FPRAS via MCMC where it mixes torpidly (and very little effort)
FFPRAS VIA MCMC WHERE IT MIXES TORPIDLY (and very little effort)
JIN-YI CAI AND TIANYU LIUAbstract. Is Fully Polynomial-time Randomized Approximation Scheme (FPRAS) for a problem via an MCMCalgorithm possible when it is known that rapid mixing provably fails? We introduce several weight-preservingmaps for the eight-vertex model on planar and on bipartite graphs, respectively. Some are one-to-one, whileothers are holographic which map superpositions of exponentially many states from one setting to another, ina quantum-like many-to-many fashion. In fact we introduce a set of such mappings that forms a group in eachcase. Using some holographic maps and their compositions we obtain FPRAS for the eight-vertex model atparameter settings where it is known that rapid mixing provably fails due to an intrinsic barrier. This FPRASis indeed the same MCMC algorithm, except its state space corresponds to superpositions of the given states,where rapid mixing holds. FPRAS is also given for torus graphs for parameter settings where natural Markovchains are known to mix torpidly. Our results show that the eight-vertex model is the first problem with theprovable property that while NP-hard to approximate on general graphs (even
E-mail addresses : [email protected], [email protected] .Supported by NSF CCF-1714275. a r X i v : . [ c s . CC ] O c t . IntroductionLet ๐บ be any 4-regular graph. We label four incident edges of each vertex from 1 to 4. The eight-vertexmodel on ๐บ is defined as follows. The states consist of even orientations , i.e. all orientations having an evennumber of arrows into (and out of) each vertex. There are eight permitted types of local configurationsaround a vertexโhence the name eight-vertex model (see Figure 1). Figure 1. Valid configurations of the eight-vertex model.Classically, the eight-vertex model is defined by statistical physicists on a square lattice region where eachvertex of the lattice is connected by an edge to four nearest neighbors. In general, the eight configurations1 to 8 in Figure 1 are associated with eight possible weights ๐ค , โฆ , ๐ค . Denote the set of these eight localconfigurations by ๐ . By physical considerations, the total weight of a state remains unchanged if all arrowsare flipped, assuming there is no external electric field. In this case we write ๐ค = ๐ค = ๐ , ๐ค = ๐ค = ๐ , ๐ค = ๐ค = ๐ , and ๐ค = ๐ค = ๐ . This complementary invariance is known as the arrow reversal symmetry orthe zero field assumption .Even in the zero-field setting, this model is already enormously expressive. The special case when ๐ = 0 is the six-vertex model, which itself has sub-models such as the ice ( ๐ = ๐ = ๐ ), KDP, and Rys ๐น models; on the square lattice, some other important models such as the dimer and zero-field Ising modelscan be reduced to it [Bax72]. Together with ferromagnetic Ising and monomer-dimer models, the six-vertex and eight-vertex models are among the most studied models in statistical physics โ . Beyond physics,Kuperberg gave a simplified proof of the famous alternating-sign matrix (ASM) conjecture in combinatoricsusing a connection to the six-vertex model [Kup96]. Recently, the six-vertex model played an importantrole in explicating the phase transition of the Potts model and the random cluster model on the squarelattice [DCGH +
16, RS19]. After the eight-vertex model was introduced in 1970 by Sutherland [Sut70], andFan and Wu [FW70], Baxter [Bax71, Bax72] achieved a good understanding of the zero-field case in thethermodynamic limit on the square lattice (in physics this understanding of the limiting case is calledโexactly solvedโ).In this paper, we assume the arrow reversal symmetry and our algorithmic and complexity resultsfurther assume that ๐, ๐, ๐, ๐ โฅ 0 (as is the case in classical physics), unless otherwise explicitly stated. Forany 4-regular graph ๐บ (not just the grid or even planar graph, however for plane graphs the edges arelocally labeled from 1 to 4 cyclically), the partition function of the eight-vertex model on ๐บ with parameters (๐, ๐, ๐, ๐) is defined as(1.1) ๐ (๐บ; ๐, ๐, ๐, ๐) = โ ๐โ ๎ป ๐ (๐บ) ๐ ๐ +๐ ๐ ๐ +๐ ๐ ๐ +๐ ๐ ๐ +๐ , where ๎ป ๐ (๐บ) is the set of all even orientations of ๐บ , and ๐ ๐ is the number of vertices in type ๐ in ๐บ ( ,locally depicted as in Figure 1) under an even orientation ๐ โ ๎ป ๐ (๐บ) .In terms of exact complexity, a dichotomy is given for the eight-vertex model on general 4-regular graphsfor all eight (possibly complex) parameters [CF17]. This is studied in the context of a classification programfor the complexity of counting problems [CC17], where the eight-vertex model serves as an important basiccase for Holant problems defined by not necessarily symmetric constraint functions. It is shown that every โ A search in
Google Scholar for โsix- and eight-vertex modelsโ returns โAbout 153,000 resultsโ. etting is either P-time computable (and some are surprising) or ๐, ๐, ๐, ๐ are nonnegative realnumbers, the problem of computing the partition function of the eight-vertex model is ๐ = ๐ = ๐ = ๐ (this is equivalent to the unweighted case); (2) three of ๐, ๐, ๐, ๐ are zero; or (3) two of ๐, ๐, ๐, ๐ are zero and the other two are equal. The full classification of the exact complexity for the eight-vertexmodel on planar graphs is still open, but in the full version of this paper we will show that in our settingwhere ๐, ๐, ๐, ๐ are nonnegative, the problem that is ๐ + ๐ = ๐ + ๐ or (2) one of ๐, ๐ is zero and one of ๐, ๐ is zero.Recently, the approximate complexity of counting and sampling of the eight-vertex model (and its specialcase, the six-vertex model) has been studied [GR10, Liu18, CLL19, FR19, CLLY20, CL20]. Interestingly, theseresults conform to the phase transition phenomenon in physics. In order to state the previous results andpresent our work, we adopt the following notations assuming ๐, ๐, ๐, ๐ โฅ 0 . โ ๎ = { (๐, ๐, ๐, ๐) | ๐ โค ๐ + ๐ + ๐, ๐ โค ๐ + ๐ + ๐, ๐ โค ๐ + ๐ + ๐, ๐ โค ๐ + ๐ + ๐} ; โ ๎ = { (๐, ๐, ๐, ๐) | ๐ + ๐ โค ๐ + ๐, ๐ + ๐ โค ๐ + ๐, ๐ + ๐ โค ๐ + ๐} ; โ ๎ = { (๐, ๐, ๐, ๐) | ๐ โค ๐ + ๐ + ๐ , ๐ โค ๐ + ๐ + ๐ , ๐ โค ๐ + ๐ + ๐ , ๐ โค ๐ + ๐ + ๐ } . Remark . ๎ โ ๎ and ๎ โ ๎ .Physicists have shown an order-disorder phase transition for the eight-vertex model on the squarelattice between parameter settings outside ๎ and those inside (see Baxterโs book [Bax82] for more details).Physicists call ๎ the disordered phase , and its complement ๎ , which consists of 4 disjoint regions inwhich one of (๐, ๐, ๐, ๐) dominates, the ordered phases . In [CLLY20] and [CL20], it was shown that: (1)approximating the partition function of the eight-vertex model on general 4-regular graphs outside ๎ isNP-hard, (2) approximating the partition function of the eight-vertex model on general 4-regular graphsoutside ๎ is at least as hard as approximately counting perfect matchings on general graphs (in short wesay it is โ for general 4-regular graphs in ๎ โ ๎ , and (4) there is an FPRASfor planar 4-regular graphs in a subregion of ๎ โ ๎ . Note that all previous positive results are confinedwithin ๎ , in particular within the disordered phase ๎ . See Figure 2.Previous FPRAS results in [CLLY20] are based on the method of Markov chain Monte Carlo (MCMC) .A nice upper bound on the mixing time of a specific Markov chain can be achieved only in ๎ (which isa subregion of the disordered phase ๎ ) using a canonical path argument . The canonical path argumentwas introduced for perfect matchings [JS93] and extends well to problems which are believed to havestrong connections with perfect matchings [McQ13]. We proved in [CL20] that computing ๐ (๐, ๐, ๐, ๐) for any (๐, ๐, ๐, ๐) โ ๎ can be reduced to the problem of counting perfect matchings by expressing thelocal constraints (๐, ๐, ๐, ๐) using โmatchgatesโ whereas such expression provably fails for (๐, ๐, ๐, ๐) โ ๎ .Moreover, in the ordered phases of the eight-vertex model (the four disjoint parts of ๎ ), torpid mixingresults of natural Markov chains were established on grid/torus graphs [GR10, Liu18, FR19].In this paper, we give the first FPRAS for ๐ (๐, ๐, ๐, ๐) for (๐, ๐, ๐, ๐) outside ๎ on planar and on bipartitegraphs. We introduce a special edge-2-coloring model, called the even-coloring model (Section 2), andsimultaneously set up two different kinds of relations between the partition functions of the eight-vertexmodel and the even-coloring model. The first kind of relations exploit the property of planar and of bipartitegraphs, and are one-to-one weight-preserving mappings between the states of the eight-vertex model ๎ป ๐ (๐บ) and the states of the even-coloring model ๎ฏ ๐ (๐บ) ; the second is by the method of holographic transformation introduced by Valiant [Val08] which can be thought of as an (exponentially many)-to-(exponentially many)map between the two state spaces (Section 3). This, magically, allows us to identify the partition functionsof the eight-vertex model on the same graph under totally different parameter settings. Interestingly, we โ Suppose ๐ โถ ฮฃ โ โ โ is a function mapping problem instances to real numbers. A fully polynomial randomized approximationscheme (FPRAS) [KL83] for a problem is a randomized algorithm that takes as input an instance ๐ฅ and ๐ > 0 , running in timepolynomial in the length |๐ฅ| and ๐ โ1 , and outputs a number ๐ (a random variable) such that Pr [(1 โ ๐)๐ (๐ฅ) โค ๐ โค (1 + ๐)๐ (๐ฅ)] โฅ . igure 2. A Venn diagram of the approximation complexity of the eight-vertex model ongeneral 4-regular graphs.show that these maps and their compositions among different parameter settings under which the partitionfunction is preserved have group structures . For planar graphs, this group is isomorphic to the symmetrygroup ๐ on three elements (see Section 4); for bipartite graphs, this group is isomorphic to the dihedralgroup ๐ท of order 12 (the symmetry group of a regular hexagon, see Section 5).Therefore, although the Markov chain on a graph under certain parameter settings outside ๎ is notrapidly mixing, after โmixing upโ the state space using a combination of two maps, the Markov chain turnsout to be rapidly mixing. Indeed, as a consequence, this โindirectโ MCMC leads to FPRAS for new regionsin the disordered phase ๎ and, for the first time, in the ordered phases ๎ for planar graphs and for bipartitegraphs. Theorem 1.1.
Let ๐บ be a 4-regular plane graph. There is an FPRAS for ๐ (๐บ; ๐, ๐, ๐, ๐) for (๐, ๐, ๐, ๐) in asubregion of ๎ โ ๎ โ ๎ and in a subregion of ๎ . We have proved that on general 4-regular graphs, approximating the eight-vertex model in ๎ โ ๎ is ๎ is NP-hard [CLLY20]. Therefore, we have found a family of problems (withparameters ranging in a region of parameter space) having the following provable properties: For theeight-vertex model in the subregion of ๎ given in Theorem 1.1 (described more explicitly in Corollary 4.3),computing ๐ (๐, ๐, ๐, ๐) is(1) NP-hard to approximate on general 4-regular graphs [CLLY20], and(2) has an FPRAS on planar 4-regular graphs (this paper).This separation of complexity for general and for planar graphs should be compared and contrasted withthe FKT algorithm [TF61, Kas61, Kas67] for exact counting of perfect matchings, but here for approximatecounting. Previously the combined results of [GvV15] and [HVV15] proved a similar result for ๐ -coloringsfor general versus planar graphs. We note that it was shown in [GJM15] that approximating the partitionfunctions of many two-state spin systems remain NP-hard on planar graphs. A similar result was shown in[GJ12] for approximating the Tutte polynomial ๐ (๐บ; ๐ฅ, ๐ฆ) in a large portion of the (๐ฅ, ๐ฆ) plane. We can alsoprove that for the subregion, ๐ (๐, ๐, ๐, ๐) is heorem 1.2. Let ๐บ be a 4-regular bipartite graph. There is an FPRAS for ๐ (๐บ; ๐, ๐, ๐, ๐) for (๐, ๐, ๐, ๐) in asubregion of ๎ โ ๎ and in a subregion of ๎ . Note that the subregions mentioned in Theorem 1.2 are disjoint from those mentioned in Theorem 1.1.We have proved that on general 4-regular graphs, approximating the eight-vertex model in ๎ โ ๎ is ๎ is NP-hard [CLLY20]. Therefore, we have found a family of problems (withparameters ranging in a region of