Improved Inapproximability Results for Counting Independent Sets in the Hard-Core Model

Impro v ed Inappro ximability Results for Coun ting Indep enden t Sets in the Hard-Core Mo del Andreas Galanis ∗ Qi Ge † Daniel ˇ Stefank o vi ˇ c † Eric Vigo da ⋆ Linji Y ang ⋆ August 17, 20 1 8 Abstract W e study the computational complexit y of approximately coun ting the num be r of independent sets of a graph with maximum degree ∆. More gener ally , for a n input gra ph G = ( V , E ) and an activity λ > 0, w e are in terested in the quantit y Z G ( λ ) defined as the sum o ver indep enden t sets I weigh ted as w ( I ) = λ | I | . In statistical ph ysics, Z G ( λ ) is the partition function for the hard-core mo del, whic h is an idea lized mo del of a gas where the particles ha ve non-neg ligible size. Recen tly , an in teresting phase transition was sho wn to o ccur for the complexity of approximating the partition function. W eitz show ed a n FP AS for the partition function for any g raph of maximum degree ∆ when ∆ is constan t a nd λ < λ c ( T ∆ ) := (∆ − 1) ∆ − 1 / (∆ − 2) ∆ . The quant ity λ c ( T ∆ ) is the critical p oin t for the s o-called uniqueness thre shold o n the infinite, regular tree o f degree ∆. On the other side, Sly prov ed that there do es not exist efficient (randomized) approximation algorithms for λ c ( T ∆ ) < λ < λ c ( T ∆ ) + ε (∆), unless NP=RP , for some function ε (∆) > 0 . W e remov e the upp er bo und in the assumptions of Sly’s result for ∆ 6 = 4 , 5, that is, we sho w that there do es no t exist efficien t ra ndomized approximation algor ithms for all λ > λ c ( T ∆ ) for ∆ = 3 a nd ∆ ≥ 6 . Sly’s inapproximability result uses a clev er r eduction, combined with a second-moment analysis of Mossel, W eitz a nd W or mald which prove to rpid mixing of the Gla uber dynamics for sampling from the asso ciated Gibbs distribution on almost every r egular graph o f deg ree ∆ for the same rang e of λ as in Sly’s result. W e extend Sly’s result by improving up on the tech nical work of Mossel et al., via a more detailed analysis of indep enden t sets in random regular g raphs. 1 In tro duction F or a graph G = ( V , E ) and activit y λ > 0, the hard-core mo del is d efined on the set I ( G ) of indep endent sets of G where set I ∈ I ( G ) has weigh t w ( I ) := λ | I | . The so-c alled partition function for the mo del is defined as: Z G ( λ ) := X I ∈I ( G ) w ( I ) = X I ∈I ( G ) λ | I | . The Gibbs distr ibution µ is o v er th e set I ( G ) wh ere µ ( I ) = w ( I ) / Z G ( λ ). The case λ = 1 is esp eciall y in teresting fr om a com binatorial p ersp ectiv e, s ince the partition fun ctio n is the num b er of indep endent sets in G and the Gibbs distribu tio n is uniformly distributed o v er the set of indep enden t sets. The h ard-core mo del has receiv ed considerable at tent ion in sev eral fields. In stat istical physics, it is stud ied as an idealized mo del of a gas w here the gas particles ha ve non-negligible size so ∗ School of Computer Science, Georgia Institute of T ec hnology , A tlanta GA 30332. Email: { agalanis ,vigo da,lj yang } @cc.gatec h.edu. Researc h supp orted in part by NSF gran t CCF-083029 8 and CCF-0910584 . † Department of Computer Science, Universit y of Ro c hester, Ro c hester, NY 146 27. Email: { qge,stefank o } @cs.rochester.edu. Researc h supp orted in part by NSF gran t CCF-09104 15. 1 neigh b oring sites cannot sim ultaneously b e o ccupied [ 3 , 1 ]. The activit y λ corresp onds to the fugacit y of the gas. The mod el also arose in op erations researc h in the study of comm unication net w orks [ 7 ]. W e study t he computational complexit y of appro ximating the partition f unction. V alia nt [ 15 ] pro ve d that exactly computing the n umber of indep enden t sets of an in put graph G = ( V , E ) is #P-complete. Greenhill [ 4 ] prov ed that eve n when the input is restricted to graphs with maxim um degree 3, it is still #P-complete . Hence, our fo cus is on approxima ting the p artiti on function. W eitz [ 16 ] ga v e an FP AS ( fully polynomial-time appro ximation sc heme) fo r the partitio n function of graphs with maxim um degree ∆ when ∆ is constan t and λ < λ c ( T ∆ ) := (∆ − 1) ∆ − 1 / (∆ − 2) ∆ . The activit y λ c ( T ∆ ) is the critical activit y f or the threshold of uniquen ess/non-uniqueness o f the infi nite- v olume Gibbs measures on th e infin ite ∆-regular tree [ 7 ]. Recen tly , S ly [ 11 ] prov ed that, unless N P = R P , for ev ery ∆ ≥ 3, there exists a f unction ε ( ∆) > 0 such that for graphs with maxim um degree ∆ there do es not exist an FPRAS (fully-p olynomial time randomized appro ximation sc heme) for the partition function at act ivit y λ satisfying: λ c ( T ∆ ) < λ < λ c ( T ∆ ) + ε (∆) . ( ⋆ ) It wa s conjectured in Sly [ 11 ] and Mossel et al. [ 10 ] th at the inappro ximabilit y r esult h olds for all λ > λ c ( T ∆ ). W e almost resolv e this conjecture, that is w e p ro v e the conjecture for all ∆ with th e exception of ∆ ∈ { 4 , 5 } . Theorem 1. U nless N P = RP, ther e do es not exist an FPRAS for the p artition fu nction of the har d-c or e mo del for gr aphs of maximum de gr e e at most ∆ at activity λ when: • ∆ = 3 and λ > λ c ( T 3 ) = 4 ; or • ∆ ≥ 6 and λ > λ c ( T ∆ ) ; or • ∆ = 4 and λ ∈ ( λ c ( T 4 ) = 1 . 6875 , 2 . 01538] ∪ (4 , + ∞ ) ; or • ∆ = 5 and λ ∈ ( λ c ( T 5 ) = 256 / 243 , 1 . 45641] ∪ (1 . 6875 , 2 . 01538] ∪ (4 , + ∞ ) . Sly’s work utilizes earlier w ork of Mossel et al. [ 10 ] which studied the Glaub er d ynamics. The Glaub er dynamics is a simp le Mark o v chain ( X t ) that is used to sample fr om the Gibbs distribu tio n (and hence to approximat e the partition function via no w standard tec hniques, see [ 6 , 12 ]). F or an inpu t graph G = ( V , E ) and activit y λ > 0, the state space of th e c hain is I ( G ). F rom a state X t ∈ I ( G ), the transitions X t → X t +1 are defined b y the follo wing sto c hastic process: • C hoose a ve rtex v un iformly at random from V . • L et X ′ = ( X t ∪ { v } with probabilit y λ/ (1 + λ ) X t \ { v } with p robabilit y 1 / (1 + λ ) . • I f X ′ ∈ I ( G ), then set X t +1 = X ′ , ot herw ise set X t +1 = X t . It is straigh tforward to v erify that the Glaub er d ynamics is ergod ic, and the un ique stationary distribution is the Gibbs distrib ution. The m ixing time T mix is the m inim um n umb er of s te ps T from the w orst initial state X 0 , so that the distrib ution of X T is within (total) v ariatio n distance ≤ 1 / 4 of the stationary distr ibution. The c hain is said to b e r apid ly mixing if the mixing time is p olynomial in n = | V | , and it is said to b e torpid ly mixing if th e mixing time is exp onen tial in n 2 (for the purp oses of this pap er, that means T mix = exp(Ω( n ))) . W e refer the reader to L evin et al. [ 8 ] f or a m ore thorough in tro duction to the Glaub er d ynamics. Mossel et al. [ 10 ] p ro v ed that the Glaub er dynamics is torpidly mixing, for all ∆ ≥ 3, for graph s with maxim um degree ∆ wh en λ satisfies ( ⋆ ). This result impro v ed up on earlier work of Dy er et al. [ 2 ] wh ic h held for larger λ , but not do wn to the critical activit y λ c ( T ∆ ). The torpid m ixing result of Mossel et al. [ 10 ] follo ws immediate ly (via a condu cta nce argument) from their main result that for a rand om ∆ -regula r bipartite graph, for λ satisfying ( ⋆ ), an indep enden t set dr a wn from th e Gibb s distribution is “unbala nced” with high probabilit y . The pro of of Mossel et al. [ 10 ] is a tec hnically in vo lv ed second momen t ca lculation that Sly [ 11 ] calls a “tec hnical tour de force”. Our main contribution is to impr o v e up on Mossel et al.’s result, most notably , exte nd ing it to all λ > λ c ( T ∆ ) for ∆ = 3. Our impro v ed anal ysis comes f rom u sing a sligh tly different parameteriz ation of the second moment of th e p artit ion fu nction, w hic h brings in symmetry , and allo ws for simpler pr oofs. T o formally state our results f or indep enden t sets of random regular graphs, we need to partitio n the set of indep end en t sets as follo ws. F or a bipartite graph G = ( V 1 ∪ V 2 , E ) where | V 1 | = | V 2 | = n , for δ > 0, for i ∈ { 1 , 2 } , let I δ i = { I ∈ I ( G ) : | I ∩ V i | > | I ∩ V 3 − i | + δ n } denote the indep endent sets that are un balanced a nd “biased” tow ards V i . Let I δ B = { I ∈ I ( G ) : | I ∩ V i | ≤ | I ∩ V 3 − i | + δ n } denote the set of nearly b ala nced indep endent sets. Let G ( n, ∆) d enote the probability d istribution ov er bip artit e graphs with n + n vertic es formed b y taking the u nion of ∆ random p erfect matc hings. Strictly sp eaking, this distribution is ov er m ulti-graphs. Ho wev er, for indep enden t sets the m ultiplicit y of an edge d oes not matt er so w e can view G ( n, ∆) as a distribu tio n o v er simple graphs with maxim um degree ∆. Moreo v er, since our results hold asymptotica lly almost surely (a.a.s.) ov er G ( n, ∆), as noted in [ 10 , Section 2.1], b y standard argumen ts (see the note after the pro of of [ 9 , Theorem 4]), our results also hold a.a.s. for the uniform distribu tio n ov er bipartite ∆-regular graphs. Theorem 2. F or ∆ = 3 and any λ > λ c ( T ∆ ) , ther e exist a > 1 and δ > 0 such that, asymptotic al ly almost sur ely, for a gr ap h G chosen fr om G ( n, ∆) , the Gibbs distribution µ satisfies: µ ( I δ B ) ≤ a − n min { µ ( I δ 1 ) , µ ( I δ 2 ) } . Ther efor e, the Glaub er dynamics i s torpid ly mixing. This prov es Conjecture 2.4 of [ 10 ] for the case d = 3. F or ∆ ≥ 4 we can extend Mossel et al.’s results to a larger range of λ than previously kno wn. Theorem 3. F or al l ∆ ≥ 4 ther e exists ε (∆) > 0 , for any λ wher e λ c ( T ∆ ) < λ < λ c ( T ∆ ) + ε (∆) , ther e exist a > 1 and δ > 0 such that, asympto tic al ly almo st sur ely, for a gr aph G chosen fr om G ( n, ∆) , the Gibbs distribution µ satisfies: µ ( I δ B ) ≤ a − n min { µ ( I δ 1 ) , µ ( I δ 2 ) } . Ther efor e, the Glaub er dynamics i s torpid ly mixing. The function ε (∆) satisfies: • If ∆ ≥ 6 , ε (∆) ≥ λ c ( T ∆ ) − λ c ( T ∆+1 ) . 