γ d 0 d Y i =1 β ( R v i + δ v i ) + 1 R v i + δ v i + γ . (6) And for all v ertices v 6∈ B ∗ M ( L ), w e u se th e naiv e b ounds that R v = 0 and δ v = ∞ . Since γ > Γ > 1 > β ≥ 0, th e range of the recursion is (0 , 1] as long as the inp u ts are p ositiv e. Th us for all free vertice s v ∈ B ∗ M ( L ), it holds that 0 < R v ≤ R v + δ v ≤ 1. Due to the monotonicit y of the recursion, denoted b y r the r o ot of the tree, R r and R r + δ r are lo wer and upp er b ounds resp ectiv ely f or all R τ Λ T where τ Λ = σ Λ in B ∗ M ( L ). Theorem 5 is th en implied b y that δ r ≤ exp( − Ω( L )). Let f ( x 1 , . . . , x d ) = β d 1 γ d 0 Q d i =1 β x i +1 x i + γ . Then the recursions (6) can b e written as that R v + δ v = f ( R v 1 , . . . , R v d ) and R v = f ( R v 1 + δ v 1 , . . . , R v d + δ v d ). Due to the Mean V alue Th eorem, th er e exist f R i ∈ [ R v i , R v i + δ v i ], 1 ≤ i ≤ d , su c h that δ v = f ( R v 1 , . . . , R v d ) − f ( R v 1 + δ v 1 , . . . , R v d + δ v d ) = −∇ f ( f R 1 , . . . , f R d ) · ( δ v 1 , . . . , δ v d ) = β d 1 γ d 0 (1 − β γ ) · d Y i =1 β f R i + 1 f R i + γ · d X i =1 δ v i ( β f R i + 1)( f R i + γ ) . A straigh tforward estimation giv es that δ v max 1 ≤ i ≤ d { δ v i } ≤ β d 1 γ d 0 (1 − β γ ) · d Y i =1 β f R i + 1 f R i + γ · d X i =1 1 ( β f R i + 1)( f R i + γ ) . If th is r atio is b oun ded by a constan t less than 1, th en the gap δ sh rinks b y a constan t factor for eac h step of recursion, th us an exp onenti al deca y wo u ld ha ve b een established. Ho we ver, suc h a 15 step-wise guarant ee of deca y h olds in general only wh en the γ is substanti ally greater than Γ( β ). A sim ulation sh o ws that wh en γ is sufficiently close to Γ( β ), the gap δ ma y indeed increase for some sp ecific d and R i . W e then apply an amortized analysis to sho w that ev en though the gap δ may o ccasionally increase, it deca ys exp onential ly in a long r u n. 5.1 Amortized analysis of correlation deca y W e use the p oten tial metho d to analyze the amortized b eh avior of correlatio n deca y . Th e p oten tial function is d efined as Φ( R ) = R D +1 2 D ( β R + 1) , where D is the crucial d wh ic h generates th e highest un iqueness thresh old as formally defined in Section 4 W e will analyze th e deca y r ate of δ Φ instead of δ . This is done by introd ucing a monotone f unction ϕ ( R ), wh ich is implicitly defin ed by its deriv ative ϕ ′ ( R ) = 1 Φ( R ) . W e denote th at y v = ϕ ( R v ) and y v + ǫ v = ϕ ( R v + δ v ). Recall that R v + δ v = f ( R v 1 , . . . , R v d ) and R v = f ( R v 1 + δ v 1 , . . . , R v d + δ v d ) where f ( x 1 , . . . , x d ) = β d 1 γ d 0 Q d i =1 β x i +1 x i + γ . Th en y v = ϕ ( R v ) = ϕ ( f ( R v 1 + δ v 1 , . . . , R v d + δ v d )) = ϕ  f ( ϕ − 1 ( y v 1 + ǫ v 1 ) , . . . , ϕ − 1 ( y v d + ǫ v d ))  ; and y