parameter space) having the following provable properties: For theeight-vertex model in the subregion of ๎ given in Theorem 1.2 (described more explicitly in Corollary 5.3),computing ๐ (๐, ๐, ๐, ๐) is(1) NP-hard to approximate on general 4-regular graphs [CLLY20], and(2) has an FPRAS on bipartite 4-regular graphs (this paper).Previously the only problem having similar properties is the antiferromagnetic Ising model on general versusbipartite graphs [SS14, CGG + + (โค/๐โค) ร (โค/๐โค) with even ๐, ๐ also have this property and arebipartite, the results in Theorem 1.1 and Theorem 1.2 also hold in torus graphs (with even side lengths).Finally, we note that the techniques introduced in this paper have other applications. First, the mapsbetween partition functions under different parameter settings established in this paper are not onlyuseful for giving approximation algorithms. The same maps are useful in our understanding of the exactcomputational complexity of the eight-vertex model on planar graphs. Second, the techniques are useful forstudying the partition functions of other edge-orientation problems (e.g. other vertex models in statisticalphysics) or edge-coloring problems (e.g. Holant problems) on planar/bipartite graphs.2. The even-coloring modelWe introduce the following edge-2-coloring model on 4-regular graphs called the even-coloring model : avalid configuration of this model assigns either green or red to every edge such that the number of greenedges incident to any vertex is even (zero, two, or four). Similar to the eight-vertex model, there are alsoeight valid local configurations around a vertex (shown in Figure 3), and configurations 1 to 8 in Figure 3can be associated with weights ๐ค โฒ1 , โฆ , ๐ค โฒ8 respectively. Denote the set of these eight local configurations by ๐ EC . Figure 3. Valid configurations of the even-coloring model.To set up a correspondence between the even-coloring model and the eight-vertex model satisfyingarrow reversal symmetry, we consider the even-coloring model with weights that satisfy the color reversal ymmetry . That is, the weight of a local configuration at a vertex remains unchanged if the color on everyincident edge is changed. In this case we write ๐ค โฒ1 = ๐ค โฒ2 = ๐ค , ๐ค โฒ3 = ๐ค โฒ4 = ๐ฅ , ๐ค โฒ5 = ๐ค โฒ6 = ๐ฆ , and ๐ค โฒ7 = ๐ค โฒ8 = ๐ง .Given a 4-regular graph ๐บ , we label four incident edges of each vertex from 1 to 4. The partition function ofthe even-coloring model with parameters (๐ค, ๐ฅ, ๐ฆ, ๐ง) on ๐บ is defined as(2.1) ๐ EC (๐บ; ๐ค, ๐ฅ, ๐ฆ, ๐ง) = โ ๐โ ๎ฏ ๐ (๐บ) ๐ค ๐ +๐ ๐ฅ ๐ +๐ ๐ฆ ๐ +๐ ๐ง ๐ +๐ , where ๎ฏ ๐ (๐บ) is the set of all even colorings of ๐บ , and ๐ ๐ is the number of vertices in type ๐ in ๐บ ( ,locally depicted as in Figure 3) under the even-coloring ๐ โ ๎ฏ ๐ (๐บ) .3. Holographic TransformationGiven a 4-regular graph ๐บ = (๐ , ๐ธ) , the edge-vertex incidence graph ๐บ โฒ = (๐ ๐ธ , ๐ ๐ , ๐ธ โฒ ) is a bipartite graphwhere (๐ข ๐ , ๐ข ๐ฃ ) โ ๐ ๐ธ ร ๐ ๐ is an edge in ๐ธ โฒ iff ๐ โ ๐ธ in ๐บ is incident to ๐ฃ โ ๐ . We model an orientation( ๐ค โ ๐ฃ ) on an edge ๐ = {๐ค, ๐ฃ} โ ๐ธ from ๐ค into ๐ฃ in ๐บ by assigning to (๐ข ๐ , ๐ข ๐ค ) โ ๐ธ โฒ and to (๐ข ๐ , ๐ข ๐ฃ ) โ ๐ธ โฒ in ๐บ โฒ . A configuration of the eight-vertex model on ๐บ is a on ๐บ โฒ , namely ๐ โถ ๐ธ โฒ โ {0, 1} ,where for each ๐ข ๐ โ ๐ ๐ธ its two incident edges are assigned 01 or 10, and for each ๐ข ๐ฃ โ ๐ ๐ the sum of values โ ๐ (๐ ๐ ) โก 0 (mod 2) , over the four incident edges of ๐ข ๐ฃ . Thus we model the even orientation rule of ๐บ on all ๐ฃ โ ๐ by requiring โtwo-0-two-1/four-0/four-1โ locally at each vertex ๐ข ๐ฃ โ ๐ ๐ .The โone-0-one-1โ requirement on the two edges incident to a vertex in ๐ ๐ธ is a binary Diseqal-ity constraint, denoted by (โ ) . The values of a 4-ary constraint function ๐ can be listed in a matrix ๐(๐ ) = [ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ๐ ] , called the constraint matrix of ๐ . For the eight-vertex model satisfying the evenorientation rule and arrow reversal symmetry, the constraint function ๐ at every vertex ๐ฃ โ ๐ ๐ in ๐บ โฒ hasthe form ๐(๐ ) = [ ๐ 0 0 ๐0 ๐ ๐ 00 ๐ ๐ 0๐ 0 0 ๐ ] , if we locally index the left, down, right, and up edges incident to ๐ฃ by 1, 2, 3, and4, respectively according to Figure 1. Thus computing the partition function ๐ (๐บ; ๐, ๐, ๐, ๐) is equivalentto evaluating โ ๐โถ๐ธ โฒ โ{0,1} โ ๐ขโ๐ ๐ธ (โ ) (๐ | ๐ธ โฒ (๐ข) ) โ ๐ขโ๐ ๐ ๐ (๐ | ๐ธ โฒ (๐ข) ) , where ๐ธ โฒ (๐ข) denotes the incident edges of ๐ข โ ๐ ๐ธ โช ๐ ๐ . In fact, in this way we express the partition functionof the eight-vertex model as the Holant sum in the framework for Holant problems: ๐ (๐บ; ๐, ๐, ๐, ๐) = Holant (๐บ โฒ ; โ | ๐ ) where we use Holant (๐ป ; ๐ | ๐ ) to denote the Holant sum โ ๐โถ๐ธโ{0,1} โ ๐ขโ๐ ๐ (๐ | ๐ธ(๐ข) ) โ ๐ขโ๐ ๐ (๐ |
๐ธ(๐ข) ) on abipartite graph ๐ป = (๐ , ๐ , ๐ธ) for the Holant problem Holant (๐ | ๐ ) . Each vertex in ๐ (or ๐ ) is assigned theconstraint function ๐ (or ๐ , respectively). The constraint function ๐ is written as a row vector, whereas theconstraint function ๐ is written as a column vector, both as truth tables. (See [CC17] for more on Holantproblems.) The following proposition says that an invertible holographic transformation does not changethe complexity of the Holant problem in the bipartite setting. Proposition 3.1 ([Val08]) . Suppose
๐ โ โ is an invertible matrix. Let ๐ = arity(๐) and ๐ = arity(๐ ) . Define ๐ โฒ = ๐ (๐ โ1 ) โ๐ and ๐ โฒ = ๐ โ๐ ๐ . Then for any bipartite graph ๐ป , Holant (๐ป ; ๐ | ๐ ) =
Holant (๐ป ; ๐ โฒ | ๐ โฒ ) . We denote Holant (๐บ; ๐ ) =
Holant (๐บ โฒ ; = | ๐ ) . For the even-coloring model, if we view a green-red edgecoloring by a 0-1 assignment to the edges such that an edge ๐ is assigned if it is colored green and assigned if it is colored red, then the partition function of the even-coloring model ๐ EC (๐บ; ๐ค, ๐ฅ, ๐ฆ, ๐ง) is exactly thevalue of the Holant problem Holant (๐บ; [ ๐ง 0 0 ๐ค0 ๐ฅ ๐ฆ 00 ๐ฆ ๐ฅ 0๐ค 0 0 ๐ง ]) . he following two lemmas show that the eight-vertex model and the even-coloring model are connectedvia suitable holographic transformations in unexpected ways as Holant problems. Lemma 3.2.