3 • F or ∆ = 5 , ε (5) ≥ . 40291 2 . • F or ∆ = 4 , ε (4) ≥ . 32788 7 . In the follo wing section we lo ok at the first and second momen ts of the partition fu nction. W e then state tw o tec hn ic al lemmas (Lemma 10 and Lemma 11 ) from wh ic h Theorems 1 , 2 and 3 easily follo w using w ork of Sly [ 11 ] and Mossel et al. [ 10 ]. In Section 4 w e prov e the tec hnical lemmas. S ome of our p roofs use Mathematica to pr o v e inequalities inv olving r ati onal f unctions, this is discuss ed in Section 2.5 . 2 Ov erview Pro ceeding as in [ 10 ] and [ 11 ], roughly sp eaking, to pro v e Th eorems 1 , 2 and 3 we need to pr o v e that there exist graphs G whose p artit ion function is close to the exp ected v alue (w here the exp ectation is ov er a random G ∼ G ( n, ∆)). At the h ea rt of the argum en t lies a careful analysis of the first tw o momen ts of the partition function. The aim of this S ect ion is to give a brief tec hnical ov erview of the analysis. F or more details, the reader is referr ed to [ 10 ] and [ 11 ]. In Section 2.1 , w e defin e the quan tities w hic h will b e p rev alen t throughout the text. In S ec- tions 2.2 and 2.3 , w e revisit th e fir st and second moments and state ou r main tec h nical Lemma. W e then pro v e Theorems 1 , 2 and 3 in Section 2.4 . In Section 2.5 , we cla rify our use of computer assistance for the p roofs of some tec hnical inequalities. 2.1 Phase T r an sition R evisited Recall, for the infinite ∆-regular tree T ∆ , Kelly [ 7 ] sh o w ed that there is a p hase tran sitio n at λ c ( T ∆ ) = (∆ − 1) ∆ − 1 / (∆ − 2) ∆ . F ormally , this phase transition can b e defined in the follo wing manner. Let T ℓ denote th e complete tree of degree ∆ and conta ining ℓ lev els. Let p ℓ denote th e marginal pr obabilit y that the ro ot is o ccupied in the Gibbs distrib ution on T ℓ . Note, f or even ℓ the ro ot is in the maximum in dep enden t set, whereas for o dd ℓ the ro ot is not in th e maximum indep endent set. Our int erest is in comparing the m argi nal probabilit y for the ro ot in ev en versus o dd sized trees. Hence, let p + = lim ℓ →∞ p 2 ℓ and p − = lim ℓ →∞ p 2 ℓ +1 . One can pro v e these limit s exist by analyzing appropriate recurr ences. The p hase transition on the tree T ∆ captures whether p + equals (or not) p − . F or all λ ≤ λ c ( T ∆ ), w e ha ve p + = p − , and let p ∗ := p + = p − . On the other side, for all λ > λ c ( T ∆ ), we hav e p + > p − . Mossel et al. [ 10 ] exhib ite d the critical role th ese quantit ies p + and p − pla y in th e analysis of th e Gibbs distribution on random graphs G ( n, ∆). 2.2 First Momen t of the Partition F unction Let G ∼ G ( n , ∆). F or α, β ≥ 0, let I α,β G = { I ∈ I G | | I ∩ V 1 | = αn, | I ∩ V 2 | = β n } , that is, α is the fraction of the v ertices in V 1 that are in the indep endent set and β is the fr ac tion of the v ertices in V 2 for an indep endent set of a bipartite graph G . W e consider only v alues of ( α, β ) in th e region R = { ( α, β ) | α, β ≥ 0 and α + β ≤ 1 } , 4 since it is straigh tforw ard to see that for a graph G ∼ G ( n, ∆), it deterministic al ly holds that I α,β G = ∅ wh enev er ( α, β ) / ∈ R . F or ( α, β ) ∈ R , define also Z α,β G = X I ∈I α,β G λ ( α + β ) n , i.e., Z α,β G is the total weigh t of indep endent sets in I α,β G . The fir st momen t of Z α,β G is E G [ Z α,β G ] = λ ( α + β ) n  n αn  n β n   (1 − β ) n αn   n αn  ! ∆ ≈ exp( n Φ 1 ( α, β )) , where Φ 1 ( α, β ) = ( α + β ) ln( λ ) + H ( α ) + H ( β ) + ∆(1 − β ) H  α 1 − β  − ∆ H ( α ) , and H ( x ) = − x ln x − ( 1 − x ) ln(1 − x ) is the entrop y fun ctio n. The asym ptoti c order of E G [ Z α,β G ] follo ws ea sily by Stirling’s appro ximation. The first moment w as analyzed in the wo rk of Dy er et al. [ 2 ]. W e use the follo wing lemma from Mossel et al. [ 10 ] that relates the prop erties of the first momen t to p ∗ , p + and p − . The most imp ortan t asp ect of this lemma is th at in the non-u niqueness region E G [ Z α,β G ] is maximized wh en α 6 = β ([ 2 ]) and more sp ec ifically for ( α, β ) = ( p + , p − ) ([ 1 0 ]). Lemma 4 (Lemma 3.2 and Prop ositi on 4.1 of Mossel et al. [ 10 ]) . The fol lowing holds: 1. the stationary p oint ( α, β ) of Φ 1 over R is the solution to β = φ ( α ) and α = φ ( β ) , wher e φ ( x ) = (1 − x ) 1 −  x λ (1 − x )  1 / ∆ ! , (1) and the solutions ar e exactly ( p + , p − ) , ( p − , p + ) , and ( p ∗ , p ∗ ) when λ > λ c ( T ∆ ) , and the u nique solution is ( p ∗ , p ∗ ) when λ ≤ λ c ( T ∆ ) ; 2. when λ ≤ λ c ( T ∆ ) , ( p ∗ , p ∗ ) is the unique maximum of Φ 1 over R , and when λ > λ c ( T ∆ ) , ( p + , p − ) and ( p − , p + ) ar e the maxima of Φ 1 over R , and ( p ∗ , p ∗ ) is not a lo c al maximum; 3. al l lo c al maxima of Φ 1 satisfy α + β + ∆(∆ − 2) αβ ≤ 1 ; 4. p + , p − , p ∗ satisfy p − < p ∗ < p + and wh en λ → λ c ( T ∆ ) fr om ab ove, we have p ∗ , p − , p + → 1 / ∆ . F or eve ry ∆ ≥ 3, defi ne the regio n R ∆ = { ( α, β ) | α, β > 0 and α + β + ∆(∆ − 2) αβ ≤ 1 } . P art 3 of Lemma 4 establishes that the lo cal m axima of Φ 1 lie in R ∆ . Note that for ∆ ≥ 3, we hav e R ∆ ⊂ R . Hence, the local maxima for all ∆ ≥ 3 lie in the in terior of R . 2.3 Second Momen t of the Partition F unction The s ec ond momen t of Z α,β G satisfies ([ 10 ]) E G [( Z α,β G ) 2 ] ≈ exp( n · max γ ,δ,ε Φ 2 ( α, β , γ , δ, ε )) , 5 where Φ 2 ( α, β , γ , δ, ε ) = 2( α + β ) ln( λ ) + H ( α ) + H 1 ( γ , α ) + H 1 ( α − γ , 1 − α ) + H ( β ) + H 1 ( δ , β ) + H 1 ( β − δ, 1 − β ) + ∆  H 1 ( γ , 1 − 2 β + δ ) − H ( γ ) + H 1 ( ε, 1 − 2 β + δ − γ ) + H 1 ( α − γ − ε, β − δ ) − H 1 ( α − γ , 1 − γ ) + H 1 ( α − γ , 1 − β − γ − ε ) − H 1 ( α − γ , 1 − α )  , (2) and H ( x ) = − x ln( x ) − (1 − x ) ln(1 − x ), H 1 ( x, y ) = − x (ln( x ) − ln( y )) + ( x − y )(ln( y − x ) − ln( y )). T o make Φ 2 w ell defined, the v ariables ha v e to satisfy ( α, β ) ∈ R and γ , δ, ε ≥ 0 , α − γ − ε ≥ 0 , β − δ ≥ 0 , 1 − 2 β + δ − γ − ε ≥ 0 , 1 − α − β − ε ≥ 0 , β − δ + ε + γ − α ≥ 0 . (3) Lemma 4 tells us th at in the non-uniqu eness regio n (whic h is the region of in terest in this pap er) the fi rst momen t is m a ximized wh en ( α, β ) is ( p + , p − ) (or symm etric ally , ( p − , p + )). T o sho w that these unbiased configur at ions dominate the Gibbs d istribution with high p robabilit y (as desired for Theorems 2 an d 3 w e will apply the second momen t method, as used in [ 10 ]. T o that end , w e need to analyze the second momen t for ( α, β ) = ( p + , p − ), and show that E G [( Z α,β G ) 2 ] = O  ( E G [ Z α,β G ]) 2  . T o do that w e need to sho w that for for ( α, β ) = ( p + , p − ), E G [( Z α,β G ) 2 ] is roughly d et ermined by uncorr el ated pairs of confi guratio ns. This crux of this is to sho w that Φ 2 is maximized when γ = α 2 and δ = β 2 , whic h is d eta iled in the up coming condition whic h w as prop osed in [ 11 ]. Condition 1 (Cond itio n 1.2 1 of Sly [ 11 ]) . Ther e exists a c onstant χ > 0 such that when | p + − α | < χ and | p − − β | < χ then g α,β ( γ , δ, ε ) := Φ 2 ( α, β , γ , δ, ε ) achieves its unique maximum in the r e gi on ( 3 ) at the p oint ( γ ∗ , δ ∗ , ε ∗ ) = ( α 2 , β 2 , α (1 − α − β )) . As mentio ned earlier, Condition 1 implies that for ( α, β ) = ( p + , p − ), E G [( Z α,β G ) 2 ] = O  ( E G [ Z α,β G ]) 2  . While the implici t constan t in the latter equalit y is bigge r than one, an applicatio n of t he small graph conditioning m et ho d [ 5 , 17 ] sho ws that for ( α, β ) = ( p + , p − ), Z α,β G is concen trated aroun d its ex- p ected v alue (up to a m ultiplicativ e arb itrarily s mall p olynomial f ac tor). This yields a lo w er b ound on the partition function Z α,β G for ( α, β ) = ( p + , p − ). On the other hand , it is str ai ghtfo rward to an upp er b ound on th e partition f unction for balanced configurations α = β . C onsequen tly , one obtains Theorems 2 and 3 a s detailed b elo w in Section 2.4 , whic h implies that the Gibbs distribution is unbala nced w ith high p robabilit y . Sly [ 11 ] uses rand om regular bipartite graph s as a gadget in his r eduction and utilize s this bimod alit y prop ert y of the Gibbs distribution. Before stating our n ew results on when Condition 1 holds, it is useful to remin d the reader the previously kno wn v alues of λ f or whic h Condition 1 h olds: • ∆ ≥ 3, and λ c ( T ∆ ) < λ < λ c ( T ∆ ) + ε (∆) for some (small) ε ( ∆) > 0, ([ 10 , Lemma 6.10, Lemma 5.1]); • ∆ = 6 and λ = 1, ([ 11 , Sectio n 5]). Let λ 1 / 2 ( T ∆ ) be the sm al lest v alue of λ such that φ ( φ (1 / 2)) = 1 / 2 ( φ is the function d efined in Lemma 4 ). Equiv alen tly λ 1 / 2 ( T ∆ ) is the minim um solution of  1 + (1 /λ ) 1 / ∆  1 − 1 / ∆  1 − (1 /λ ) 1 / ∆  1 / ∆ = 1 . (4) 1 The num b ering in this pap er fo r results from Sly’s w ork [ 11 ] refer to the arXiv version of his paper. 