v + ǫ v = ϕ ( R v + δ v ) = ϕ ( f ( R v 1 , . . . , R v d )) = ϕ  f ( ϕ − 1 ( y v 1 ) , . . . , ϕ − 1 ( y v d ))  . By the Mean V alue Theorem, there exists an e R ∈ [ R v , R v + δ v ] su c h that ǫ v = ϕ ( R v + δ v ) − ϕ ( R v ) = δ v · ϕ ′ ( e R ) = δ v Φ( e R ) . (7) By the Mean V alue Theorem, there exist f R i ∈ [ R v i , R v i + δ v i ], 1 ≤ i ≤ d , su c h that ǫ v = ϕ ( f ( R v 1 , . . . , R v d )) − ϕ ( f ( R v 1 + δ v 1 , . . . , R v d + δ v d )) = −∇ ϕ  f ( f R 1 , . . . , f R d )  · ( δ v 1 , . . . , δ v d ) = (1 − β γ ) ·  β d 1 γ d 0 Q d i =1 β f R i +1 f R i + γ  D − 1 2 D β β d 1 γ d 0 Q d i =1 β f R i +1 f R i + γ + 1 · d X i =1 δ v i ( β f R i + 1)( f R i + γ ) ≤ d γ ( d 0 + d 1 + d ) D − 1 2 D , (8) where (8) is trivially implied b y that f R i ∈ (0 , 1], γ > 1 and β γ < 1. By the Mean V alue Theorem, there exist e y i ∈ [ y v i , y v i + ǫ v i ] and due to the monotonicit y of ϕ ( · ), corresp onding f R i ∈ [ R v i , R v i + δ v i ] 16 that e y i = ϕ ( f R i ), 1 ≤ i ≤ d , su c h that ǫ v = ϕ  f ( ϕ − 1 ( y 1 ) , . . . , ϕ − 1 ( y d ))  − ϕ  f ( ϕ − 1 ( y 1 + ǫ 1 ) , . . . , ϕ − 1 ( y d + ǫ d ))  = −∇ ϕ  f ( ϕ − 1 ( e y 1 ) , . . . , ϕ − 1 ( e y d ))  · ( ǫ 1 , . . . , ǫ d ) = (1 − β γ )  β d 1 γ d 0 Q d i =1 β f R i +1 f R i + γ  D − 1 2 D β β d 1 γ d 0 Q d i =1 β f R i +1 f R i + γ + 1 · d X i =1 f R i D +1 2 D · ǫ v i f R i + γ ≤ max 1 ≤ i ≤ d { ǫ v i } · (1 − β γ )  β d 1 γ d 0 Q d i =1 β f R i +1 f R i + γ  D − 1 2 D β β d 1 γ d 0 Q d i =1 β f R i +1 f R i + γ + 1 · d X i =1 f R i D +1 2 D f R i + γ . (9) Since f R i ∈ (0 , 1], γ > 1, and β γ < 1, (9) trivially implies that ǫ v ≤ max 1 ≤ i ≤ d { ǫ v i } · d β d 0 γ d 1 d Y i =1 β f R i + 1 f R i + γ ! D − 1 2 D ≤ d γ ( d 0 + d 1 + d ) D − 1 2 D · m ax 1 ≤ i ≤ d { ǫ v i } , (10) On the other hand , we kno w that β ≤ √ β Γ < D − 1 D +1 (due to Lemma 15 in Sectio n 4). It is easy to v erify that function x D − 1 2 D β x +1 is monotonically increasing w hen x ≤ 1. Then the follo wing is also implied b y (9): ǫ v ≤ α ( d ; f R 1 , . . . , f R d ) · max 1 ≤ i ≤ d { ǫ v i } . (11) where the f unction α ( d ; x 1 , . . . , x d ) captures the amortized deca y , d efined as α ( d ; x 1 , . . . , x d ) = (1 − β γ )  Q d i =1 β x i +1 x i + γ  D − 1 2 D β Q d i =1 β x i +1 x i + γ + 1 · d X i =1 x D +1 2 D i x i + γ . (12) Our goal is to u pp er b oun d the α ( d ; x 1 , . . . , x d ) assum ing th e uniqueness condition. A conca ve analysis reduces the upp er b ound to the symmetric cases that all x i are equal. Lemma 16 L et 0 ≤ β < 1 , γ > Γ( β ) , and β γ < 1 . Then for any d ≥ 1 and any x 1 , . . . , x d ∈ (0 , 1] , ther e exists an x ∈ (0 , 1] , such that α ( d ; x 1 , . . . , x d ) is maximize d when al l x i = x . Pro of: W e denote y i = ln( β x i +1 x i + γ ), then x i = 1 − β γ e y i − β − γ and α ( d ; x 1 , . . . , x