Let ๐บ be a 4-regular graph and let ๐ ๐ = [ โ1 1 1 โ11 โ1 1 โ11 1 โ1 โ11 1 1 1 ] . Then ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐ค, ๐ฅ, ๐ฆ, ๐ง) where [ ๐ค๐ฅ๐ฆ๐ง ] = ๐ ๐ [ ๐๐๐๐ ] .Proof. Using the binary disequality function (โ ) for the orientation of any edge, we can express the partitionfunction of the eight-vertex model ๐บ as a Holant problem on its edge-vertex incidence graph ๐บ โฒ , ๐ (๐บ; ๐, ๐, ๐, ๐) = Holant (๐บ โฒ ; โ | ๐ ) , where ๐ is the 4-ary signature with ๐(๐ ) = [ ๐ 0 0 ๐0 ๐ ๐ 00 ๐ ๐ 0๐ 0 0 ๐ ] . Note that, writing the truth table of (โ ) = (0, 1, 1, 0) as a vector and multiplied by a tensor power of the matrix ๐ โ1 , where ๐ = [ ] we get (โ )(๐ โ1 ) โ2 =(1, 0, 0, 1) , which is exactly the truth table of the binary equality function (= ) . Then according to Proposi-tion 3.1, by the ๐ -transformation, we getHolant (๐บ โฒ ; โ | ๐ ) = Holant (๐บ โฒ ; โ โ (๐ โ1 ) โ2 | ๐ โ4 โ ๐ )= Holant (๐บ โฒ ; = | ๐ โ4 ๐ )= Holant (๐บ; ๐ โ4 ๐ ) , and a direct calculation shows that ๐(๐ โ4 ๐ ) = [ ๐+๐+๐+๐ 0 0 โ๐+๐+๐โ๐0 ๐โ๐+๐โ๐ ๐+๐โ๐โ๐ 00 ๐+๐โ๐โ๐ ๐โ๐+๐โ๐ 0โ๐+๐+๐โ๐ 0 0 ๐+๐+๐+๐ ] . (cid:3) Readers are referred to Appendix A for a more insightful explanation on why the arity-4 constraintfunction ๐ is transformed to a real-valued constraint function, under the complex-valued ๐ -transformation. Lemma 3.3.
Let ๐บ be a 4-regular graph and let ๐ ๐ป๐ = [ โ1 1 1 11 โ1 1 11 1 โ1 11 1 1 โ1 ] . Then ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐ค, ๐ฅ, ๐ฆ, ๐ง) where [ ๐ค๐ฅ๐ฆ๐ง ] = ๐ ๐ป๐ [ ๐๐๐๐ ] .Proof. For the eight-vertex model as a Holant problem Holant (๐บ โฒ ; โ | ๐ ) , we perform a holographictransformation by the matrix [ ] . We note that this is the composition of a ๐ -transformation andan ๐ป -transformation where ๐ = [ ] and ๐ป = [ ] , namely [ ] = ๐ป ๐ . Then ๐ (๐บ; ๐, ๐, ๐, ๐) = Holant (๐บ โฒ ; โ | ๐ )= Holant (๐บ โฒ ; (โ ) โ ((๐ป ๐ ) โ1 ) โ2 | (๐ป ๐ ) โ4 โ ๐ )= Holant (๐บ โฒ ; = | (๐ป ๐ ) โ4 ๐ )= Holant (๐บ; (๐ป ๐ ) โ4 ๐ ) . Here (โ ) โ ((๐ป ๐ ) โ1 ) โ2 = (โ ) โ (๐ โ1 ) โ2 โ (๐ป โ1 ) โ2 = (= ) โ (๐ป โ1 ) โ2 = (= ) , because ๐ป is orthogonal. Now adirect calculation shows that ๐((๐ป ๐ ) โ4 ๐ ) = [ ๐+๐+๐โ๐ 0 0 โ๐+๐+๐+๐0 ๐โ๐+๐+๐ ๐+๐โ๐+๐ 00 ๐+๐โ๐+๐ ๐โ๐+๐+๐ 0โ๐+๐+๐+๐ 0 0 ๐+๐+๐โ๐ ] . (cid:3) . Planar graphs Lemma 4.1.
Let ๐บ be a 4-regular plane graph. Then ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐, ๐, ๐, ๐) .Proof. It is well known that a connected planar graph is Eulerian if and only if its dual is bipartite [Wel69].Partition functions are multiplicative over connected components, so we may assume ๐บ is connected. As ๐บ is planar and 4-regular, the dual ๐บ โ is bipartite. Hence, we can color the faces of ๐บ using two colors,say black and white, so that any two adjacent faces (i.e., they share an edge) are of different colors. Fordefiniteness, we assume that the outer face of ๐บ is colored white. See Figure 4a for an example. Everyedge separates one face colored white and another face colored black, so each edge is on a unique whiteface. This shows that the binary relation on the set of edges defined by being on the same white face is anequivalence relation. BB BB B BW WW W (a) (b)
Figure 4. A proper 2-coloring of the faces of a planar 4-regular graph and its canonicalorientation ๐ .Based on the above facts, there is a canonical orientation ๐ which orients all the edges along any (non-outer) white face clockwise . This is also the same as to orient edges along every black face counterclockwise .See Figure 4b for a pictorial illustration. Observe that ๐ is an Eulerian orientation of ๐บ (at every vertex thein-degree equals the out-degree) and in ๐ every vertex is in the 5th local configuration in Figure 6 (andequivalently the 6th local configuration in Figure 7). (a) (b) Figure 5. An even orientation ๐ โฒ and its corresponding even coloring.For an arbitrary orientation ๐ โฒ of ๐บ , we say an edge ๐ is green if ๐ โฒ (๐) = ๐ (๐) , and ๐ is red otherwise. Thenfor any even orientation ๐ โฒ of ๐บ , we can show that this assignment of green-red colors is an even coloring f ๐บ . (See an illustration in Figure 5: Figure 5a is an even orientation and Figure 5b is its correspondingeven coloring.) Every even orientation ๐ โฒ gives an even coloring because: โ Under this coloring, the canonical orientation ๐ receives the all-green coloring, which is an evencoloring itself. โ For any even orientation ๐ โฒ , the local configuration of ๐ โฒ at any vertex differs from the localconfiguration of ๐ at the same vertex on an even number of edges. These edges receive the red colorand the others receive green. Therefore, at each vertex the color assignment will be in one of thestates shown in Figure 3. Figure 6