6 The f oll o wing Lemma is the tec h nical core of this w ork. Lemma 5. Condition 1 holds for 1. ∆ = 3 and λ > λ c ( T ∆ ) , and 2. ∆ > 3 and λ ∈ ( λ c ( T ∆ ) , λ 1 / 2 ( T ∆ )] , Lemma 5 is prov ed in Section 4 . As a corollary of Lemma 5 w e get that Condition 1 h olds for the range of λ sp ecified in Theorems 2 an d 3 . Corollary 6. Condition 1 hold s for: 1. F or ∆ = 3 and λ > λ c ( T 3 ) . 2. F or ∆ ≥ 6 and λ c ( T ∆ ) < λ ≤ λ 1 / 2 ( T ∆ ) and λ 1 / 2 ( T ∆ ) > λ c ( T ∆ − 1 ) . 3. F or ∆ = 5 and λ c ( T 5 ) < λ ≤ λ c ( T 5 ) + . 402912 . 4. F or ∆ = 4 , λ c ( T 4 ) < λ ≤ λ c ( T 4 ) + . 327887 . Pr o of. Pa rt 1 is iden tical to the first b ullet in Lemma 5 . P art 2 follo ws from the second bullet of Lemma 5 and the fact that for ∆ ≥ 6 , it hol ds λ 1 / 2 ( T ∆ ) > λ c ( T ∆ − 1 ). T o see this, by ( 4 ), we ha v e th at 1 − (1 /λ 1 / 2 ( T ∆ )) 1 / ∆ > 0, whic h imp lie s that λ 1 / 2 ( T ∆ ) > 1. F or ∆ ≥ 6, w e ha ve λ c ( T ∆ ) < 1. Hence, f or ∆ ≥ 7, we ha v e λ 1 / 2 ( T ∆ ) > λ c ( T ∆ − 1 ). F or ∆ = 6, the cla im follo ws f rom the fact that λ c ( T 5 ) = 256 / 243 < λ 1 / 2 ( T 6 ) ≈ 1 . 2 3105. F or ∆ = 5, note that λ 1 / 2 ( T 5 ) − λ c ( T 5 ) > 1 . 4 5641 − 256 / 243 > . 402912 , whic h pro v es P art 3 . F or ∆ = 4, note that λ 1 / 2 ( T 4 ) − λ c ( T 4 ) > 2 . 0 15387 − 27 / 16 = . 327887 , whic h pro v es P art 4 . Theorems 2 and 3 no w follo w from Corollary 6 as outlined earlier, a nd detailed in the follo wing subsection. 2.4 Pro ofs of Main Theorems W e now pro ceed to pro v e Theorems 2 an d 3 . Pr o ofs of The or ems 2 and 3 . The pr oof is essen tially the same as th e pro of of [ 10 , Th eo rem 2.2] with m inor mo difications. W e include the pro of for the sake of co mpleteness. Cho ose δ > 0 s uc h that for X := min | x − p + |≤ δ, | y − p − |≤ δ Φ 1 ( x, y ) , Y := max | x − y |≤ δ Φ 1 ( x, y ) it holds that τ := X − Y > 0. T o see th at this is p ossible, n ot e that Φ 1 is con tin uous and hence it is uniformly con tin uous at an y clo sed and b ounded region. S ince Φ 1 exhibits a global maxim um at ( p + , p − ), th e existence of δ f ol lo ws. 7 By Ma rko v’s in equalit y , we obtain that a.a.s. µ ( I δ B ) = P x,y : | x − y |≤ δ Z x,y G Z G ≤ exp( n ( Y + τ 4 )) Z G . (5) T o b ound min { µ ( I δ 1 ) , µ ( I δ 2 ) } , we need the follo wing result from [ 10 ]. While their result is only stated for ( α, β ) close to (1 / ∆ , 1 / ∆), it can readily b e v erified (as Sly also obs erv ed, e.g., see the discussion b efore Le mma 3.4 in [ 11 ]) that their pro of holds in a neigh b ourho o d of ( p + , p − ), wheneve r Condition 1 h olds. Lemma 7 (Th eo rem 3.4 of Mossel et al. [ 10 ]) . 2 L et ∆ ≥ 3 . Supp ose that Condition 1 holds, then for al l sufficiently lar ge n ther e exist ( α n , β n ) wher e α n = p + + o (1) , β n = p − + o (1) , and nα n and nβ n ar e inte gers , (6) and it holds a.a.s. that Z α n ,β n G ≥ 1 n E G [ Z α n ,β n G ] . F or all n large enough, there exist ( α n , β n ) wh ere | α n − p + | ≤ 1 /n , | β n − p − | ≤ 1 /n and nα n and nβ n are in tege rs, and therefore ( 6 ) holds. F rom Lemma 7 and Corollary 6 it follo ws that a.a .s. Z α,β G ≥ exp( n ( X − τ 4 )) and consequen tly µ ( I δ 1 ) ≥ exp( n ( X − τ 4 )) Z G . Since b oth Φ 1 and Φ 2 are symmetric with resp ect to α, β , a sim ila r statemen t to Lemma 7 holds with th e roles of p + , p − in terc hanged, so that min { µ ( I δ 1 ) , µ ( I δ 2 ) } ≥ exp( n ( X − τ 4 )) Z G . (7) Com bining ( 5 ) and ( 7 ), w e obtain µ ( I δ B ) ≤ exp( − nτ 2 ) min { µ ( I δ 1 ) , µ ( I δ 2 ) } . (8) This co mpletes the p roof. The torpid m ixing of the Glaub er dyn amic s claimed in Theorems 2 and 3 follo ws from ( 8 ) by Claim 2. 3 in [ 2 ], whic h is a standard conductance a rgument. F or Theorem 1 , w e will use Lemma 5 combined with the w ork of Sly [ 11 ], but w e need one ad- ditional ingredien t. The follo wing com binatorial resu lt will b e u sed to extend the inapproximabilit y result f or ∆ = 3 to a range of λ for ∆ ≥ 6. Lemma 8. L et G b e a gr aph of ma ximum de gr e e ∆ and let k > 1 b e an inte ger. Consider the gr aph H obtaine d fr o m G b e r epla cing e ach v ertex by k c opies of th at vertex and e ach e dge by the c omplete bip a rtite gr ap h b etwe en the c orr esp o nding c opies. Then, Z G ((1 + λ ) k − 1) = Z H ( λ ) . 2 The stated version of the theorem d iffers sligh tly from the version in [ 10 ]. In particular, in [ 10 ] ( α, β ) is fix ed, in th e sense t h at it’s ind ependent of n . Here, ( α, β ) depends on n . I n the applicatio n of th is t h eorem in the pro of of Theorem 2.2 in [ 10 ], it is unclear how they ded uce the existence of a fix ed ( α, β ) and this is why w e mo dified the statement o f the theorem. Their pro of of Theorem 3.4 in [ 10 ] still goes th rou gh for this slightly modified version. 8 Pr o of. Consider the map f : I H → I G that m aps an indep enden t s et I of H to an ind epend en t s et J of G s uc h that v ∈ J if and only if at least one of the k copies of v in H are in I . F or an indep enden t set J of G the total co ntribution of sets in f − 1 ( J ) to Z H ( λ ) is ((1 + λ ) k − 1) | J | , since for eac h v ∈ J w e can c ho ose any non-empty subset of its k copies in H . W e are no w r ea dy to pr o v e Theorem 1 . Pr o of of The or em 1 . Sly’s reduction [ 11 , Section 2] establishes that for any λ > λ c ( T ∆ ) suc h that Condition 1 h olds and also the follo wing t wo inequ al ities hold: (∆ − 1) p + p − ≤ (1 − p + )(1 − p − ) and p + < 3 5 (1 − p − ) , (9) then there do es n ot exist (assu ming NP 6 =RP ) an FPRAS for the partition fu nctio n of the hard-core mo del with activit y λ . F or the t w o inequalitie s in ( 9 ), as S ly [ 11 ] p oin ts out, they are unn ec essary . First off, the inequalit y (∆ − 1) p + p − ≤ (1 − p + )(1 − p − ) is imp lie d b y Part 3 of L emma 4 . And the condition p + < 3 5 (1 − p − ) is used to s implify the pr oof of Lemm a 4.2 in [ 11 ]. F or completeness, a sligh t modifi ca tion of S ly’s Lemma 4.2 without ( 9 ) in the h yp othesis is pro v ed in Section 3 . Corollary 6 establishes that Condition 1 holds for a range of activities λ for eac h ∆ ≥ 3. By the discus sion ab o v e, we obtain that there d oes not exist (assuming NP 6 =RP) an FPRAS for the partition function of the hard-core mod el with activit y • λ ∈ ( λ c ( T 3 ) , ∞ ] for ∆ = 3. • λ ∈ ( λ c ( T ∆ ) , λ c ( T ∆ − 1 )] f or ∆ ≥ 6. • λ ∈ ( λ c ( T 5 ) , λ c ( T 5 ) + . 402912] for ∆ = 5. • λ ∈ ( λ c ( T 4 ) , λ c ( T 4 ) + . 327887] for ∆ = 4. Lemma 8 (used with k = 2 an d ∆ = 3) implies that there do es n ot exist (assumin g NP 6 =RP) an FPRAS for the partition function of the hard-core mo del with ac tivit y λ > √ 5 − 1 in graphs of maxim um degree 6. The range of λ for ∆ and ∆ − 1 where we can p ro v e hardness (that is, ( λ c ( T ∆ ) , λ c ( T ∆ − 1 )]) o v erlap for ∆ ≥ 6. This is usefu l s ince th e hardn ess for ∆ − 1 automatical ly give s hard ness f or ∆. Th us for ∆ ≥ 6 we hav e th e h ardness result on the set ( λ c ( T ∆ ) , λ c ( T ∆ − 1 )] ∪ · · · ∪ ( λ c ( T 6 ) , λ c ( T 5 )] ∪ ( λ c ( T 5 ) , λ c ( T 5 ) + 0 . 40291 2] ∪ ( √ 5 − 1 , ∞ ) = ( λ c ( T ∆ ) , ∞ ) . This co nclud es the proof of the theo rem for ∆ ≥ 6. F or ∆ = 4, w e ha v e the hardness result on the set ( λ c ( T 4 ) , 2 . 01 538] ∪ (4 , + ∞ ). F or ∆ = 5, w e ha v e the hardness result on the set ( λ c ( T 5 ) , 1 . 45 641] ∪ (1 . 687 5 , 2 . 0153 8] ∪ (4 , + ∞ ) . 2.5 On the Use of Computational Assistance W e use Mathematica to prov e sev eral inequalities inv olvi ng rational functions in regions b oun ded by rational f unctions. Su c h inequalities are kn o wn to b e decidable by T arski’s quan tifier elimination [ 14 ], the particular v ersion of Collins algebraic d ec omp osition (CAD) used by Mathematica’s Reso lve command is describ ed in [ 13 ]. The algorithms are guaran teed to return correct answ ers—they do not suffer from precision issu es since they use in terv al arithmetic (a real num b er is represen ted using an interv al whose end p oin ts are rational n umb ers). 