d ) = (1 − β γ ) exp  D − 1 2 D P d i =1 y i  β exp  P d i =1 y i  + 1 · d X i =1  1 − β γ e y i − β − γ  D +1 2 D 1 − β γ e y i − β = exp  D − 1 2 D P d i =1 y i  β exp  P d i =1 y i  + 1 · d X i =1 f ( y i ) , 17 where f ( y ) =  1 − β γ e y − β − γ  D +1 2 D ( e y − β ). It holds that f ′ ( y ) = e y  1 − β γ e y − β − γ  D +1 2 D 1 + D +1 2 D (1 − β γ ) γ e y − 1 ! , f ′′ ( y ) = e y  1 − β γ e y − β − γ  D +1 2 D 4 D 2 ( e y − β )( γ e y − 1) 2 · g ( y , D ) , where g ( y , D ) = e y ( β γ − 1) 2 − 2(1 − β γ )( e y − β )(1 − γ e y ) D − (2 β + 2 β 2 γ − e y (1 + 10 β γ + β 2 γ 2 ) + 6 γ e 2 y (1 + β γ ) − 4 γ e 3 y ) D 2 . The fact e y ∈ ( β , 1 γ ) implies that the sign of f ′′ ( y ) is the same as that of g ( y , D ). In the follo w, w e sho w that g ( y , D ) is alw a ys negativ e. The co efficient of D in g ( y ) is obviously negativ e gi v en that e y ∈ ( β , 1 γ ). No w we show that the co efficien t of D 2 in g ( y ) is also negativ e. T o sh o w this, the condition e y ∈ ( β , 1 γ ) is n ot sufficient. W e su b stitute y i = ln( β x i +1 x i + γ ) bac k and recall that x i ∈ (0 , 1), w e h a ve 2 β + 2 β 2 γ − e y (1 + 10 β γ + β 2 γ 2 ) + 6 γ e 2 y (1 + β γ ) − 4 γ e 3 y = ( β γ − 1) 2 ( γ + x ) 3 · ( γ 2 − x 2 + γ (1 − β γ ) x + 3 γ x + 4 β γ x 2 + β x 3 ) > 0 . Since b oth the co efficien ts of D and D 2 are negativ e, we can choose D = 1, in whic h case, f ′′ ( y ) = − γ e y < 0 . Denote that ¯ y = 1 d P d i =1 y i . Due to the Jensen’s In equalit y , P d i =1 f ( y i ) ≤ d f ( ¯ y ). Therefore, α ( d ; x 1 , . . . , x d ) = exp  D − 1 2 D P d i =1 y i  β exp  P d i =1 y i  + 1 · d X i =1 f ( y i ) ≤ exp  d ( D − 1) 2 D ¯ y  β exp ( d ¯ y ) + 1 · d f ( ¯ y ) . Let x satisfy that ¯ y = ln( β x +1 x + γ ), i.e. x = 1 − β γ e ¯ y − β − γ . It is then easy to v erify that x ∈ (0 , 1] since all x i ∈ (0 , 1] and y i = ln( β x i +1 x i + γ ) is monoto n e with resp ect to x i . Th er efore, α ( d ; x 1 , . . . , x d ) is maximized when all x i = x ∈ (0 , 1]. W e then deal with the symmetric case. Let α ( d, x ) = α ( d ; x, . . . , x | {z } d ) = d (1 − β γ ) x D +1 2 D ( β x + 1) d ( D − 1) 2 D ( x + γ ) 1+ d ( D − 1) 2 D  β  β x +1 x + γ  d + 1  . 18 Let f ( x ) =  β x +1 x + γ  d b e the symmetric version of the recursion (1). Observ e that α ( d, x ) = Φ( x ) Φ( f ( x )) | f ′ ( x ) | , which is exactly the amortized d eca y ratio in the symmetric case. Recall the formal d efi nitions of D and X in Defin ition 14 in Section 4. O ur main disco v ery is the follo wing lemma which states th at at the uniquen ess thr eshold γ = Γ( β ), the v alue of α ( d, x ) is maximized at d = D and x = X with α ( D , X ) = 1. It is in d ebt to the magic of the p oten tial metho d to observ e suc h a harmoniously b eautiful coincidence b et we en amortized correlation deca y and phase