Figure 7We claim that this color assignment on the edges of ๐บ gives a bijection ๐ Planar from ๐ to ๐ EC . Giventhe black and white coloring of the faces of ๐บ , there are two types of vertices in ๐บ , either the one inFigure 6 or the other one in Figure 7. The correspondence of local configurations from Figure 6 to Figure 3 is (1, 2, 3, 4, 5, 6, 7, 8) โ (3, 4, 1, 2, 7, 8, 5, 6) ; the correspondence of local configurations from Figure 7 to Figure 3is (1, 2, 3, 4, 5, 6, 7, 8) โ (4, 3, 2, 1, 8, 7, 6, 5) . Consequently, ๐ Planar induces a one-to-one correspondencebetween ๎ป ๐ (๐บ) and ๎ฏ ๐ (๐บ) .Although there are two types of maps, both of them map {1, 2} to {3, 4} , {3, 4} to {1, 2} , {5, 6} to {7, 8} ,and {7, 8} to {5, 6} . In terms of the weights of local configurations: the arrow reversal symmetry of theeight-vertex model induces an equivalence relation {1, 2}, {3, 4}, {5, 6}, {7, 8} in Figure 1; the color reversalsymmetry of the even-coloring model induces an equivalence relation {1, 2}, {3, 4}, {5, 6}, {7, 8} in Figure 3.Thus there is a uniform and consistent way to assign a mapping of parameters. One can see that ๐ Planar isweight-preserving if the even-coloring model has parameter setting (๐ค, ๐ฅ, ๐ฆ, ๐ง) = (๐, ๐, ๐, ๐) . It follows that ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐, ๐, ๐, ๐). (cid:3) Now that we have set up multiple equations (Lemma 3.2, Lemma 3.3, and Lemma 4.1) between ๐ and ๐ EC under different mappings in terms of the parameter settings, we can combine them and obtainequations between ๐ under different parameter settings.Before that, we make the following observation. Since in any even orientation of a 4-regular graph ๐บ ,the number of sinks (Figure 1-7) must be equal to the number of sources (Figure 1-8) and thus their sum is lways even, we know that in the eight-vertex model under parameter setting (๐, ๐, ๐, ๐) , the weight of astate is unchanged if we flip ๐ to โ๐ . Therefore, we have(4.1) ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ (๐บ; ๐, ๐, ๐, โ๐). (In particular, for non-negative ๐, ๐, ๐, ๐ , even though โ๐ makes an appearance on the right-hand-side, thisequation says the value ๐ (๐บ; ๐, ๐, ๐, โ๐) โฅ 0 .) Obviously the approximation complexity for computing ๐ (๐, ๐, ๐, โ๐) is the same as that for ๐ (๐, ๐, ๐, ๐) . Notation.
Given a set of 4-tuples ๐ , let ๐ ๐ (๐) = {(๐, ๐, ๐, โ๐) | (๐, ๐, ๐, ๐) โ ๐} . Notation.
Given two invertible matrices ๐ and ๐ , denote by โจ๐ , ๐ โฉ the group of matrices generatedby ๐ and ๐ , where the group operation is matrix multiplication. Theorem 4.2.
Let ๐บ be a 4-regular plane graph and let ๐ Pl ๐ = [ ] , ๐ Pl ๐ป๐ = [ ] . Thenfor any ๐ โ โจ๐ Pl ๐ , ๐ Pl ๐ป๐ โฉ , ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ (๐บ; ๐ โฒ , ๐ โฒ , ๐ โฒ , ๐ โฒ ) where [ ๐ โฒ ๐ โฒ ๐ โฒ ๐ โฒ ] = ๐ [ ๐๐๐๐ ] .Remark . One can check that โจ๐ Pl ๐ , ๐ Pl ๐ป๐ โฉ is isomorphic to the symmetry group ๐ and the groupelements are shown in Table 1. Proof.
First we prove the theorem for ๐ Pl ๐ . From Lemma 3.2, we know that ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐ค, ๐ฅ, ๐ฆ, ๐ง) where [ ๐ค๐ฅ๐ฆ๐ง ] = [ โ1 1 1 โ11 โ1 1 โ11 1 โ1 โ11 1 1 1 ] [ ๐๐๐๐ ] . From Lemma 4.1, we know that ๐ EC (๐บ; ๐ค, ๐ฅ, ๐ฆ, ๐ง) = ๐ (๐บ; ๐ โฒ , ๐ โฒ , ๐ โฒ , ๐ โฒ ) where [ ๐ โฒ ๐ โฒ ๐ โฒ ๐ โฒ ] = [ ] [ ๐ค๐ฅ๐ฆ๐ง ] . Therefore, we have ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ (๐บ; ๐ โฒ , ๐ โฒ , ๐ โฒ , ๐ โฒ ) where [ ๐ โฒ ๐ โฒ ๐ โฒ ๐ โฒ ] =[ ] โ [ โ1 1 1 โ11 โ1 1 โ11 1 โ1 โ11 1 1 1 ] [ ๐๐๐๐ ] = ๐ Pl ๐ [ ๐๐๐๐ ] .The proof for ๐ Pl ๐ป๐ is similar. Instead of combining Lemma 3.2 and Lemma 4.1, we simply need tocombine Lemma 3.3 and Lemma 4.1 and notice that ๐ Pl ๐ป๐ = [ ] โ [ โ1 1 1 11 โ1 1 11 1 โ1 11 1 1 โ1 ] .Since the theorem is proved for the two invertible matrices ๐ Pl ๐ and ๐ Pl ๐ป๐ , it is also true for the group ofmatrices generated by these two matrices using their inverse and matrix multiplication. (cid:3) Notation.
In order to state the results in this section and the next section, we adopt the following notationsassuming ๐, ๐, ๐, ๐ โฅ 0 . โ ๎ญ = {(๐, ๐, ๐, ๐) | ๐ โค ๐ + ๐ + ๐} , ๎ฎ = {(๐, ๐, ๐, ๐) | ๐ โค ๐ + ๐ + ๐} , ๎ฏ = {(๐, ๐, ๐, ๐) | ๐ โค ๐ + ๐ + ๐} , ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ โค ๐ + ๐ + ๐} ; โ ๎ญ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ + ๐ โค ๐ + ๐} , ๎ฎ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ + ๐ โค ๐ + ๐} , ๎ฏ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ + ๐ โค ๐ + ๐} . Remark . ๎ญ๎ฐ โ ๎ญ โ ๎ฐ , ๎ฎ๎ฐ โ ๎ฎ โ ๎ฐ , ๎ฏ๎ฐ โ ๎ฏ โ ๎ฐ . ๎ = ๎ญ โ ๎ฎ โ ๎ฏ โ ๎ฐ . ๎ = ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ . Inaddition, we abuse the notation and use ๎ฏ = {(๐, ๐, ๐, ๐) | ๐ โฅ ๐ + ๐ + ๐} , ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ โฅ ๐ + ๐ + ๐} , ๎ญ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ + ๐ โฅ ๐ + ๐} , ๎ฎ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ + ๐ โฅ ๐ + ๐} , and ๎ฏ๎ฐ = {(๐, ๐, ๐, ๐) | ๐ + ๐ โฅ ๐ + ๐} in Table 1 and Table 2. Corollary 4.3.