9 3 Non-Reconstruction revisited In this section, we rep ro v e Lemma 4.2 from Sly [ 11 ] without ( 9 ) in the hyp o thesis. This will allo w us to fo cus on pro ving Lemma 5 . Recall, T ∆ is the infinite ∆-regular tree, and p + , p − denote the marginal pr obabilit ies that the ro ot is occupied for the limit of even a nd o dd, resp ectiv ely , sized trees. Let ˆ T ∆ denote the infinite (∆ − 1)-ary tree r ooted at ρ . (Thus, these t w o trees only d iffer at the ro ot.) F or ℓ ∈ N , let ˆ T ℓ denote the tree with branching factor ∆ − 1 and con taining ℓ lev els. L et q ℓ denote the marginal probabilit y that th e ro ot is occup ied in th e Gibbs d istribution on ˆ T ℓ . Analogous to p + and p − , let q + = lim ℓ →∞ q 2 ℓ and q − = lim ℓ →∞ q 2 ℓ +1 . The d ensities q + , q − are rela ted to p + , p − b y: q + = p + 1 − p − , q − = p − 1 − p + . There are t wo semi-translation in v ariant measures ˆ µ + and ˆ µ − on ˆ T ∆ obtained by taking the w eak limit of the hard-core measure of ev en-sized trees ˆ T 2 ℓ and o dd-sized trees ˆ T 2 ℓ +1 , resp ectiv ely . In these measures ˆ µ + , ˆ µ − on the infi nite tree, q + and q − , r espectiv ely , are the marginal p robabilitie s that the ro ot is o ccupied. These measures ˆ µ + and ˆ µ − can also b e generated b y a broadcasting pro cess, see [ 11 , Sectio n 4]. F or v ∈ ˆ T ∆ , den ot e by S v,ℓ the vertice s at lev el ℓ in the su btree of ˆ T ∆ ro oted at v . Let X ρ,ℓ, + denote the marginal pr obabilit y that the ro ot ρ is occup ied in an indep endent set X generated by the follo wing pr ocess: w e fir st sample an indep endent set ˆ X from the measure ˆ µ + , then we condition on the configuration ˆ X S on S ρ,ℓ , and fi nally we sample an in dep enden t set X from th e h ard-core measure on ˆ T ℓ conditioned on X S = ˆ X S . Note th at the confi guratio n of the v ertices in S ρ,ℓ is a random v ecto r, so th at X ρ,ℓ, + is a r andom v ariable. Define similarly X ρ,ℓ, − for ˆ µ − . W e may extend this definition to an arbitrary v ertex v ∈ ˆ T ∆ at distance D from th e ro ot ρ , by setting X v,ℓ , + = X ρ,ℓ, + if D is ev en or X v,ℓ , + = X ρ,ℓ, − if D is o dd. Thus, X v,ℓ , + is the pr obabilit y that, in an appropriate translation of ˆ µ + , v is o ccupied conditioning on the configu ratio n of S v,ℓ . Define similarly X v,ℓ , − for ˆ µ − . In Sly [ 11 ], it is p ro v ed that X ρ,ℓ, + ( X ρ,ℓ, − ) is strongly concen trated around q + ( q − , resp ectiv ely) under th e condition that p + < 3 5 (1 − p − ). This concen tration is us ed b y Sly for establishing the prop erties of the gadget he uses in his redu ctio n. W e n eed to repr o v e the concen tration without the condition p + < 3 5 (1 − p − ). Lemma 9 (see Lemma 4.2 in Sly [ 11 ]) . W hen λ > λ c ( T ∆ ) , ther e exist c on stants ζ 1 ( λ, ∆) , ζ 2 ( λ, ∆) > 0 , for al l suffici e ntly lar ge ℓ , P h    X ρ,ℓ, + − q +    ≥ exp( − ζ 1 ℓ ) i ≤ exp( − exp ( ζ 2 l )) . P h    X ρ,ℓ, − − q −    ≥ exp( − ζ 1 ℓ ) i ≤ exp( − exp ( ζ 2 l )) . Pr o of. W e follo w th e pr oof of [ 11 ]. F or a v ertex v ∈ ˆ T ∆ , let N ( v ) = N 1 ( v ) den ot e its c hildren, and for i ∈ N , let N i ( v ) denote its descendan ts i leve ls b elo w. F or 1 ≤ i ≤ ℓ and s ∈ { + , −} , let X N i ( v ) ,ℓ − i,s denote the v ector { X w ,ℓ − i,s : w ∈ N i ( v ) } . By sta nd ard tr ee recursions, it can b e pro ve d that for s ∈ { + , −} , X v,ℓ ,s = h ( X N ( v ) ,ℓ − 1 ,s ) := λ Q w ∈ N ( v ) (1 − X w ,ℓ − 1 ,s ) 1 + λ Q w ∈ N ( v ) (1 − X w ,ℓ − 1 ,s ) . 10 In our fr amew ork, one m a y establish a con traction pr operty for h , but it is slightl y more straight - forw ard to lo ok at depth t w o of the recursion. Note that 1 − X v,ℓ ,s = 1 1 + λ Q w ∈ N ( v ) (1 − X w ,ℓ − 1 ,s ) so that X v,ℓ ,s = 1 − 1 1 + λ Q w ∈ N ( v ) 1 1+ λ Q z ∈ N ( w ) (1 − X z ,ℓ − 2 ,s ) = λ λ + Q w ∈ N ( v )  1 + λ Q z ∈ N ( w ) (1 − X z ,ℓ − 2 ,s )  (10) =: r ( X N 2 ( ρ ) ,ℓ − 2 ,s ) . By recursiv ely app lying ( 10 ), w e obtain that X ρ,ℓ,s =: g ( X N 2 L ( ρ ) ,ℓ − 2 L,s ) . Note that r ( X N 2 ( ρ ) ,ℓ − 2 ,s ) = g ( X N 2 ( ρ ) ,ℓ − 2 ,s ). F or the rest of the pro of, we fo cus on the case s = + and v b eing at ev en d istance from the ro ot ρ , the other ca ses b eing almost iden tical (since w e are lo oking at depth t w o of the recursion). Assume X N 2 ( v ) ,ℓ − 2 , + , X ′ N 2 ( v ) ,ℓ − 2 , + are t w o v ectors whic h a re equal except at one v ertex z ∗ ∈ N 2 ( v ). T o prov e the new v ersion of the lemma, it suffices to obtain the follo win g con traction prop ert y . W e will pro ve for all ℓ sufficient ly large , for all v ∈ ˆ T ∆ , for all p airs X N 2 ( v ) ,ℓ − 2 , + , X ′ N 2 ( v ) ,ℓ − 2 , + that differ at a single vertex z ∗ , | r ( X N 2 ( v ) ,ℓ − 2 , + ) − r ( X ′ N 2 ( v ) ,ℓ − 2 , + ) | ≤  1 (∆ − 1) 2 + 1 10  | X z ∗ ,ℓ − 2 , + − X ′ z ∗ ,ℓ − 2 , + | . (11) T o accomplish this, it s uffices to pro v e that for ℓ large enough, it holds that:     ∂ r ∂ X z ∗ ,ℓ − 2 , + ( X N 2 ( v ) ,ℓ − 2 , + )     < 1 (∆ − 1) 2 + 1 10 . (12) Let w ∗ ∈ N ( v ) b e the parent of z ∗ . Then ( 12 ) is equiv alen t to: λ 2 Q z ∈ N ( w ∗ ) \{ z ∗ } (1 − X z ,ℓ − 2 , + ) Q w ∈ N ( v ) \{ w ∗ }  1 + λ Q z ∈ N ( w ) (1 − X z ,ℓ − 2 , + )  h λ + Q w ∈ N ( v )  1 + λ Q z ∈ N ( w ) (1 − X z ,ℓ − 2 , + ) i 2 < 1 (∆ − 1) 2 + 1 10 . Note that as ℓ → + ∞ , since v is at ev en d ista nce fr om ρ , X v,ℓ , + con v erges almost su rely to q + (see [ 11 , equatio n (4.1)] and [ 11 , proof of Lemma 4.2]) . Hence, a s ℓ → + ∞ , w e ha v e that     ∂ r ∂ X z ∗ ,ℓ − 2 , + ( X N 2 ,ℓ − 2 , + )     → λ 2 (1 − q + ) ∆ − 2  1 + λ (1 − q + ) ∆ − 1  ∆ − 2 h λ + (1 + λ (1 − q + ) ∆ − 1 ) ∆ − 1 i 2 =: γ . Pic k ℓ 0 = ℓ 0 (∆ , λ ) so that for all ℓ ≥ ℓ 0 w e ha v e th at     ∂ r ∂ X z ∗ ,ℓ − 2 , + ( X N 2 ( v ) ,ℓ − 2 , + ) − γ     ≤ 1 10 . 11 W e will see n ext that γ = q + q − and hence by P art 3 of Lemma 4 , it f ol lo ws that γ ≤ 1 (∆ − 1) 2 whic h pro ve s ( 12 ). T o see that γ = q + q − , n ot e that λ (1 − q + ) ∆ − 1 = q − 1 − q − and λ (1 − q − ) ∆ − 1 = q + 1 − q + (one can d eriv e these equalities by p + = φ ( p − ) , p − = φ ( p + ), see Part 1 of Lemma 4 ). It is now a matter of a few algebra su bstitutions to c hec k that γ = q + q − . This pro v es ( 11 ). Recursively applying this relation ( 11 ) implies that for all ℓ − L > ℓ 0 , for ∆ ≥ 3, | g ( X N L ( v ) ,ℓ − L, + ) − g ( X ′ N L ( v ) ,ℓ − L, + ) | ≤  1 2  ⌊ L/ 2 ⌋ | X z ∗ ,ℓ − L, + − X ′ z ∗ ,ℓ − L, + | . (13) Ha ving established ( 13 ), the rest of Sly’s pro of of Lemma 4.2 go es th rough. 4 Analysis of the Second Momen t In this sectio n, w e p ro v e Lemma 5 . W e break its pro of into t w o sligh tly ea sier components, namely Lemmata 10 and 11 stated b elo w, dep ending on the v al ue of p + . Notic e that Lemm a 10 determines explicitly a region of ( α, β ) for wh ic h Con ditio n 1 h olds. Lemma 10. L et ∆ ≥ 3 and let ( α, β ) ∈ R ∆ , α, β > 0 , and α, β ≤ 1 / 2 . Then g α,β ( γ , δ, ε ) := Φ 2 ( α, β , γ , δ, ε ) achieves its u nique maximum in the r e gion ( 3 ) at the p oint ( γ ∗ , δ ∗ , ε ∗ ) . Lemma 11. Fix ∆ = 3 and λ > λ c ( T ∆ ) . L et p + and p − b e the c orr esp onding pr ob abilities. Assume that 1 / 2 ≤ p + < 1 . Ther e exists a c onstan t χ > 0 such that for | p + − α | < χ and | p − − β | < χ , g α,β ( γ , δ, ε ) := Φ 2 ( α, β , γ , δ, ε ) achieves its unique m aximum in the r e gion ( 3 ) at the p oint ( γ ∗ , δ ∗ , ε ∗ ) . Pr o of of L emma 5 . W e b egin by proving the second part of the lemma (∆ > 3 case) and then we pro ve the first part (∆ = 3 case). Fix any ∆ ≥ 3 and λ > λ c ( T ∆ ). Let p + , p − b e the corresp onding marginal probabilities that th e ro ot is o ccupied in the m ea sur es µ + , µ − . By the third item of Lemma 4 , we ha ve that ( α, β ) = ( p + , p − ) is con tained in the interio r of R ∆ . Hence, the same holds for ev ery ( α, β ) in a small enough neighborh oo d of ( p + , p − ). Hence, p ro vided that p + ≤ 1 / 2, Lemm a 10 v erifies the condition for any suc h p oin t. It is n o w easy to see that p + is increasing in λ , and hence p + ≤ 1 / 2 iff λ ∈ ( λ c ( T ∆ ) , λ 1 / 2 ( T ∆ )]. This pro v es th e second part of Lemma 5 . The fir st part of Lemm a 5 is prov ed analogously . When p + ≤ 1 / 2 , one uses Lemma 10 as ab o v e, while wh en p + ≥ 1 / 2 the condition redu ce s to Lemma 11 . Th us w e ma y f ocus our attent ion on proving Lemmata 10 and 11 . While the analysis at some p oin t for Lemma 11 requires tigh ter arguments, the t w o pro ofs share man y common prepro cessing steps. The rest of this section is dev oted to these common prepro cessing steps and th e pro ofs of Lemmata 10 and 11 are give n in Section 5 . 12 4.1 The Pa rtial Deriv ativ es The deriv ativ es of Φ 2 with resp ect to γ , δ , ε can easily b e compu te d and are also giv en in [ 10 , Pro of of Lemma 6.1]: exp  ∂ Φ 2 ∂ γ  = (1 − 2 β + δ − γ − ε ) ∆ ( α − γ − ε ) ∆ (1 − 2 α + γ ) ∆ − 1 (1 − β − γ − ε ) ∆ ( β − α + γ − δ + ε ) ∆ ( α − γ ) ∆ − 2 γ , (14) exp  ∂ Φ 2 ∂ δ  = ( β − α − δ + γ + ε ) ∆ (1 − 2 β + δ ) ∆ − 1 (1 − 2 β + δ − γ − ε ) ∆ ( β − δ ) ∆ − 2 δ , (15) exp  ∂ Φ 2 ∂ ε  = (1 − 2 β + δ − γ − ε ) ∆ ( α − γ − ε ) ∆ (1 − α − β − ε ) ∆ ε ∆ ( β − α − δ + γ + ε ) ∆ (1 − β − γ − ε ) ∆ . (16) 4.2 Excluding the Boundary of the Region W e argue that for ( α, β ) ∈ R ∆ , the maxim um of g α,β cannot o ccur on the b oundary of the r eg ion defined by ( 3 ). Lemma 12. F or ev ery ∆ ≥ 3 and ( α, β ) ∈ R ∆ , g α,β ( γ , δ, ε ) := Φ 2 ( α, β , γ , δ, ε ) attains its maximum in the interior of ( 3 ). Pr o of. W e follo w the pro of of L emma 6.1 of [ 10 ] (wh ere the s ame result is prov ed in the s pecial case α = β = 1 ∆ ). W e will prov e that g α,β ( γ , δ, ε ) attains its maximum in th e interior of ( 3 ) by showing th at at least one deriv ati ve in ( 14 )–( 16 ) go es to infinity (+ or − according to the d irect ion), as we appr oa c h one of the boun daries d efined b y ( 3 ) from the in terior of ( 3 ). F or ∆ ≥ 3, note that ( α, β ) ∈ R ∆ implies α + β < 1. Without loss of generalit y , w e assu me that α ≥ β . Hence β < 1 / 2. W e hav e ( 15 ) go es to + ∞ as δ → 0, an d ( 16 ) goes to + ∞ as ε → 0. W e also ha v e ( 16 ) go es to −∞ as γ + ε → α , ( 15 ) goes to −∞ as δ → β , ( 16 ) goes to −∞ as γ + ε − δ → 1 − 2 β , ( 16 ) go es to − ∞ as ε → 1 − α − β , and ( 16 ) go es to + ∞ as γ + ε − δ → α − β . When α ≥ 1 / 2, the condition γ = 0 is not a b oun dary , as γ ≥ 0 is implied b y th e conditions δ ≥ 0, 1 − α − β − ε ≥ 0, and β − δ + ε + γ − α ≥ 0. On th e other h and, w hen α < 1 / 2 , w e h a v e ( 14 ) go es to + ∞ as γ → 0. 