transition of uniqueness. Lemma 17 L et 0 ≤ β < 1 and γ = Γ( β ) . It holds that sup d ≥ 1 0 0 is indep enden t of d . Thus, ∂ α ( d, x ) ∂ d = C 1 z D − 1 2 D 2 D ( β z + 1) 2 · g ( z ) , where the f unction g ( z ) is defin ed as g ( z ) = 2 D ( β z + 1) − (( D + 1)( β z + 1) − 2 D ) ln z . It is ob vious that C 1 z D − 1 2 D 2 D ( β z +1) 2 > 0, th u s the sign of ∂ α ( d,x ) ∂ d is go verned b y g ( z ). Note that 0 < z =  β x +1 x +Γ  d < 1. Then d g ( z ) d z = 1 z (( D − 1)( β z + 1) − ( D + 1) β z ln z ) > ( D − 1)( β z + 1) ≥ 0 Therefore, g ( z ) is strictly increasing w ith resp ect to z . Due to Lemma 15, it holds that ln  β X +1 X +Γ  = 2( β X +1) ( D +1)( β X +1) − 2 D . Thus, g ( X ) = 2 D ( β X + 1) − D (( D + 1)( β X + 1) − 2 D ) ln  β X + 1 X + Γ  = 0 . 19 Therefore, ∂ α ( d,x ) ∂ d < 0 wh en z < X ; ∂ α ( d,x ) ∂ d = 0 wh en z = X ; and ∂ α ( d,x ) ∂ d > 0 wh en z > X . Note that z =  β x +1 x +Γ  d is monotonically decreasing with resp ect to d since  β x +1 x +Γ  < 1 Γ < 1. Let ρ ( x ) = ln X ln( β x +1) − ln( x +Γ) . It is then easy to v erify that ∂ α ( d, x ) ∂ d      < 0 if d > ρ ( x ) , = 0 if d = ρ ( x ) , > 0 if d < ρ ( x ) . Therefore, for any d and x , α ( d, x ) ≤ α ( ρ ( x ) , x ). Recall th at α ( d, x ) = (1 − β Γ) x D +1 2 D x +Γ · dz D − 1 2 D ( β z +1) , where z =  β x +1 x +Γ  d . When d = ρ ( x ) = ln X ln( β x +1) − ln ( x +Γ) , it holds that z =  β x +1 x +Γ  d = X . Therefore, α ( ρ ( x ) , x ) = C 2 · x D +1 2 D ( x + Γ)(ln( β x + 1) − ln( x + Γ)) , where C 2 = (1 − β Γ) X D − 1 2 D ln X ( β X + 1) is indep enden t of x , and C 2 < 0 since 0 < X < 1. d α ( ρ ( x ) , x ) d x = C 2 x − D +1 2 D · h ( x ) 2 D ( x + Γ) 2 ( β x + 1)  ln  β x +1 x +Γ  2 , where h ( x ) = 2 D (1 − β Γ) x − ( β x + 1)(2 D x − ( D + 1)( x + Γ)) · ln  β x + 1 x + Γ  . It is easy to see that C 2 x − D +1 2 D 2 D ( x +Γ) 2 ( β x +1) ( ln ( β x +1 x +Γ )) 2 < 0 and h ( x ) is m onotonically increasing. Due to Lemma 15, ln  β X +1 X +Γ  = 2 D (1 − β Γ) X ( β X +1)(2 DX − ( D +1)( X +Γ)) , thus h ( X ) = 0. Therefore, d α ( ρ ( x ) ,x ) d x is monotonically decreasing w ith resp ect to x and d α ( ρ ( x ) ,x ) d x    x = X = 0, w hic h implies th at for any x , α ( ρ ( x ) , x ) ≤ α ( ρ ( X ) , X ). Due to (2), it holds that X =  β X +1 X +Γ  D , th us ρ ( X ) = ln X ln ( β X +1 X + Γ ) = D , hence α ( ρ ( X ) , X ) = α ( D , X ). In conclusion, assuming 0 ≤ β < 1 and γ = Γ( β ), for an y d ≥ 1 and 0 < x ≤ 1, it holds that α ( d, x ) ≤ α ( ρ ( x ) , x ) ≤ α ( ρ ( X ) , X ) = α ( D , X ) = 1 . As a consequ en ce of the ab o ve lemma, a strict upp er b ound is obtained as follo w s. Lemma 18 F or 0 ≤ β < 1 and Γ( β ) < γ < 1 β , ther e exists a c onsta nt α < 1 such that for any d ≥ 1 and 0 < x ≤ 1 , it holds that α ( d, x ) ≤ α . 