Let ๐บ be a 4-regular plane graph and let ๐ Pl ๐ and ๐ Pl ๐ป๐ be defined as in Theorem 4.2. Then forany ๐ โ โจ๐ Pl ๐ , ๐ Pl ๐ป๐ โฉ , there is an FPRAS for ๐ (๐บ; ๐, ๐, ๐, ๐) if ๐ [ ๐๐๐๐ ] โ ๎ โ ๎ . Thus we know that for any
๐ โ โจ๐ Pl ๐ , ๐ Pl ๐ป๐ โฉ , there is an FPRAS for ๐ (๐บ; ๐, ๐, ๐, ๐) for (๐, ๐, ๐, ๐) in asubregion of ๐ โ1 ( ๎ ) . Note that ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฏ โ ๎ โ ๎ and ๎ฏ โ ๎ . With the help of Table 1, onecan see that Theorem 1.1 is implied by Corollary 4.3 โก . โก In fact, to prove Theorem 1.1 we need (๐ Pl ๐ป๐ ) โ1 ( ๎ โ ๎ ) โ ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฏ โ ๎ and one can check that this is true. able 1. Elements of โจ๐ Pl ๐ , ๐ Pl ๐ป๐ โฉ and preimages of ๎ = ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ under cor-responding maps. A substantial subregion of each preimage admits FPRAS on planar4-regular graphs. The last column lists the approximation complexity of the eight-vertexmodel on general 4-regular graphs in the corresponding preimage regions.Element Matrix Preimage Approximation ๐ผ [ ] ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ FPRAS in ๎ ๐ Pl ๐ 12 [ ] ๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฏ ) (๐ Pl ๐ ) [ ] ๎ฏ NP-hard ๐ Pl ๐ป๐ 12 [ ] ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฏ ๐ Pl ๐ ๐ Pl ๐ป๐ 12 [ ] ๐ ๐ ( ๎ฏ ) NP-hard (๐ Pl ๐ ) ๐ Pl ๐ป๐ [ ] ๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ ) FPRAS in ๐ ๐ ( ๎ )
5. Bipartite graphs
Lemma 5.1.
Let ๐บ be a bipartite graph. Let ๐บ โฒ denote its edge-vertex incidence graph. Suppose ๐ satisfiesarrow reversal symmetry. Then Holant (๐บ โฒ ; โ | ๐ ) = Holant (๐บ โฒ ; = | ๐ ) = Holant (๐บ; ๐ ) . In particular, if ๐บ is 4-regular, then ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐, ๐, ๐, ๐) .Proof. For any bipartite graph
๐บ = (๐ฟ, ๐ , ๐ธ) , there is a canonical orientation ๐ which is to orient all the edgesfrom ๐ to ๐ฟ . Let ๐บ be a 4-regular bipartite graph. Observe that in ๐ every vertex in ๐ฟ has local configurationFigure 1-7 and every vertex in ๐ has local configuration Figure 1-8.For an arbitrary orientation ๐ โฒ of ๐บ , we say an edge ๐ is green if ๐ โฒ (๐) = ๐ (๐) , and ๐ is red otherwise. Thiscoloring assignment on the edges of ๐บ gives a bijection ๐ Bipartite from ๐ to ๐ EC . For the vertices in ๐ฟ ,this can be seen in the entry-wise correspondence (1, 2, 3, 4, 5, 6, 7, 8) โ (1, 2, 3, 4, 5, 6, 7, 8) from Figure 1to Figure 3; for the vertices in ๐ , the correspondence of local configurations from Figure 1 to Figure 3is (1, 2, 3, 4, 5, 6, 7, 8) โ (2, 1, 4, 3, 6, 5, 8, 7) . Consequently ๐ Bipartite defines a one-to-one correspondencebetween ๎ป ๐ (๐บ) and ๎ฏ ๐ (๐บ) . Again because both maps respect the same equivalence relation induced bythe arrow reversal symmetry, this one-to-one correspondence ๐ Bipartite is weight-preserving if the even-coloring model has the parameter setting (๐ค, ๐ฅ, ๐ฆ, ๐ง) = (๐, ๐, ๐, ๐) . It follows that ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ EC (๐บ; ๐, ๐, ๐, ๐). The idea of the above mapping can be easily extended to general (not necessarily 4-regular) graphs.For a bipartite graph
๐บ = (๐ฟ, ๐ , ๐ธ) and its edge-vertex incidence graph ๐บ โฒ = (๐ ๐ธ , ๐ฟ โช ๐ , ๐ธ โฒ ) , every vertex ๐ฃ ๐ โ ๐ ๐ธ has degree 2 and connects a vertex ๐ โ ๐ฟ with a vertex ๐ โ ๐ . To see a one-to-one weight-preservingmapping from valid configurations in Holant (๐บ โฒ ; โ | ๐ ) to valid configurations in Holant (๐บ โฒ ; = | ๐ ) , onesimply flips the assignment on every edge {๐ฃ ๐ , ๐} such that ๐ฃ ๐ โ ๐ ๐ธ and ๐ โ ๐ . (cid:3) For any bipartite regular graph
๐บ = (๐ฟ, ๐ , ๐ธ) , we know that |๐ฟ| = |๐ | and hence the total number ofvertices is always even. In the eight-vertex model under parameter setting (๐, ๐, ๐, ๐) , the weight of a state s unchanged if we flip the sign of the weight on every vertex. Therefore, for bipartite 4-regular graphs, inaddition to (4.1), we also have(5.1) ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ (๐บ; โ๐, โ๐, โ๐, โ๐). Notation.
Given a set of 4-tuples ๐ , let ๐ (๐) = {(โ๐, โ๐, โ๐, โ๐) | (๐, ๐, ๐, ๐) โ ๐} . Theorem 5.2.
Let ๐บ be a 4-regular bipartite graph and let ๐ Bi ๐ = [ โ1 1 1 โ11 โ1 1 โ11 1 โ1 โ11 1 1 1 ] , ๐ Bi ๐ป๐ = [ โ1 1 1 11 โ1 1 11 1 โ1 11 1 1 โ1 ] .Then for any ๐ โ โจ๐ Bi ๐ , ๐ Bi ๐ป๐ โฉ , ๐ (๐บ; ๐, ๐, ๐, ๐) = ๐ (๐บ; ๐ โฒ , ๐ โฒ , ๐ โฒ , ๐ โฒ ) where [ ๐ โฒ ๐ โฒ ๐ โฒ ๐ โฒ ] = ๐ [ ๐๐๐๐ ] .Remark . One can check that โจ๐ Bi ๐ , ๐ Bi ๐ป๐ โฉ is isomorphic to the dihedral group ๐ท and the group elementsare shown in Table 2. Proof.
The proof is similar to that of Theorem 4.2. Instead of combining the holographic maps in Lemma 3.2,Lemma 3.3 with the planar map in Lemma 4.1, we need to combine them with the bipartite map inLemma 5.1. (cid:3)
Corollary 5.3.