4.3 Eliminating one v ariabl e Fix ∆ , α, β , γ , δ and view Φ 2 as a function of ε . W e maximize with resp ect to ε . In this setting, it w as p ro v ed [ 10 , Lemma 6.3], that the only maximizer of the f unction Φ 2 in the interior of ( 3 ) is obtained b y solving ∂ Φ 2 ∂ ε = 0 and is giv en b y: ˆ ε := ˆ ε ( α, β , γ , δ ) = 1 2 (1 + α − β − 2 γ − √ D ) , (17) where D = (1 + α − β − 2 γ ) 2 − 4( α − γ )(1 − 2 β − γ + δ ) = ( α + β − 1) 2 + 4( α − γ )( β − δ ) . Define ˆ η := ˆ η ( α, β , γ , δ ) = 1 2 (1 − α + β − 2 δ − √ D ) , ( 18) 13 and note that α − γ − ˆ ε = β − δ − ˆ η = 1 2 ( − (1 − α − β ) + √ D ) , ( α − γ − ˆ ε )(1 − α − β − ˆ ε − ˆ η ) = ˆ ε ˆ η . (19) The new parameter ˆ η (not used in [ 10 ]) is symmetric with ˆ ε , i.e, the constrain ts and formulas we ha v e are inv ariant under a symmetry that sw aps α, γ , ˆ ε with β , δ , ˆ η . T his w ill allo w for simpler argumen ts (using the symmetry). F rom the previous d iscussion and equation ( 17 ), w e ma y eliminate v ariable ε of our c onsideration. Of course, this introdu ces some complexit y due to the r adica l √ D , b ut still this is manage able. Let f ( γ , δ ) := g α,β ( γ , δ, ˆ ε ) = Φ 2 ( α, β , γ , δ, ˆ ε ) . T o p ro v e that ( γ ∗ , δ ∗ , ε ∗ ) is the un ique global maximum of g α,β in th e in terior of the regio n defined b y ( 3 ), it su ffices to pro ve that ( γ ∗ , δ ∗ ) is th e uniqu e global maxim um of f for ( γ , δ ) in the interio r of the follo wing region, which con tains the ( γ , δ )-pro jection of the region defined by ( 3 ): 0 ≤ γ ≤ α, 0 ≤ δ ≤ β , 0 ≤ 1 − 2 β + δ − γ , 0 ≤ 1 − 2 α + γ − δ . (20) Eac h inequalit y in ( 20 ) is implied b y the inequalities in ( 3 ), the only non-trivial case being the last inequalit y whic h is the sum of 1 − α − β − ε ≥ 0 and β − δ + ε + γ − α ≥ 0. The fi rst deriv ativ es of f with r espect to γ , δ are ∂ f ∂ γ ( γ , δ ) = ∆ ln W 11 + ln W 12 , (21) ∂ f ∂ δ ( γ , δ ) = ∆ ln W 21 + ln W 22 , (22) where W 11 = ( α − γ − ˆ ε ) ˆ ε (1 − 2 α + γ ) ˆ η ( α − γ ) 2 = ˆ ε (1 − 2 α + γ ) (1 − α − β − ˆ ε )( α − γ ) , W 12 = ( α − γ ) 2 (1 − 2 α + γ ) γ , W 21 = ( β − δ − ˆ η ) ˆ η (1 − 2 β + δ ) ˆ ε ( β − δ ) 2 = ˆ η (1 − 2 β + δ ) (1 − α − β − ˆ η )( β − δ ) , W 22 = ( β − δ ) 2 (1 − 2 β + δ ) δ . Note that the r igh tmost equalities in the definition of W 11 and W 21 follo w from ( 18 ) and ( 19 ). F or ev ery ∆ ≥ 3, and ( α, β ) ∈ R ∆ , w e ha v e that ( γ ∗ , δ ∗ ) is a stationary p oin t of f (this follo ws from th e fact that for γ = α 2 and δ = β 2 , the inequalities on the right- hand s ides in Lemma 13 b ecome equalities, and fr om ( 21 ), ( 22 ) w e ha v e th at the deriv ativ es of f v anish). 4.4 Restricting the Region T o determine whether ( 21 ) and ( 22 ) are zero it will b e useful to un derstand conditions that mak e W 11 , W 12 , W 21 , W 22 greater or equal to 1. Th e f ol lo wing lemma giv es suc h conditions. The pro of is giv en in Section 6.1 . Lemma 13. F or every ( α, β ) ∈ R , and ( γ , δ ) in the interior of ( 20 ), W 11 ≥ 1 ⇐ ⇒ (1 − α ) 2 δ + β 2 (2 α − 1 − γ ) ≥ 0 , W 12 ≥ 1 ⇐ ⇒ γ ≤ α 2 , W 21 ≥ 1 ⇐ ⇒ (1 − β ) 2 γ + α 2 (2 β − 1 − δ ) ≥ 0 , W 22 ≥ 1 ⇐ ⇒ δ ≤ β 2 . 14 By considering the sign of ( 21 ) and ( 22 ) and Lemma 13 , we ha v e that the stationary p oints of f (and h ence of g α,β ) ca n only b e in 0 < γ ≤ α 2 , 0 < δ ≤ β 2 , (1 − α ) 2 δ + β 2 (2 α − 1 − γ ) ≤ 0 , (1 − β ) 2 γ + α 2 (2 β − 1 − δ ) ≤ 0 , (23) or α 2 ≤ γ < α, β 2 ≤ δ < β , (1 − α ) 2 δ + β 2 (2 α − 1 − γ ) ≥ 0 , (1 − β ) 2 γ + α 2 (2 β − 1 − δ ) ≥ 0 . (24) Note that for γ = α 2 , δ = β 2 one has W 11 = W 12 = W 21 = W 22 = 1 (Lemma 13 holds with equalities as w ell instead of inequalities), so that ( α 2 , β 2 ) is alw a ys a stationary p oin t f or f ( γ , δ ). 4.5 The Hessian T o p ro v e that f has a unique maxim um, we are go ing to argue that f is strictly co ncav e in eac h of the regio ns defined b y ( 23 ) and ( 2 4 ). It will thus b e cr ucia l to stud y the Hessian of f . Let H denote th e Hessian of f , i.e., H = ∂ f ∂ 2 γ ( γ , δ ) ∂ f ∂ γ ∂ δ ( γ , δ ) ∂ f ∂ δ∂ γ ( γ , δ ) ∂ f ∂ 2 δ ( γ , δ ) ! . Our goal is to express H in a helpf ul explicit form. In this v ein, it will b e con v enient to define the follo w ing quantit ies. R 1 = 1 − α − β 1 − α − β − ˆ ε − ˆ η , R 2 = √ D 1 − 2 α + γ , R 3 = 2( α − γ − ˆ ε ) α − γ , R 4 = √ D γ , R 5 = 2(1 − β − γ − ˆ ε ) α − γ , R 6 = √ D 1 − 2 β + δ , R 7 = 2( α − γ − ˆ ε ) β − δ , R 8 = √ D δ , R 9 = 2(1 − β − γ − ˆ ε ) β − δ . H can no w b e w ritte n in a relativ ely nice form with resp ect to the R i . Namely , ∂ f ∂ 2 γ ( γ , δ ) = 1 √ D h ( − R 1 + R 2 + R 3 )∆ − R 2 − R 3 − R 4 − R 5 i , (25) ∂ f ∂ 2 δ ( γ , δ ) = 1 √ D h ( − R 1 + R 6 + R 7 )∆ − R 6 − R 7 − R 8 − R 9 i , (26) ∂ f ∂ γ ∂ δ ( γ , δ ) = ∂ f ∂ δ ∂ γ ( γ , δ ) = ∆ R 1 √ D . (27) Insp ecting the R i w e obtain the follo w ing ob serv ation. Observ at ion 14. R 1 , . . . , R 9 ar e p ositive when ( α, β ) ∈ R ∆ and ( γ , δ ) in the interior of ( 20 ). Observ ation 14 and equation ( 27 ) immediately yield: Observ at ion 15. F or every ( α, β ) ∈ R ∆ , and ( γ , δ ) in the interior of ( 20 ), ∂ f ∂ γ ∂ δ ( γ , δ ) = ∂ f ∂ δ ∂ γ ( γ , δ ) > 0 . In Secti on 6.2 we prov e the follo wing tec h nical in equali t y on the R i . Lemma 16. F or every ( α, β ) ∈ R , and ( γ , δ ) in the interior of ( 20 ), R 1 > R 2 + R 3 , and R 1 > R 6 + R 7 . 15 Applying Lemma 16 and Ob serv ation 14 , ( 25 ) and ( 26 ) give th e f oll o wing straightforw ard corol- lary . Corollary 17. F or every ( α, β ) ∈ R , and ( γ , δ ) in the interior of ( 20 ), ∂ f ∂ 2 γ ( γ , δ ) < 0 , ∂ f ∂ 2 δ ( γ , δ ) < 0 . Corollary 17 implies that the sum of the eigen v alues of M is negativ e. Hence, H is n eg ativ e definite iff the d ete rmin an t of H is negat ive . W e ha v e the follo wing expression for det( H ). det( H ) = ∂ f ∂ 2 γ ( γ , δ ) · ∂ f ∂ 2 δ ( γ , δ ) − ∂ f ∂ γ ∂ δ ( γ , δ ) · ∂ f ∂ δ ∂ γ ( γ , δ ) = 1 D  (∆ − 1) 2 h ( − R 1 + R 2 + R 3 )( − R 1 + R 6 + R 7 ) − R 2 1 i +(∆ − 1) h ( − R 1 + R 2 + R 3 )( − R 1 − R 8 − R 9 ) + + ( − R 1 + R 6 + R 7 )( − R 1 − R 4 − R 5 ) − 2 R 2 1 i + h ( − R 1 − R 8 − R 9 )( − R 1 − R 4 − R 5 ) − R 2 1 i  . (28) 5 Concluding the Pro ofs of Lemmata 10 and 11 In this section, we giv e the pr oofs of Lemmata 10 and 11 . W e first recap w hat w e ha v e accomplished in S ect ion 4 f or ev ery ( α, β ) ∈ R ∆ . 1. T he function g α,β ( γ , δ, ε ) attains its maximum in the in terior of the reg ion ( 3 ). S ee Section 4.2 . 2. T o study the (lo cal) m axima of g in the in terior of t he reg ion ( 3 ), it suffices to study the maxima of the function f ( γ , δ ) = g α,β ( γ , δ, ˆ ε ) in t he in terior o f the region ( 20 ). More explicitly , if f ( γ , δ ) has a unique m axi mum in the interio r of the region ( 20 ) at ( γ ∗ , δ ∗ ) = ( α 2 , β 2 ), then g α,β has a unique maximum in the in terior of the regi on ( 3 ) at ( γ ∗ , δ ∗ , ε ∗ ) =  α 2 , β 2 , α (1 − α − β )  . The function f is differen tiable in the in terior of the regio n ( 20 ). See Sectio n 4.3 . 3. T he p oin t ( γ ∗ , δ ∗ ) = ( α 2 , β 2 ) is alw a ys s ta tionary for f . E v ery stationary p oint of f lies in one of the t w o regions defined b y (i) ( 20 ) and ( 23 ), (ii) ( 20 ) and ( 24 ). S ee Section 4.4 . 4. T he function f is strictly co ncav e iff det( H ) > 0. See S ec tion 4.5 . By the ab o v e d iscussion, if for some ( α, β ) ∈ R ∆ it holds that f is strictly conca v e in eac h of the regions (i) ( 20 ) and ( 23 ), (ii) ( 20 ) and ( 24 ), then g α,β has a unique maxim um at ( γ ∗ , δ ∗ , ε ∗ ) =  α 2 , β 2 , α (1 − α − β )  . Thus, it suffices to c heck that det( H ) > 0 in eac h of these regions. T his is essen tially the wa y we derive Lemmata 10 and 11 . Hence, the main c hallenge is pr o ving that det( H ) > 0. Th is can b e done sligh tly m ore easily in the regio n ( 20 ) and ( 23 ). Indeed, in Section 6.3 w e pro v e the follo wing lemma. Lemma 18. det( H ) > 0 for every ∆ ≥ 3 , ( α, β ) ∈ R ∆ , ( γ , δ ) in the interior of ( 20 ) and ( γ , δ ) in ( 23 ). Pro ving det( H ) > 0 in the in tersecti on of the regions ( 20 ) and ( 23 ) is tric kier and is essen tially the reason we do not obta in our hardness result for ∆ = 4 , 5. At this p oi nt, it is con v enien t to split the analysis for eac h of the lemmas. 