20 Pro of: Let α β ,γ = sup d ≥ 1 0 0 is ind ep endent of γ . Let h ( γ ) = ( x + γ ) d ( D +1) 2 D ( β ( β x +1) d +( x + γ ) d ) . d h ( γ ) d γ = d ( D + 1)( x + γ ) d (3 D + 1) 2 D − 1 2 D ( β ( β x + 1) d + ( x + γ ) d ) 2 ·  β  β x + 1 x + γ  − D − 1 D + 1  < d ( D + 1)( x + γ ) d (3 D + 1) 2 D − 1 2 D ( β ( β x + 1) d + ( x + γ ) d ) 2  β Γ − D − 1 D + 1  < 0 , where the s econd to the last inequalit y holds b ecause x > 0 and γ > Γ, and the last inequalit y is due to Lemma 15. Th e fact that d h ( γ ) d γ < 0 imp lies that h ( γ ) is s tr ictly decreasing. Thus, α ( d, x ) = C 3 ·  1 − β γ x + γ  · h ( γ ) is s trictly decreasing with resp ect to γ o ver γ ∈ (Γ , 1 β ). Let α γ ( d, x ) denote the α ( d, x ) with parameter γ . W e can assu m e th at there exist finite d ∗ ≥ 1 and constant 0 < x ∗ ≤ 1 ac h ieving that α γ ( d ∗ , x ∗ ) = sup d ≥ 1 0 1 which dep e nd only on β and γ , for every vertex v ∈ B M ( L ) , assuming that v ha s d 0 childr en fixe d to b e b lu e , d 1 childr en fixe d to b e gr e en, and d fr e e childr en v 1 , . . . , v d , it hol ds that ǫ v ≤ M α ⌈ log M ( d 0 + d 1 + d +1) ⌉− 1 ; (13) ǫ v ≤ α ⌈ log M ( d 0 + d 1 + d +1) ⌉ · max 1 ≤ i ≤ d { ǫ v i } . (14) 21 Pro of: W e c h o ose α to b e th e one in Lemma 19, and M > 1 to satisfy k γ k D − 1 2 D ≤ α ⌈ log M k ⌉ for k ≥ M . (15) Due to (8), ǫ v ≤ d γ ( d 0 + d 1 + d ) D − 1 2 D ≤ M α ⌈ log M ( d 0 + d 1 + d +1) ⌉− 1 , where the last inequalit y follo ws from (15) if d 0 + d 1 + d ≥ M and the case d 0 + d 1 + d < M is trivial since d γ ( d 0 + d 1 + d ) D − 1 2 D < d ≤ M . T h u s (13 ) is p ro ved. Due to (11), ǫ v ≤ α ( d ; f R 1 , . . . , f R d ) · max 1 ≤ i ≤ d { ǫ v i } where R v i ≤ f R i ≤ R v i + δ v i . S ince v ∈ B M ( L ), its children v i ∈ B ∗ M ( L ). As we d iscussed in th e b eginning of this section, 0 < R v i ≤ R v i + δ v i ≤ 1, th us f R i ∈ (0 , 1]. Then due to Lemma 19, there is a constan t α < 1, ǫ v ≤ α ( d ; f R 1 , . . . , f R d ) · max 1 ≤ i ≤ d { ǫ v i } ≤ α · max 1 ≤ i ≤ d { ǫ v i } . (16) Th us, (14) h olds trivially when d 0 + d 1 + d < M . As for d 0 + d 1 + d ≥ M , d ue to (10), ǫ v ≤ d γ ( d 0 + d 1 + d ) D − 1 2 D · max i { ǫ v i } ≤ α ⌈ log M ( d 0 + d 1 + d +1) ⌉ · max 1 ≤ i ≤ d { ǫ v i } . Therefore, (14) is prov ed. Pro of of Theorem 5. W e p ro ve by stru ctural in duction in B M ( L ). The hyp othesis is ∀ v ∈ B M ( L ) , ǫ v ≤ M α L − L M ( v ) − 1 . F or the basis, w e consider those v ertices v ∈ B M ( L ) whose children are in B ∗ M ( L ) \ B M ( L ). The fact that the children of v are not in B M ( L ) implies that L M ( v ) + ⌈ lo g M ( d 0 + d 1 + d + 1) ⌉ > L , where d 0 + d 1 + d is the num b er of c h ildren of v . Then due to (13) of Lemma 20, ǫ v ≤ M α ⌈ log M ( d 0 + d 1 + d +1) ⌉− 1 ≤ M α L − L M ( v ) − 1 . The induction step is s traigh tforwa rd. F or every child v i of v , L M ( v i ) = L M ( v ) + ⌈ log M ( d 0 + d 1 + d + 1) ⌉ . Sup p ose that th e ind u ction hyp othesis is true for all v i . Due to (14) of Lemma 20, ǫ v ≤ α ⌈ log M ( d 0 + d 1 + d +1) ⌉ · m ax 1 ≤ i ≤ d { ǫ v i } ≤ α ⌈ log M ( d 0 + d 1 + d +1) ⌉ + L − L M ( v i ) − 1 = α L − L M ( v ) − 1 . The h yp othesis is p ro ved. Finally , f or the ro ot r of the tree, L M ( r ) = 0, th us due to (7), there exists an e R ∈ [ R r , R r + δ r ] ⊆ (0 , 1] s u c h that δ r = Φ( e R ) · ǫ r ≤ e R D +1 2 D ( β e R + 1) · M α L − L M ( r ) − 1 ≤ 2 M α L − 1 . As we d iscussed in the b eginning of this section, this imp lies Th eorem 5 . 22 6 A tigh t analysis for β = 0 In this section, we giv e a sligh tly improv ed and tight analysis (sin ce w e also h a ve a hardness result) of the algorithm wh en β = 0. In the d efinition of Γ( β ), we tak e the maxim um o ver all the p ossible real d ≥ 1. As degrees of graphs, only those inte ger v alues hav e physical meanings an d w e also b eliev e that the maxim um v alue o ver all the integ er d gives the right b oundary b et wee n tractable and hard. In the follo wing, w e s h o w how to extend our resu lt to integral d for the sp ecial case of β = 0. Recall that 0 ≤ β < 1, ˆ x satisfies ˆ x =  β ˆ x +1 ˆ x + γ  d . The int eger version of Γ( β ) can b e formally defined as Γ ∗ ( β ) = inf  γ ≥ 1    ∀ d ∈ { 1 , 2 , 3 , . . . } , d (1 − β γ )( β ˆ x + 1) d − 1 ( ˆ x + γ ) d +1 ≤ 1  . F or β = 0, we can solve it and ha ve that Γ ∗ (0) = max d ∈{ 1 , 2 , 3 ,... } ( d − 1) d − d d +1 . It is ea sy to v erify that ( d − 1) d − d d +1 is monotonously in creasing when d ≤ 11, decreasing when d ≥ 12 and reac h ing the maxim u m when d = 11. Th erefore Γ ∗ (0) = 10 · 11 − 11 12 . W e notice that Γ ∗ (0) = 10 · 11 − 11 12 ≈ 1 . 110171 4 and the con tin uous version Γ(0) ≈ 1 . 11017 15. The in tegralit y gap is almost negligible, esp ecially wh en compared to the previous b est b ou n dary for γ when β = 0 pro vided b y th e heat-bath random w alk algorithm in [41], which is appro ximately 1.32. Theorem 21 L et A =  0 1 1 γ  , wher e γ > Γ ∗ (0) = 10 · 11 − 11 12 . Ther e is an FPT AS for Z A ( G ) . Pro of: The algorithm is exactly the same as the alg orithm in Sect ion 3. What we need is to establish a correlat ion d eca y . F or this, we use a sp ecial p oten tial function by substituting β = 0 and D with D ∗ = 11. Therefore the p oten tial function is Φ( R ) = R D ∗ +1 2 D ∗ = R 6 11 . The analysis remain the same as b efore, except Lemma 17, w hic h is the only place assuming con tinuous d in the old a nalysis. W e need to repro ve Lemma 17 for in tegral d . The symmetric amortized d eca y α ( d, x ) is now written as α ∗ ( d, x ) = dx 6 11 ( x + γ ) 1+ 5 d 11 . W e are ab out to sh o w that if γ > Γ ∗ (0), th ere is a constan t α < 1 suc h that α ∗ ( d, x ) ≤ α < 1 for all 0 ≤ x < 1. Also b y the strict monotonicit y , w e only need to pr o ve (b y substituting γ with Γ ∗ (0)) α ∗ ( d, x ) = dx 6 11 ( x + Γ ∗ (0)) 1+ 5 d 11 ≤ 1 . T ake the p artial d eriv ativ e of α ∗ o v er x , we hav e ∂ α ∗ ∂ x = − d 11 x 5 11 ( x + Γ ∗ (0)) 2+ 5 d 11 ((5 + 5 d ) x − 6Γ ∗ (0)) . 