Let ๐บ be a 4-regular bipartite graph and let ๐ Bi ๐ and ๐ Bi ๐ป๐ be defined as in Theorem 5.2. Thenfor any ๐ โ โจ๐ Bi ๐ , ๐ Bi ๐ป๐ โฉ , there is an FPRAS for ๐ (๐บ; ๐, ๐, ๐, ๐) if ๐ [ ๐๐๐๐ ] โ ๎ โ ๎ . Thus we know that for any
๐ โ โจ๐ Bi ๐ , ๐ Bi ๐ป๐ โฉ , there is an FPRAS for ๐ (๐บ; ๐, ๐, ๐, ๐) for (๐, ๐, ๐, ๐) in asubregion of ๐ โ1 ( ๎ ) . Note that ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฐ โ ๎ โ ๎ and ๎ฐ โ ๎ . With the help of Table 2, onecan see that Theorem 1.2 is implied by Corollary 5.3.6. Concluding remarksAll the FPRAS results obtained in this paper come from the algorithm for ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ . It isopen if there exists an FPRAS/FPTAS for all (๐, ๐, ๐, ๐) โ ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ . Assuming such an algorithmexists, our maps in Section 4 would imply that all ๎ญ๎ฐ โ ๎ฎ๎ฐ is approximable on planar graphs, and ourmaps in Section 5 would imply that all ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ and ๎ฐ are approximable on bipartite graphs. Wenote that the approximation in ๎ญ , ๎ฎ , and ๎ฏ is proved to be NP-hard even on bipartite graphs [CLLY20].In Section 4, the canonical orientation has the same weight ๐ on every vertex and we are able to obtainalgorithms for the eight-vertex model under parameter settings where ๐ is relatively large, e.g. the region ๎ฏ ; in Section 5, the canonical orientation has the same weight ๐ on every vertex and we are able to obtainalgorithms for parameter settings where ๐ is relatively large, e.g. region ๎ฐ . In general, the paradigmproposed in this paper can be applied to the study of the eight-vertex model on other classes of graphs inadditional to planar/bipartite/torus graphs. In particular, the methodology can be readily extended to anyclass of graphs with a โcanonicalโ even orientation where every vertex has the same weight (one of ๐ , ๐ , ๐ ,or ๐ ). able 2. Elements of โจ๐ Bi ๐ , ๐ Bi ๐ป๐ โฉ and preimages of ๎ = ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ under cor-responding maps. A substantial subregion of each preimage admits FPRAS on bipartite4-regular graphs. The last column lists the approximation complexity of the eight-vertexmodel on general 4-regular graphs in the corresponding preimage regions.Element Matrix Preimage Approximation ๐ผ [ ] ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ FPRAS in ๎ ๐ Bi ๐ 12 [ โ1 1 1 โ11 โ1 1 โ11 1 โ1 โ11 1 1 1 ] ๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฐ ) (๐ Bi ๐ ) [ ] ๐ ( ๎ฐ ) NP-hard (๐ Bi ๐ ) โ๐ผ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ ) FPRAS in
๐ ( ๎ )(๐ Bi ๐ ) โ๐ Bi ๐ ๐ (๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฐ )) (๐ Bi ๐ ) โ (๐ Bi ๐ ) ๎ฐ NP-hard ๐ Bi ๐ป๐ 12 [ โ1 1 1 11 โ1 1 11 1 โ1 11 1 1 โ1 ] ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฐ ๐ Bi ๐ ๐ Bi ๐ป๐ 12 [ ] ๐ (๐ ๐ ( ๎ฐ )) NP-hard (๐ Bi ๐ ) ๐ Bi ๐ป๐ [ โ1 0 0 00 โ1 0 00 0 โ1 00 0 0 1 ] ๐ (๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ )) FPRAS in
๐ (๐ ๐ ( ๎ ))(๐ Bi ๐ ) ๐ Bi ๐ป๐ โ๐ Bi ๐ป๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ โ ๎ฐ ) (๐ Bi ๐ ) ๐ Bi ๐ป๐ โ๐ Bi ๐ ๐ Bi ๐ป๐ ๐ ๐ ( ๎ฐ ) NP-hard (๐ Bi ๐ ) ๐ Bi ๐ป๐ โ (๐ Bi ๐ ) ๐ Bi ๐ป๐ ๐ ๐ ( ๎ญ๎ฐ โ ๎ฎ๎ฐ โ ๎ฏ๎ฐ ) FPRAS in ๐ ๐ ( ๎ ) ppendix A.The readers may have noticed that even though ๐ = [ ] is a complex-valued matrix, under the ๐ -transformation not only the binary Eqality function (= ) is transformed to the binary Diseqalityfunction (โ ) , the arity 4 constraint function ๐ is also transformed to a real-valued constraint function ๐ โ4 ๐ .This is not a coincidence, but a consequence of the fact that ๐ satisfies arrow reversal symmetry.We say a real-valued constraint function ๐ satisfies arrow reversal symmetry if for all (๐ , โฆ , ๐ ๐ ) โ {0, 1} ๐ , ๐ (๐ , โฆ , ๐ ๐ ) = ๐ (๐ , ๐ , โฆ , ๐ ๐ ), where ๐ ๐ = 1 โ ๐ ๐ for all ๐ . Lemma A.1.
A real-valued ๐ of arity ๐ satisfies arrow reversal symmetry, if and only if ๐ โ๐ ๐ is real-valued.Proof. Suppose ๐ satisfies arrow reversal symmetry. Denote by ฬ๐ = ๐ โ๐ ๐ . We have ๐/2 ฬ๐ = [ ] โ๐ ๐ , andthus for all (๐ , โฆ , ๐ ๐ ) โ {0, 1} ๐ , ๐/2 ฬ๐ ๐ โฆ๐ ๐ = โ (๐ ,โฆ,๐ ๐ )โ{0,1} ๐ ๐ ๐ ,โฆ,๐ ๐ โ {(โ1) ๐ ๐ ๐ ๐ ๐ ๐ ๐ } . Hence, taking complex conjugation, ๐/2 ฬ๐ ๐ โฆ๐ ๐ = โ (๐ ,โฆ,๐ ๐ )โ{0,1} ๐ ๐ ๐ โฆ๐ ๐ โ {(โ1) ๐ ๐ ๐ ๐ (โ๐) ๐ ๐ }= โ (๐ ,โฆ,๐ ๐ )โ{0,1} ๐ ๐ ๐ โฆ๐ ๐ โ {(โ1) ๐ ๐ (1โ๐ ๐ ) (โ๐) ๐ ๐ }= 2 ๐/2 ฬ๐ ๐ โฆ๐ ๐ . Now in the opposite direction, suppose ฬ๐ is real. We have ๐ โ1 = [ ] , hence by the inversetransformation ๐/2 ๐ = [ ] โ๐ ฬ๐ , and thus for all (๐ , โฆ , ๐ ๐ ) โ {0, 1} ๐ , ๐/2 ๐ ๐ โฆ๐ ๐ = โ (๐ ,โฆ,๐ ๐ )โ{0,1} ๐ ฬ๐ ๐ ,โฆ,๐ ๐ โ {(โ1) ๐ ๐ ๐ ๐ (โ๐) ๐ ๐ } . So ๐/2 ๐ ๐ โฆ๐ ๐ = โ (๐ ,โฆ,๐ ๐ )โ{0,1} ๐ ฬ๐ ๐ ,โฆ,๐ ๐ โ {(โ1) (1โ๐ ๐ )๐ ๐ (โ๐) ๐ ๐ }= โ (๐ ,โฆ,๐ ๐ )โ{0,1} ๐ ฬ๐ ๐ โฆ๐ ๐ โ {(โ1) ๐ ๐ ๐ ๐ ๐ ๐ ๐ }= 2 ๐/2 ๐ ๐ โฆ๐ ๐ = 2 ๐/2 ๐ ๐ โฆ๐ ๐ . (cid:3) eferences [Bax71] R. J. Baxter. Eight-vertex model in lattice statistics. Phys. Rev. Lett. , 26:832โ833, Apr 1971.[Bax72] R. J. Baxter. Partition function of the eight-vertex lattice model.