16 5.1 Pro of of Lemma 10 In the setup of Lemma 10 , we h a v e ( α, β ) ∈ R ∆ and α, β ≤ 1 / 2. W e sup press the details of pro ving det( H ) > 0 as a lemma, whose pro of we defer to Section 6.4 . Lemma 19. det( H ) > 0 for every ∆ ≥ 3 , ( α, β ) ∈ R ∆ , α, β ≤ 1 / 2 , ( γ , δ ) in the interior of ( 20 ) and ( γ , δ ) in ( 24 ). Using Lemmata 18 and 19 , the pro of of Lemma 10 is immediate. Pr o of of L emma 10 . Lemm a 18 and Lemm a 19 im ply that f has a un ique maximum at ( γ ∗ , δ ∗ ) for ev ery ∆ ≥ 3, ( α, β ) ∈ R ∆ , 1 / 2 ≥ α, β and ( γ , δ ) in the interior of ( 20 ). This follo ws from the fact that det( H ) > 0 implies that the Hessian of f is negativ e defin ite in the region of in terest, i.e., where g α,β could p ossibly h a v e stationary p oin ts, w hic h in tur n imp lie s that f is strictly conca v e in the region and h ence has a unique maxim um. By the defin itio n of f , it follo ws that g α,β has a un ique maxim um at ( γ ∗ , δ ∗ , ε ∗ ). F or a more th orough outline, see the b egi nn ing of Section 5 . 5.2 Pro of of Lemma 11 In the setup of Lemma 11 , w e ha ve ∆ = 3 and p + ≥ 1 / 2. By ( 1 ), tedious but otherw ise simp le algebra give s that the solution of β = φ ( α ) and α = φ ( β ) with α 6 = β satisfies α 2 − 2 α + αβ + 1 − 2 β + β 2 = 0 . It can also b e chec k ed that when 1 > p + ≥ 1 / 2 , it holds that 0 < p − ≤ (3 − √ 5) / 4. Defin e R ′ 3 to b e the set of pairs ( α, β ) such that α 2 − 2 α + αβ + 1 − 2 β + β 2 = 0, 1 / 2 ≤ α < 1 and 0 < β ≤ (3 − √ 5) / 4. Our goal is to sh o w that det( H ) > 0 for every ( α, β ) ∈ R ′ 3 , ( γ , δ ) in the int erior of ( 20 ) and ( γ , δ ) in ( 24 ). W e can rewrite det( H ) using the form ula in ( 28 ) as det( H ) = 1 D h 3 R 1 ( U 1 + U 2 ) + U 1 U 2 i = U 1 D h 3 R 1 (1 + U 2 /U 1 ) + U 2 i , (29) where U 1 = R 8 + R 9 − 2 R 6 − 2 R 7 , U 2 = R 4 + R 5 − 2 R 2 − 2 R 3 . The f ol lo wing lemma establishes tec hnical inequalities on U 1 , U 2 , R 1 , . . . , R 9 , which are cru cia l in establishin g the p ositivit y of det( H ). Its pro of is giv en in Section 6.5 . Lemma 20. W e have U 1 > 0 , R 4 > R 6 , R 5 > R 7 , R 5 > 4 R 3 , R 9 > 4 R 7 , and 3 R 8 / 2 + R 9 > 9 R 2 , for every ( α, β ) ∈ R ′ 3 , ( γ , δ ) in the interior of ( 20 ) and ( γ , δ ) in ( 24 ). With Lemma 20 at hand, we can no w pro v e th at det( H ) is p ositiv e. Lemma 21. det( M ) > 0 for every ( α, β ) ∈ R ′ 3 , ( γ , δ ) in the interior of ( 20 ) and ( γ , δ ) in ( 24 ). Pr o of. Obser v e that U 1 + 3 U 2 = ( R 8 + 2 R 9 / 3 − 6 R 2 ) + (3 R 4 − 2 R 6 ) + ( R 9 / 3 − 4 R 7 / 3) + (3 R 5 / 2 − 2 R 7 / 2) + (3 R 5 / 2 − 6 R 3 ) > 0 , 17 where the last inequalit y follo ws by Lemma 21 . O nce again by Lemma 21 , we h a v e U 1 > 0 and hence U 2 /U 1 > − 1 / 3. Th us, ( 29 ) gi ves det( H ) > U 1 D (2 R 1 + U 2 ) = U 1 D  R 4 + R 5 + 2( R 1 − R 2 − R 3 )  > 0 , where the last inequalit y follo ws from Lemma 16 and Observ ation 14 . Pr o of of L emma 11 . As in the pro of of Lemma 10 , Lemmata 18 and 21 yield th at g α,β ( γ , δ, ε ) ac hiev es its un ique maxim um in the r eg ion ( 3 ) at the p oin t ( γ ∗ , δ ∗ , ε ∗ ), for ev ery ( α, β ) ∈ R ′ 3 . W e next sh o w that g α,β ( γ , δ, ε ) := Φ 2 ( α, β , γ , δ, ε ) also ac hiev es its u nique maximum in the region ( 3 ) at the p oi nt ( γ ∗ , δ ∗ , ε ∗ ) in a small neigh b orho od of R ′ 3 . First note that Φ 2 is con tin uous. By Lemma 12 , w e ha ve for sufficiently small χ > 0, the maxim um of g α,β cannot b e ob ta ined on the b oundary of the region ( 3 ). Note that the deriv at ive s of Φ 2 are con tinuous. It follo ws that for sufficien tly sm all χ > 0, all stationary p oints of g α,β ha v e to b e close to the p oin t ( γ ∗ , δ ∗ , ε ∗ ). W e can choose χ such that det( H ) > 0 in the neigh b orho o d of the p oin t ( γ ∗ , δ ∗ , ε ∗ ), w hic h implies that g α,β has a uniqu e stationary p oin t and it is a maxim um. 6 Remaining Pro ofs of T ec hnical Lemmas 6.1 Pro of of Lemma 13 W e define W 3 to b e the numerator of W 11 min us the denominator of W 11 , and W 4 to b e the n umerator of W 21 min us the d enominato r of W 21 ; more precisely W 3 = ( α − γ − ˆ ε ) ˆ ε (1 − 2 α + γ ) − ˆ η ( α − γ ) 2 , (30) W 4 = ( β − δ − ˆ η ) ˆ η (1 − 2 β + δ ) − ˆ ε ( β − δ ) 2 . (31) Substituting the expression for ˆ ε and simplifyin g w e obtain W 3 := (( α + β − 1 + √ D ) / 2) ˆ ε (1 − 2 α + γ ) − ˆ η ( α − γ ) 2 , W 4 := (( α + β − 1 + √ D ) / 2) ˆ η (1 − 2 β + δ ) − ˆ ε ( β − δ ) 2 . By expanding W 3 and W 4 , w e hav e W 3 = W 31 + W 32 √ D , W 4 = W 41 + W 42 √ D , where W 31 = ((3 / 2) β − (1 / 2) β 2 − δ − (3 / 2) αβ + δ α ) γ + β + (5 / 2) β α 2 + (1 / 2) α 3 + (3 / 2) α − (3 / 2) α 2 − (1 / 2) β 2 − 1 / 2 + δ α − (7 / 2) αβ − δ α 2 + β 2 α, W 32 = αβ − (1 / 2) β γ − α − (1 / 2) β + (1 / 2) α 2 + 1 / 2 , W 41 = ((3 / 2) α − (1 / 2) α 2 − γ − (3 / 2) αβ + γ β ) δ + α + (5 / 2) αβ 2 + (1 / 2) β 3 + (3 / 2) β − (3 / 2 ) β 2 − (1 / 2) α 2 − 1 / 2 + γ β − (7 / 2) αβ − γ β 2 + α 2 β , W 42 = αβ − (1 / 2) αδ − β − (1 / 2) α + (1 / 2) β 2 + 1 / 2 . 18 Note that for ev ery ( α, β ) ∈ R , W 32 > 0 wh en 0 < γ < α , and W 42 > 0 wh en 0 < δ < β . T o see W 32 > 0 note that αβ − (1 / 2) β γ − α − (1 / 2) β + (1 / 2) α 2 + 1 / 2 ≥ αβ − (1 / 2) β α − α − (1 / 2) β + (1 / 2) α 2 + 1 / 2 = (1 / 2)(1 − α )(1 − α − β ) > 0, sin ce α + β < 1 ; inequality W 42 > 0 is the same (after renaming th e v ariables). Note that W 31 is a linear f unction in γ and w e ha v e dW 31 dγ = (3 / 2) β − (1 / 2) β 2 − δ − (3 / 2) αβ + δ α, whic h is p ositiv e for all ( α, β ) ∈ R and 0 < δ < β . T o see this note (3 / 2) β − (1 / 2) β 2 − δ − (3 / 2) αβ + δ α = (1 − α )((3 / 2) β − δ ) − (1 / 2) β 2 ≥ (1 − α )(1 / 2) β − (1 / 2 ) β 2 = (1 / 2) β (1 − α − β ) > 0 . Moreo v er, we h a v e W 31 is negativ e when γ = α , ( α, β ) ∈ R and 0 < δ < β (after su bstituting γ = α in to W 31 w e obtain − (1 − α )(1 − α − β ) 2 / 2 < 0). Hence, W 31 < 0 for all ( α, β ) ∈ R , 0 < γ < α and 0 < δ < β . By the same pro of, we can sho w that W 41 < 0 for all ( α, β ) ∈ R , 0 < γ < α and 0 < δ < β (note that W 31 and W 41 are the same after renaming th e v ariables). Let W 5 = ( W 31 /W 32 ) 2 − D , W 6 = ( W 41 /W 42 ) 2 − D . Note that the signs of W 3 and W 5 are o pp osite, and the signs of W 4 and W 6 are opp osite. After substituting W 31 , W 32 , W 41 , W 42 and simplifi ca tions, w e ob ta in W 5 = − 4( β − δ )( α − γ ) 2 ((1 − α ) 2 δ + β 2 (2 α − 1 − γ )) (2 αβ − β γ − 2 α − β + α 2 + 1) 2 , W 6 = − 4( β − δ ) 2 ( α − γ )((1 − β ) 2 γ + α 2 (2 β − 1 − δ )) ( β 2 − δ α + 1 − 2 β − α + 2 αβ ) 2 . Hence, W 11 ≥ 1 ⇐ ⇒ W 5 ≤ 0 ⇐ ⇒ (1 − α ) 2 δ + β 2 (2 α − 1 − γ ) ≥ 0 , and W 21 ≥ 1 ⇐ ⇒ W 6 ≤ 0 ⇐ ⇒ (1 − β ) 2 γ + α 2 (2 β − 1 − δ ) ≥ 0 . FInally , w e analyze the conditions for W 12 ≥ 1 and W 22 ≥ 1. It is straigh tforw ard to see that W 12 ≥ 1 ⇐ ⇒ ( α − γ ) 2 ≥ (1 − 2 α + γ ) γ ⇐ ⇒ γ ≤ α 2 , and W 22 ≥ 1 ⇐ ⇒ ( β − δ ) 2 ≥ (1 − 2 β + δ ) δ ⇐ ⇒ δ ≤ β 2 . 6.2 Pro of of Lemma 16 W e hav e − R 1 + R 2 + R 3 = ( β − δ )  − 1 α − γ − ˆ ε − 1 ˆ η + 1 ˆ ε  − √ D ˆ ε + √ D 1 − 2 α + γ + 2 √ D α − γ . 19 Multiplying b y the denominators we let P 1 = ( − R 1 + R 2 + R 3 ) ˆ ε ˆ η (1 − 2 α + γ )( α − γ )( α − γ − ˆ ε ) = P 11 + P 12 √ D , where P 11 = 1 − (1 / 2) δα + 17 αβ + (9 / 2) δγ − 5 β γ + 21 αγ β δ − (37 / 2) α 2 γ β δ − 5 αγ β δ 2 + (5 / 2) αγ β 2 δ +8 γ 2 αδ β − 5 α − 3 β − δ + 10 α 2 + 3 β 2 + 3 δ β − 10 α 3 + 5 α 4 − β 3 + 15 αγ β − (13 / 2) αβ δ − (27 / 2) α γ δ − (5 / 2 ) β γ δ − 4 α 2 β δ + (27 / 2 ) α 2 γ δ − 15 α 2 γ β − 24 αγ β 2 − 8 γ 2 β δ +(17 / 2) α β 2 δ − (5 / 2) β 2 γ δ − 3 αβ δ 2 + 3 β γ δ 2 + (15 / 2) α 3 β δ + 4 α 2 β δ 2 − (11 / 2) α 2 β 2 δ − (9 / 2) α 3 γ δ + 3 α 2 γ δ 2 + 5 α 3 γ β + 17 α 2 γ β 2 + 4 αγ β 3 − 3 γ 2 αδ 2 − 5 γ 2 αβ 2 + γ 2 β δ 2 − 3 β 2 δ +3 δ 2 α − 3 δ 2 γ − 3 α 2 δ 2 + (7 / 2) α 4 δ − 8 α 4 β − 15 α 3 β 2 − 4 α 2 β 3 + 3 γ 2 δ 2 − γ 2 β 3 − 33 α 2 β +(15 / 2) α 2 δ − 16 αβ 2 + 7 β 2 γ − (19 / 2) α 3 δ + 27 α 3 β + 28 α 2 β 2 + 5 γ 2 β 2 + 4 αβ 3 − 2 β 3 γ + δ β 3 − (3 / 2) αβ 3 δ + (1 / 2) β 3 γ δ − α 5 , and P 12 = − 1 − (1 / 2) δα − 9 αβ − (5 / 2) δγ + 3 β γ − 2 αγ β δ + 4 α + 2 β + δ − 6 α 2 − β 2 − 2 δ β +4 α 3 − α 4 − 6 αγ β + 4 αβ δ + 5 αγ δ − α 2 β δ − (5 / 2) α 2 γ δ + 3 α 2 γ β + 4 αγ β 2 + γ 2 β δ − (3 / 2) αβ 2 δ + (1 / 2) β 2 γ δ − αγ δ 2 + β 2 δ − δ 2 α + δ 2 γ + α 2 δ 2 + 12 α 2 β − 2 α 2 δ + 4 αβ 2 − 2 β 2 γ + (3 / 2) α 3 δ − 5 α 3 β − 4 α 2 β 2 − γ 2 β 2 . The follo wing claim is p ro v ed u sing the Resolve function of th e Mathematica system in Ap- p endix A.2 . Claim 22. P 11 > 0 and P 12 < 0 for al l ( α, β ) ∈ R and ( γ , δ ) in the interior of ( 20 ). F rom Claim 22 w e hav e that P 11 − P 12 √ D > 0 and hence sh o wing P 11 + P 12 √ D < 0 is equiv alen t to sh o wing P 2 11 − P 2 12 D < 0 wh ic h in turn is equ iv alen t to sho wing that P 2 := ( P 11 /P 12 ) 2 − D is negativ e. W e ha v e P 2 = ( P 11 /P 12 ) 2 − D = − ( β − δ ) 2 ( α − γ ) 3 P 21 P 2 12 , where P 21 = − 8 δ α + 20 αγ β δ − 10 α 2 γ β δ + 2 αγ β δ 2 + 12 αγ β 2 δ + 2 δ + 2 δ 2 − 8 δ β + 4 β 3 + 34 αβ δ − 10 β γ δ − 44 α 2 β δ + 8 αβ 2 δ − 4 β 2 γ δ − 14 αβ δ 2 − 2 β γ δ 2 + 18 α 3 β δ + 6 α 2 β δ 2 − 10 α 2 β 2 δ +9 α 2 γ δ 2 − 16 αγ β 3 − 18 αγ δ 2 − 2 β 2 δ − 15 δ 2 α + 9 δ 2 γ + 24 α 2 δ 2 + 2 α 4 δ + 16 α 2 β 3 +4 γ 2 β 3 + 12 α 2 δ − 8 α 3 δ − 16 αβ 3 + 8 β 3 γ − 4 δ β 3 + 6 αβ 3 δ − 2 β 3 γ δ + 8 δ 2 β + 2 β 2 δ 2 +8 δ 3 α − 4 δ 3 α 2 − 11 δ 2 α 3 − 3 δ 2 αβ 2 + δ 2 γ β 2 − 4 δ γ 2 β 2 − 4 δ 3 . The follo wing claim is p ro v ed u sing the Resolve function of th e Mathematica system in Ap- p endix A.3 . Claim 23. P 21 > 0 for al l ( α, β ) ∈ R and ( γ , δ ) in the interior of ( 20 ). F rom Claim 23 w e hav e that P 2 < 0 which implies P 11 + P 12 √ D < 0 and th is completes the pro of of Lemma 16 . 