23 F or a fix ed d , when x < 6Γ ∗ (0) 5+5 d , α ∗ ( d, x ) is monoto nous increasing with x and when x > 6Γ ∗ (0) 5+5 d , α ∗ ( d, x ) is mon otonous decreasing with x . S o α ∗ ( d, x ) reac h its maxim um w hen x = 6Γ ∗ (0) 5+5 d . Su b- stituting this in to α ∗ ( d, x ), w e h a ve α ∗ ( d, x ) ≤ ˆ α ( d ) = 2 1 11 3 6 11 11 5(1+ d ) 12 d (1 + d ) 5 11 (11 + 5 d )(10 + 12 1+ d ) 5 d 11 . W e can verify that ˆ α ( d ) is monotonously in creasing when d ≤ 11 and decreasing when d ≥ 12 and it reac h its maxim um wh en d = 11. T he maximum is ˆ α (11) = 1. This completes the pro of. F or β = 0, it is ve r y related to the hardcore mo d el. W e can mak e us e of the hardness result in [58] and [31] to get a tigh t h ardness r esult as follo ws. Theorem 22 L et A =  0 1 1 γ  , wher e γ < Γ ∗ (0) = 10 · 11 − 11 12 . Ther e i s no FPRAS for Z A ( G ) unless N P = RP . Pro of: The starting p oint is th e h ard ness result for h ardcore m o del in [58]. F or hardcore mo d el, the partition fun ction is Z λ ( G ) = X S ∈ I ( G ) λ | S | , where the sum m ation go es o ver all the indep end en t set of G . F or β = 0, nonzero terms in the summation Z A ( G ) = X σ ∈ 2 V Y ( i,j ) ∈ E A σ ( i ) ,σ ( j ) ha ve a one-to-one corresp on d ing with all the indep endent sets of G . The term ind exed b y σ is nonzero iff σ − 1 (0) is an indep end en t set of G . S o Z A ( G ) can b e rewritten as Z A ( G ) = γ | E | X S ∈ I ( G ) Y v ∈ S γ − d ( v ) , where d ( v ) is the degree of vertex v . If G is a d -regular graph , this sum mation can b e further rewritten as Z A ( G ) = γ | E | X S ∈ I ( G ) ( γ − d ) | S | . Since γ | E | is a global factor whic h can b e easily compu ted, th e computation for Z A ( G ) of d -regular graph G is equiv alent to the partition function of the hardcore mod el on G with fugac ity parameter γ − d . In [58] and [31], it is prov ed that there is n o FPRAS for the partition f unction for hardcore mo del on graphs with maxim um degree d when the fugacit y p arameter λ > ( d − 1) d − 1 ( d − 2) d unless NP= RP , when d ≥ 6. If we can strength th e h ard ness r esult to d -regular graph, we can u se th e equiv alence relation to get a hard ness result for the the t wo -sp in sys tem mo del when β = 0 and γ − d > ( d − 1) d − 1 ( d − 2) d . Let d = 12 , the inequalit y giv es γ < 10 · 11 − 11 12 , as what w e claimed. In the f ollo wing, we show that their h ardness pro of for hardcore mo del indeed already works for d -regular graph. T o p ro ve the h ardness of the hardcore mod el. A reduction from