Annals of Physics , 70(1):193 โ 228, 1972.[Bax82] R. J. Baxter.
Exactly Solved Models in Statistical Mechanics . Academic Press Inc., San Diego, CA, USA, 1982.[CC17] Jin-Yi Cai and Xi Chen.
Complexity Dichotomies for Counting Problems , volume 1. Cambridge University Press, 2017.[CF17] Jin-Yi Cai and Zhiguo Fu. Complexity classification of the eight-vertex model.
CoRR , abs/1702.07938, 2017.[CGG +
16] Jin-Yi Cai, Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Mark Jerrum, Daniel ล tefankoviฤ, and Eric Vigoda.
Journal ofComputer and System Sciences , 82(5):690โ711, 2016.[CL20] Jin-Yi Cai and Tianyu Liu. Counting perfect matchings and the eight-vertex model. In
Proceedings of the 47thInternational Colloquium on Automata, Languages, and Programming , ICALP โ20, page TBD, 2020.[CLL19] Jin-Yi Cai, Tianyu Liu, and Pinyan Lu. Approximability of the six-vertex model. In
Proceedings of the Thirtieth AnnualACM-SIAM Symposium on Discrete Algorithms , SODA โ19, pages 2248โ2261, 2019.[CLLY20] Jin-Yi Cai, Tianyu Liu, Pinyan Lu, and Jing Yu. Approximability of the eight-vertex model. In
Proceedings of the 35thComputational Complexity Conference , CCC โ20, page TBD, 2020.[DCGH +
16] Hugo Duminil-Copin, Maxime Gagnebin, Matan Harel, Ioan Manolescu, and Vincent Tassion. Discontinuity of thephase transition for the planar random-cluster and potts models with ๐ > 4 . CoRR , abs/1611.09877, 2016.[DGGJ04] Martin Dyer, Leslie Ann Goldberg, Catherine Greenhill, and Mark Jerrum. The relative complexity of approximatecounting problems.
Algorithmica , 38(3):471โ500, Mar 2004.[FR19] Matthew Fahrbach and Dana Randall. Slow mixing of Glauber dynamics for the six-vertex model in the ordered phases.In Dimitris Achlioptas and Lรกszlรณ A. Vรฉgh, editors,
Approximation, Randomization, and Combinatorial Optimization.Algorithms and Techniques (APPROX/RANDOM 2019) , volume 145 of
Leibniz International Proceedings in Informatics(LIPIcs) , pages 37:1โ37:20, Dagstuhl, Germany, 2019. Schloss DagstuhlโLeibniz-Zentrum fuer Informatik.[FW70] Chungpeng Fan and F. Y. Wu. General lattice model of phase transitions.
Phys. Rev. B , 2:723โ733, Aug 1970.[GJ12] Leslie Ann Goldberg and Mark Jerrum. Inapproximability of the Tutte polynomial of a planar graph.
ComputationalComplexity , 21(4):605โ642, Dec 2012.[GJM15] Leslie Ann Goldberg, Mark Jerrum, and Colin McQuillan. Approximating the partition function of planar two-statespin systems.
Journal of Computer and System Sciences , 81(1):330 โ 358, 2015.[GR10] Sam Greenberg and Dana Randall. Slow mixing of Markov chains using fault lines and fat contours.
Algorithmica ,58(4):911โ927, Dec 2010.[GvV15] Andreas Galanis, Daniel ล tefankoviฤ, and Eric Vigoda. Inapproximability for antiferromagnetic spin systems in thetree nonuniqueness region.
J. ACM , 62(6), December 2015.[HVV15] Thomas P. Hayes, Juan C. Vera, and Eric Vigoda. Randomly coloring planar graphs with fewer colors than themaximum degree.
Random Structures & Algorithms , 47(4):731โ759, 2015.[JS93] Mark Jerrum and Alistair Sinclair. Polynomial-time approximation algorithms for the ising model.
SIAM Journal onComputing , 22(5):1087โ1116, 1993.[Kas61] P.W. Kasteleyn. The statistics of dimers on a lattice: I. The number of dimer arrangements on a quadratic lattice.
Physica , 27(12):1209 โ 1225, 1961.[Kas67] P.W. Kasteleyn. Graph theory and crystal physics. In F. Harary, editor,
Graph Theory and Theoretical Physics , pages43โ110. Academic Press, 1967.[KL83] Richard M. Karp and Michael Luby. Monte-Carlo algorithms for enumeration and reliability problems. In
Proceedingsof the 24th Annual Symposium on Foundations of Computer Science , SFCS โ83, pages 56โ64, Washington, DC, USA,1983. IEEE Computer Society.[Kup96] Greg Kuperberg. Another proof of the alternative-sign matrix conjecture.
International Mathematics Research Notices ,1996(3):139โ150, 1996.[Liu18] Tianyu Liu. Torpid mixing of Markov chains for the six-vertex model on โค . In Approximation, Randomization, andCombinatorial Optimization. Algorithms and Techniques , APPROX/RANDOM 2018, pages 52:1โ52:15, 2018.[McQ13] Colin McQuillan. Approximating Holant problems by winding.
CoRR , abs/1301.2880, 2013.[RS19] Gourab Ray and Yinon Spinka. A short proof of the discontinuity of phase transition in the planar random-clustermodel with ๐ > 4 . CoRR , abs/1904.10557, 2019.[SS14] Allan Sly and Nike Sun. Counting in two-spin models on d-regular graphs.
The Annals of Probability , 42(6):2383โ2416,2014.[Sut70] Bill Sutherland. Two dimensional hydrogen bonded crystals without the ice rule.
Journal of Mathematical Physics ,11(11):3183โ3186, 1970.[TF61] H. N. V. Temperley and Michael E. Fisher. Dimer problem in statistical mechanics-an exact result.
The PhilosophicalMagazine: A Journal of Theoretical Experimental and Applied Physics , 6(68):1061โ1063, 1961.[Val08] Leslie G. Valiant. Holographic algorithms.
SIAM J. Comput. , 37(5):1565โ1594, February 2008. Wel69] D.J.A. Welsh. Euler and bipartite matroids.
Journal of Combinatorial Theory , 6(4):375 โ 377, 1969., 6(4):375 โ 377, 1969.