20 6.3 Pro of of Lemma 18 W e will use the follo wing tec h nical lemma in the pro of of Lemma 18 . Lemma 24. F or every ( α, β ) ∈ R , and ( γ , δ ) in the interior of ( 20 ), we have ( − R 1 + R 2 + R 3 )( − R 1 + R 6 + R 7 ) − R 2 1 < 0 , (32) ( − R 1 + R 2 + R 3 )( − R 1 − R 8 − R 9 ) − R 2 1 ≥ 0 , (33) and ( − R 1 + R 6 + R 7 )( − R 1 − R 4 − R 5 ) − R 2 1 ≥ 0 . (34) Pr o of of L emma 24 . W e first pr o v e inequalit y ( 32 ). W e ha v e that ( − R 1 + R 2 + R 3 )( − R 1 + R 6 + R 7 ) = ( R 1 − R 2 − R 3 )( R 1 − R 6 − R 7 ) < R 2 1 , where the last inequ ali t y uses Lemma 16 and the p ositivit y of the R i (Observ ation 14 ). This establishes ( 32 ). W e next pro ve ( 33 ), noting that inequalit y ( 34 ) follo ws by an analogo us argum en t. The left-hand side of ( 33 ) m ultiplied by the denominators and simplified is P 5 = (( − R 1 + R 2 + R 3 )( − R 1 − R 8 − R 9 ) − R 2 1 ) δ ( α − γ ) ˆ ε ˆ η (1 − 2 α + γ )( α − γ − ˆ ε ) / √ D = P 51 + P 52 √ D, where P 51 = − 1 − 4 β γ 2 δ α + 3 δ α − 17 αβ − 3 δγ + 5 β γ − 14 αβ γ δ + 11 α 2 β γ δ + 5 α + 3 β − 10 α 2 − 3 β 2 − 15 αβ γ + 9 δ αγ − 3 αβ δ + 10 α 2 β δ + 15 α 2 β γ + 24 αβ 2 γ + 3 β γ δ + 4 β γ 2 δ − 9 δ α 2 γ − 7 α 3 β δ − 5 α 3 β γ − 17 α 2 β 2 γ − 4 αβ 3 γ + 5 β 2 γ 2 α + 3 δα 3 γ + 33 α 2 β − 9 δα 2 + 16 αβ 2 − 27 α 3 β − 28 α 2 β 2 − 7 β 2 γ − 5 β 2 γ 2 + 9 δ α 3 − 4 αβ 3 + 8 α 4 β + 15 α 3 β 2 + 4 α 2 β 3 + 2 β 3 γ + β 3 γ 2 − 3 δ α 4 + 10 α 3 − 5 α 4 + β 3 + α 5 , and P 52 = 1 − 2 δ αγ + 6 αβ γ − 3 α 2 β γ − 4 αβ 2 γ + δ α 2 γ − 4 α + 6 α 2 − 2 β − 4 α 3 + α 4 + β 2 + 9 αβ − 3 β γ − δ α + δ γ + 2 δ α 2 − 12 α 2 β − 4 αβ 2 + 5 α 3 β + 4 α 2 β 2 + 2 β 2 γ + β 2 γ 2 − δ α 3 . The follo wing claim is p ro v ed u sing the Resolve function of th e Mathematica system in Ap- p endix A.4 . Claim 25. P 51 < 0 , P 52 > 0 for al l al l ( α, β ) ∈ R and ( γ , δ ) in the interior of ( 20 ). F rom Claim 25 w e ha v e that P 52 √ D − P 51 > 0 and hence the sign of P 5 = P 51 + P 52 √ D is the same a s the sign of D P 2 52 − P 2 51 whic h is the same as the sign of D − ( P 51 /P 52 ) 2 = 4( β − δ )( α − γ ) 3  (1 − α ) 2 δ + β 2 (2 α − 1 − γ )  2 P 2 52 > 0 . Hence P 5 > 0 wh ic h implies ( 33 ). 21 Pr o of of L emma 18 . F rom α + β + ∆(∆ − 2) αβ ≤ 1 w e h a v e (∆ − 1) 2 ≤ (1 − α )(1 − β ) / ( αβ ) = : R 11 . (35) Lemma 24 implies that the co efficie nt of (∆ − 1) in ( 28 ) is p ositiv e and the co efficien t of (∆ − 1) 2 in ( 28 ) is negativ e. Hence, also us ing ( 35 ), w e h a v e det( M ) ≥ 1 D n R 11  ( − R 1 + R 2 + R 3 )( − R 1 + R 6 + R 7 ) − R 2 1  + +  ( − R 1 − R 8 − R 9 )( − R 1 − R 4 − R 5 ) − R 2 1  o > 1 D  − R 1 R 2 R 11 − R 1 R 6 R 11 + R 1 R 4 + R 1 R 8 + + R 5 R 9 + R 3 R 7 R 11 + R 1 R 5 + R 1 R 9 − R 1 R 3 R 11 − R 1 R 7 R 11 + + R 2 R 7 R 11 + R 3 R 6 R 11  . W e will sho w − R 1 R 2 R 11 − R 1 R 6 R 11 + R 1 R 4 + R 1 R 8 > 0 , (36) and R 5 R 9 + R 3 R 7 R 11 + R 1 R 5 + R 1 R 9 − R 1 R 3 R 11 − R 1 R 7 R 11 + R 2 R 7 R 11 + R 3 R 6 R 11 > 0 . (37) T o p ro v e ( 36 ), note that − R 1 R 2 R 11 − R 1 R 6 R 11 + R 1 R 4 + R 1 R 8 = √ D (1 − α − β ) 1 − α − β − ˆ ε − ˆ η  1 γ + 1 δ − (1 − α )(1 − β ) αβ (1 − 2 α + γ ) − (1 − α )(1 − β ) αβ (1 − 2 β + δ )  . The p ositivit y of ( 36 ) follo ws from the follo wing claim. Claim 26. 1 γ + 1 δ − (1 − α )(1 − β ) αβ (1 − 2 α + γ ) − (1 − α )(1 − β ) αβ (1 − 2 β + δ ) > 0 , for every ( α, β ) ∈ R 3 ⊃ R ∆ , ( γ , δ ) in the interior of ( 20 ), and ( γ , δ ) in ( 23 ). Pr o of of Claim 26 .  1 γ + 1 δ − (1 − α )(1 − β ) αβ (1 − 2 α + γ ) − (1 − α )(1 − β ) αβ (1 − 2 β + δ )  αβ γ δ (1 − 2 α + γ )(1 − 2 β + δ ) = − 2 γ δ + δ αβ + γ αβ − 4 δαβ γ − 2 δαβ 2 + δ 2 αβ − 2 δ α 2 β + 4 δα 2 β 2 − 2 δ 2 α 2 β − 2 γ αβ 2 − 2 γ α 2 β + 4 γ α 2 β 2 + γ 2 αβ − 2 γ 2 αβ 2 + 4 γ δ β − 2 γ δ β 2 + γ δ 2 β + 4 γ δ α + γ δ 2 α + γ 2 δ β − 2 γ δ α 2 + γ 2 δ α − γ δ 2 − γ 2 δ > 0 , where the last inequalit y is prov ed using Resolve fun ctio n of Mathematica system, see App endix A.5 . 22 T o sh o w ( 37 ), w e fir st note th at R 5 R 9 + R 3 R 7 R 11 + R 1 R 5 + R 1 R 9 − R 1 R 3 R 11 − R 1 R 7 R 11 + R 2 R 7 R 11 + R 3 R 6 R 11 = √ D  4(1 − β − γ − ˆ ε ) ( α − γ ) + (1 − α )(1 − β ) (1 − 2 α + γ ) αβ 2( α − γ − ˆ ε ) β − δ + (1 − α )(1 − β ) (1 − 2 β + δ ) αβ 2( α − γ − ˆ ε ) α − γ − 2( α − γ − ˆ ε ) 2 ( α − γ )( β − δ )  1 ˆ η + 1 ˆ ε   (1 − α )(1 − β ) αβ − 1 − β − γ − ˆ ε α − γ − ˆ ε  > 2 √ D ( α − γ − ˆ ε ) β − δ  (1 − α )(1 − β ) (1 − 2 α + γ ) αβ − α − γ − ˆ ε ( α − γ ) ˆ η  (1 − α )(1 − β ) αβ − 1 − β − γ − ˆ ε α − γ − ˆ ε  + 2 √ D ( α − γ − ˆ ε ) α − γ  (1 − α )(1 − β ) (1 − 2 β + δ ) αβ − α − γ − ˆ ε ( β − δ ) ˆ ε  (1 − α )(1 − β ) αβ − 1 − β − γ − ˆ ε α − γ − ˆ ε  . W e will sho w (1 − α )(1 − β ) (1 − 2 α + γ ) αβ − α − γ − ˆ ε ( α − γ ) ˆ η  (1 − α )(1 − β ) αβ − 1 − β − γ − ˆ ε α − γ − ˆ ε  > 0 , for ev ery ( α, β ) ∈ R 3 ⊃ R ∆ , ( γ , δ ) in the interior of ( 20 ), a nd ( γ , δ ) in ( 23 ). T hen by sym metry , w e ha v e (1 − α )(1 − β ) (1 − 2 β + δ ) αβ − α − γ − ˆ ε ( β − δ ) ˆ ε  (1 − α )(1 − β ) αβ − 1 − β − γ − ˆ ε α − γ − ˆ ε  > 0 . Let P 7 =  (1 − α )(1 − β ) (1 − 2 α + γ ) αβ − α − γ − ˆ ε ( α − γ ) ˆ η  (1 − α )(1 − β ) αβ − 1 − β − γ − ˆ ε α − γ − ˆ ε  αβ ˆ η ( α − γ )(1 − 2 α + γ ) = P 71 + P 72 √ D , where P 71 = (3 / 2) α 2 + 4 αβ + (1 / 2) β 2 − αδ − γ β + γ δ + 1 / 2 − (3 / 2) α − β + δαβ + (3 / 2) γ αβ + δ αβ γ − δ α 2 β − (3 / 2) γ αβ 2 − (1 / 2) γ α 2 β − γ δ β − γ δ α − (1 / 2) α 3 − (9 / 2) α 2 β − (5 / 2) αβ 2 + γ β 2 + α 2 δ + (3 / 2) α 3 β + (5 / 2) α 2 β 2 , and P 72 = (1 / 2) γ αβ − 1 / 2 + (1 / 2 ) β + α − (1 / 2) α 2 − (1 / 2) αβ − (1 / 2) α 2 β . The follo wing claim is p ro v ed u sing the Resolve function of th e Mathematica system in Ap- p endix A.6 . Claim 27. P 71 > 0 and P 72 < 0 for eve ry ( α, β ) ∈ R 3 ⊃ R ∆ , ( γ , δ ) i n the interior of ( 20 ), and ( γ , δ ) i n ( 23 ). F rom Claim 27 we hav e that the follo wing expression has the same sign as P 7 : ( P 71 /P 72 ) 2 − D = P 2 71 − P 2 72 D P 2 72 , where P 2 71 − P 2 72 D > 0 . The last inequalit y is true for ev ery ev ery ( α, β ) ∈ R 3 ⊃ R ∆ , ( γ , δ ) in the inte rior of ( 20 ), and ( γ , δ ) in ( 23 ); it is pr o v ed using the Resolve fun ct ion of the Mathematica system, see App end ix A.7 . 23 6.4 Pro of of Lemma 19 W e fir st pr o v e the follo wing upp er b oun d on (∆ − 1) 2 . Prop osition 28. F or every ∆ ≥ 3 , ( α, β ) ∈ R ∆ , ( γ , δ ) in the interior of ( 20 ), and ( γ , δ ) in ( 24 ), we have (∆ − 1) 2 ≤ (1 − α )(1 − β ) αβ < 1 − β − γ − ˆ ε α − γ − ˆ ε Pr o of of P r op osition 28 . T he fir st inequalit y is straigh tforw ard from α + β + ∆(∆ − 2) αβ ≤ 1. W e next pro v e that (1 − α )(1 − β )( α − γ − ˆ ε ) < αβ (1 − β − γ − ˆ ε ). W e ha v e αβ (1 − β − γ − ˆ ε ) − (1 − α )(1 − β )( α − γ − ˆ ε ) = 1 2 (1 − α − β )(1 − α − β + 2 αβ − √ D ) > 0 , where the last inequalit y follo ws from D = (1 − α − β ) 2 + 4( α − γ )( β − δ ) < (1 − α − β + 2 αβ ) 2 . The la st inequalit y is true when γ > α 2 and δ > β 2 (to see this note (1 − α − β + 2 αβ ) 2 − (1 − α − β ) 2 = 4( α − α 2 )( β − β 2 ) > 4( α − γ )( β − δ )). Let R 10 = (1 − β − γ − ˆ ε ) / ( α − γ − ˆ ε ). Note that R 7 R 10 = R 9 , R 3 R 10 = R 5 . Lemma 24 implies th at the coefficient of (∆ − 1) in ( 28 ) is p ositiv e and the co efficie nt of (∆ − 1) 2 in ( 28 ) is negativ e. Hence, also us ing Prop osition 28 , w e h a v e det( M ) ≥ 1 D  R 10  ( − R 1 + R 2 + R 3 )( − R 1 + R 6 + R 7 ) − R 2 1  +  ( − R 1 − R 8 − R 9 )( − R 1 − R 4 − R 5 ) − R 2 1  , > 1 D ( − R 1 R 2 R 10 − R 1 R 6 R 10 + R 1 R 4 + R 1 R 8 + R 5 R 9 + R 3 R 7 R 10 + R 2 R 9 + R 5 R 6 ) . W e will prov e th at for ev ery ( α, β ) ∈ R 3 ⊃ R ∆ , α, β ≤ 1 / 2, ( γ , δ ) in the in terior of ( 20 ), and ( γ , δ ) in ( 24 ), R 1 R 4 − R 1 R 2 R 10 + 1 2 ( R 5 R 9 + R 3 R 7 R 10 ) + R 2 R 9 > 0 , (38) and then b y symmetry , w e also hav e R 1 R 8 − R 1 R 6 R 10 + 1 2 ( R 5 R 9 + R 3 R 7 R 10 ) + R 5 R 6 > 0 . The left-hand side of ( 38 ) ca n b e written as the sum of the follo wing t w o terms: (1 − α − β ) √ D 1 − α − β − ˆ ε − ˆ η  1 γ − 1 1 − 2 α + γ  , (39) and √ D (1 − 2 α + γ ) ˆ η  2(1 − α )(1 − α − β − ˆ ε ) α − γ − (1 − α − β ) 2 ˆ ε  . (40) W e hav e ( 39 ) is n on-nega tiv e, since 1 − 2 α ≥ 0. 