the m ax-cut problem to the hardcore partition fu nction is built in [5 8 ]. The hard ins tance of the h ardcore problem in their 24 reduction is almost d -reg u lar except some v ertices with degree d − 1. It can b e easily ve r ified in their gadget that if we are starting fr om a max-cut instan t in a r egular graph, we can c ho ose the suitable parameter and b uild the redu ction to a d -regular ins tance in the hardcore mo del. So it remains to sho w that max-cut on a regular graph is already NP-hard. This can b e d one by a simple r eduction from m ax-cut on arbitrary graph to a max-cut instance of a r egular graph. Let G = ( V , E ) b e a giv en max-cut instance. Let ∆ b e the maxim u m degree of G . Then the new instance is of 2∆-regular. The new graph G ′ = ( V ′ , E ′ ) is defined as follo ws: • F or ev ery v ertex v ∈ V , we construct 1 + 2(∆ − d ( v )) ve rtices in V ′ , we name them as v and v + i , v − i for i = 1 , 2 , . . . , ∆ − d ( v ). These are all the v ertices in the new graph G ′ . • F or eve r y v ∈ V and i ∈ { 1 , 2 , . . . , ∆ − d ( v ) } , we conn ect 2∆ − 1 edges in G ′ b et ween v + i and v − i , one ed ge b et wee n v and v + i , and one edge b et ween v and v − i . • F or ev ery ( u, v ) ∈ E b e an edge of E , we conn ect tw o edges b et we en u and v in G ′ . It is easy to see that all the v ertices in graph G ′ ha ve degree 2∆. F or a max-cut for G ′ , we will alw a ys put v + i and v − i in to different sides f or every v ∈ V and i ∈ { 1 , 2 , . . . , ∆ − d ( v ). If not, one can impro v e th e cut b y mo ving one of them to the other side. Given that v + i and v − i are alw ays in differen t sides, the con trib ution in the cut for the edges b et wee n v + i and v − i , v and v + i , v and v − i are all fixed. The remaining part is identi cal to the original graph except that we doub le ev ery edge. This fin ishes the r eduction and completes the pro of. 7 Op en Questions Our analysis of correlation deca y assu m es a con tinuous degree d b ecause of the the using of differ- en tiation. An op en question is to impro ve the analysis to integral d and th e un iqu eness thresh old realized b y infi n ite ( d + 1)-regular trees b T d . It will b e v ery inte r esting to pro ve a hardness result b e- y ond this threshold and observe the similar trans ition of computational complexit y in spin systems as in the h ardcore mo del [58]. In this p ap er, we consider the tw o-state spin systems without external fields. I t will b e in ter- esting to extend our r esult to cases where th ere is an external fi eld as in [41]. Since th e hardcore mo del can b e expressed as a t wo-stat e spin system w ith an external field. This w ill giv e a u nified theory co ve r ing th e pr evious r esults f or the hardcore mo del. 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