24 F or ( 40 ), we only need to pro v e 1 − α − β − ˆ ε α − γ > (1 − α − β ) 2 ˆ ε , since 1 − 2 α ≥ 0. By ( 19 ), it is equiv alent to pro v e 1 − α − β − ˆ ε − ˆ η − (1 − α − β ) 2 > 0 . (41) T o p ro v e ( 41 ), w e hav e 1 − α − β − ˆ ε − ˆ η − (1 − α − β ) 2 = − (1 − α − β ) 2 − α − β + γ + δ + √ D , and D − ( − (1 − α − β ) 2 − α − β + γ + δ ) 2 = 2 γ + 2 δ − 2 α 2 − 2 β 2 − 6 αδ − 6 γ β + 2 γ δ − 2 β δ − δ 2 +6 αβ 2 + 6 α 2 β − 6 α 2 β 2 − 2 αγ − 4 α 3 β + 2 α 2 γ + 2 α 2 δ − 4 αβ 3 + 2 β 2 γ + 2 β 2 δ − γ 2 + 4 αβ δ + 4 αβ γ + 2 α 3 +2 β 3 − α 4 − β 4 . The follo wing claim is p ro v ed u sing the Resolve function of th e Mathematica system in Ap- p endix A.8 . Claim 29. D − ( − (1 − α − β ) 2 − α − β + γ + δ ) 2 > 0 for every ( α, β ) ∈ R , α ≤ 1 / 2 , β ≤ 1 / 2 , ( γ , δ ) in the interior of ( 20 ), and ( γ , δ ) in ( 24 ). Since − (1 − α − β ) 2 − α − β + γ + δ < 0, we can conclude that ( 41 ) is tru e. Hence, we complete the pro of of Lemma 19 . 6.5 Pro of of Lemma 20 W e hav e R 4 − R 6 = √ D · 1 − 2 β + δ − γ γ (1 − 2 β + δ ) > 0 , where the inequalit y follo ws from ( 20 ). W e hav e R 5 − R 7 = (1 − α − β )( α + β − γ − δ ) + ( β − δ − α + γ ) √ D ( α − γ )( β − δ ) . If β − δ − α + γ ≥ 0, then R 5 > R 7 . W e ma y a ssu me that β − δ − α + γ < 0. T o show R 5 > R 7 , it is equiv alen t to prov e that (1 − α − β )( α + β − γ − δ ) > ( α − γ − β + δ ) √ D . W e hav e (1 − α − β ) 2 ( α + β − γ − δ ) 2 ( α − γ − β + δ ) 2 − D = 4( α − γ )( β − δ )(1 − 2 β − γ + δ )(1 − 2 α − δ + γ ) ( α − γ − β + δ ) 2 > 0 , where the inequalit y follo ws from ( 20 ). 25 T o s ho w that R 5 > 4 R 3 and R 9 > 4 R 7 , it is suffi cie nt to prov e that 1 − β − γ − ˆ ε > 4( α − γ − ˆ ε ), whic h follo ws from Pr oposition 28 . W e next pro v e U 1 > 0. W e first p ro v e that R 8 > 2 R 6 . W e hav e R 8 − 2 R 6 = √ D · 1 − 2 β − δ δ (1 − 2 β + δ ) ≥ √ D · 1 − 3 β δ (1 − 2 β + δ ) > 0 , where the first inequalit y follo ws from ( 20 ) and the last inequalit y follo ws from the fact that β ≤ (3 − √ 5) / 4. Hence U 1 = R 8 − 2 R 6 + R 9 − 2 R 7 > 0. W e next sho w: 3 R 8 2 + R 9 > 9 R 2 . b y the fact that ˆ ε := 1 2 (1 + α − β − 2 γ − √ D ) and (1 − α − β ) 2 = αβ , the goal is then to sho w, 3 √ D 2 δ + √ αβ + √ D β − δ > 9 √ D 1 − 2 α + γ , i.e., √ αβ p αβ + 4( α − γ )( β − δ ) > 9( β − δ ) 1 − 2 α + γ − 3 β 2 δ + 1 / 2 . Since th e ab o v e inequalit y is monotone with resp ect to γ , so let γ = γ min = α 2 (1+ δ − 2 β ) (1 − β ) 2 , and the goal is to sho w: √ αβ p αβ + 4( α − γ min )( β − δ ) > 9( β − δ ) 1 − 2 α + γ min − 3 β 2 δ + 1 / 2 , or equiv alen tly , 1 p 1 + 4(1 − γ min /α )(1 − δ /β ) > 9( β − δ ) 1 − 2 α + γ min − 3 β 2 δ + 1 / 2 . Replacing the term γ min in th e left-hand side, w e get, 1 q 1 + 4(1 − α (1+ δ − 2 β ) (1 − β ) 2 )(1 − δ /β ) > 9( β − δ ) 1 − 2 α + γ min − 3 β 2 δ + 1 / 2 . Since α > 1 / 2 and δ > β 2 , replace them on the LHS, then o ur goal can b e redu ced to 1 p 3 − 2 δ/β > 9( β − δ ) 1 − 2 α + γ min − 3 β 2 δ + 1 / 2 . Let X = 1 − 2 α + γ min = (1 − 2 α )(1 − β ) 2 + α 2 (1 − 2 β + δ ) (1 − β ) 2 and A = 3 β − 2 δ β . So we are go ing to sho w 2 δ X > √ A (18 δ ( β − δ ) − 3 β X + δ X ) , i.e., 4 δ 2 X 2 > A ( 18 δ ( β − δ ) − 3 β X + δ X ) 2 , when 18 δ ( β − δ ) − 3 β X + δ X > 0. The rest of the p roof can b e chec k ed b y u sing Mathematica s ince they are all p ol ynomial co nstraints, s ee App endix A.9 . 26 References [1] J. v an den Berg an d J. E. Steif. P ercolation an d the hard-core lattice gas mo del. Sto chastic Pr o c esses and their Applic ations , 49(2):179– 197, 1994. [2] M. Dyer, A. M. F r iez e, and M. Jerr um. On co unting indep enden t sets in sp arse graphs. SIAM J. 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Cambridge Univ ersit y Press, Cambridge, 1999. 27 A Mathematica Co de Clear["G lobal‘*"] SetSyste mOptions["Ineq ualitySolvingOptions" -> "CAD" -> True]; SetSyste mOptions["Ineq ualitySolvingOptions" -> "Quadratic QE" -> False ]; SetSyste mOptions["Ineq ualitySolvingOptions" -> "LinearQE" -> False]; RR = alpha > 0 && beta > 0 && alpha + beta < 1; RR3 = alpha > 0 && beta > 0 && alpha + beta + 3*alpha*be ta < 1; Inter22 = gamma > 0 && gamma < alpha && delta > 0 && delta < beta && 1 - 2*beta + delta - gamma > 0 && 1 - 2*alpha + gamma - delta > 0; l25 = gamma > 0 && gamma < alpha^2 && delta > 0 && delta < beta^2 && (1 - alpha)^ 2*delta + beta^2* (2*alpha - 1 - gamma ) < 0 && (1 - beta)^2 *gamma + alph a^2*(2*beta - 1 - de lta) < 0; u26 = gamma > alpha^2 && gamma < alpha && delta > beta^2 && delta < beta && (1 - alpha )^2*delta + beta^ 2*(2*alpha - 1 - gam ma) > 0 && (1 - beta)^2 *gamma + alph a^2*(2*beta - 1 - de lta) > 0; DD = (1 - alpha - beta)^2 + 4*(alpha - gamma)*(b eta - delta); epsilon = 1/2*(1 + alpha - beta - 2*gamma - SQ); eta = 1/2*(1 + beta - alpha - 2*delta - SQ); W11 = (alpha - gamma - epsilon )* epsilo n*(1 - 2*alph a + g amma)/(eta*(al pha - gam ma)^2); W12 = (alpha - gamma)^2 /((1 - 2*alph a + gamma )*gamma); W21 = (beta - delta - eta)*eta *(1 - 2*beta + delta )/(epsilon*(be ta - delta)^2 ); W22 = (beta - delta)^2/ ((1 - 2*beta + delta )*delta); R1 = (1 - alpha - beta)/(1 - alpha - beta - epsilon - eta); R2 = SQ/(1 - 2*alpha + gamma); R3 = 2*(alph a - gamma - epsilon) /(alpha - gam ma); R4 = SQ/gamm a; R5 = 2*(1 - beta - gamma - epsilon )/(alpha - gamma ); R6 = SQ/(1 - 2*beta + delta); R7 = 2*(alph a - gamma - epsilon) /(beta - delt a); R8 = SQ/delt a; R9 = 2*(1 - beta - gamma - epsilon )/(beta - delta) ; R10 = (1 - beta - gamma - epsilon) /(alpha - gamma - epsilo n); R11 = (1 - alpha)*( 1 - beta) /(alpha*beta); 28 A.1 Mathematica Co de for Proving Lemma 13 W3 = Expand[ Expand[Numera tor[W11]/2 - Denomina tor[W11]/2] /. {SQ^2 -> DD}]; Exponent [W3, SQ] W31 = Coefficie nt[W3, SQ, 0] W32 = Coefficie nt[W3, SQ, 1] Resolve[ Exists[{alpha, beta, gamma}, RR && Inter22 && W32 <= 0], Reals] W4 = Expand[ Expand[Numera tor[W21]/2 - Denomina tor[W21]/2] /. {SQ^2 -> DD}]; Exponent [W4, SQ] W41 = Coefficie nt[W4, SQ, 0] W42 = Coefficie nt[W4, SQ, 1] Resolve[ Exists[{alpha, beta, delta}, RR && Inter22 && W42 <= 0], Reals] W5 = FullSim plify[(W31/W3 2)^2 - DD] W6 = FullSim plify[(W41/W4 2)^2 - DD] A.2 Mathematica Co de for Proving Claim 22 FullSimp lify[((-R1 + R2 + R3) - ((beta - delta) *(-1/(alpha - gamma - epsil on) - 1/eta + 1/eps ilon) - SQ/ep silon + SQ/(1 - 2*al pha + ga mma) + 2*SQ/(al pha - gamma)) ) /. {SQ -> Sqrt [DD]}] P1 = Expand[ FullSimplify[ ExpandAll[((beta - delta)*(- 1/(alpha - gamma - epsilon) -1/eta + 1/epsi lon)-SQ/epsilo n + SQ/(1 - 2*alpha + gamma) + 2*SQ/(al pha - gamma)) *epsilon*eta* (1 - 2*alp ha + gamm a)*(alpha - g amma)* (alpha - gamma - epsilo n)]]/.{SQ^2 -> DD, SQ^3 -> SQ*D D, SQ ^4 -> DD ^2}]; Exponent [P1, SQ] P11 = Coefficie nt[P1, SQ, 0] Resolve[ Exists[{alpha, beta, gamma, delta}, RR && Inter22 && P11 <= 0], Reals] P12 = Coefficie nt[P1, SQ, 1] Resolve[ Exists[{alpha, beta, gamma, delta}, RR && Inter22 && P12 >= 0], Reals] A.3 Mathematica Co de for Proving Claim 23 P2 = FullSim plify[(P11/P1 2)^2 - DD]; P21 = -FullSimp lify[Numerator [P2]/((beta - delta)^2*( alpha - gamma)^3 )] Resolve[ Exists[{alpha, beta, gamma, delta}, RR && Inter22 && P21 <= 0], Reals] A.4 Mathematica Co de for Proving Claim 25 P5=Expan d[FullSimplify [ Expand[( (-R1 + R2 + R3)*( -R1 - R8 - R9) - R1^ 2)*delta*(alp ha - gamma )* epsilon* eta*(1 - 2*alpha + gamma) *(alpha - gamma - epsilo n)/ SQ]/. {SQ -> Sqrt[DD] }]]/.{Sqrt[alp ha^2 + (-1 + beta)^2 + alpha (-2 + 6 beta - 4 delta) + 4 (-beta + delta ) gamm a]->SQ} Exponent [P5, SQ] P51 = Coefficie nt[P5, SQ, 0] Resolve[ Exists[{alpha, beta, gamma, delta}, RR && Inter22 && P51 >= 0]] P52 = Coefficie nt[P5, SQ, 1] 29 Resolve[ Exists[{alpha, beta, gamma, delta}, RR && Inter22 && P52 <= 0]] FullSimp lify[DD - (P51/P5 2)^2] A.5 Mathematica Co de for Proving Claim 26 Resolve[ Exists[{alpha, beta, gamma, delta}, FullSimp lify[ Expand[( -R2*R11 - R6*R11 + R4 + R8)/SQ* alpha*beta*gam ma* delta*(1 - 2*alpha + gamma )*(1 - 2*beta + delta )]] <= 0 && RR3 && l25], Reals] A.6 Mathematica Co de for Proving Claim 27 P7 = FullSim plify[((1 - alpha)*( 1 - beta) /((1 - 2* alpha + g amma)*alpha*be ta) - (alpha - gamma - epsilon)/ ((alpha - gamma)* eta)*((1 - alpha )* (1 - beta)/( alpha*beta) - (1 - beta -gamma - epsil on) /(alpha - gamma - epsilon) ))*alpha*beta* eta*(alpha - gamma)* (1 - 2*alpha + gamma)] Exponent [P7, SQ] P71 = Coefficie nt[P7, SQ, 0] Resolve[ Exists[{alpha, beta, gamma, delta}, RR3 && Inter22 && P71 <= 0], Reals] P72 = Coefficie nt[P7, SQ, 1] Resolve[ Exists[{alpha, beta, gamma, delta}, RR3 && Inter22 && P72 >= 0], Reals] A.7 Mathematica Co de for Proving Lemma 18 TP1 = Expand[P7 1^2 - P72^2*D D] Resolve[ Exists[{alpha, beta, gamma, delta}, RR3 && l25 && TP1 <= 0], Reals] A.8 Mathematica Co de for Proving Claim 29 TP2 = Expand[DD - (-(1 - alpha - beta)^2 - alpha - beta + gamma + delta)^2 ] Resolve[ Exists[{alpha, beta, gamma, delta}, RR && alpha <= 1/2 && beta <= 1/2 && u26 && TP2 <= 0], Reals] A.9 Mathematica Co de for Proving Lemma 20 X = ((1 - 2*alpha)* (1 - beta)^2 + alpha ^2*(1 - 2*bet a + d elta))/(1 - b eta)^2; A = (2*beta - 2*delta)/ beta; Resolve[ Exists[{alpha, beta, gamma, delta}, (1 - alpha - beta)^ 2 == alph a*beta && alp ha > 1/2 && alpha < 1 && beta > 0 && beta < 1 && Inter22 && u26 && 4*delta^ 2*X^2 <= A*(18*de lta*(beta - delt a) - 3*b eta*X + d elta*X)^2 && 18*delta *(beta - delta) - 3*beta *X + delt a*X > 0], Rea ls] 30