Correlation decay and partition function zeros: Algorithms and phase transitions
AA DETERMINISTIC ALGORITHM FOR COUNTING COLORINGSWITH COLORS
JINGCHENG LIU, ALISTAIR SINCLAIR, AND PIYUSH SRIVASTAVAAbstract. We give a polynomial time deterministic approximation algorithm (an FPTAS) for countingthe number of q -colorings of a graph of maximum degree ∆ , provided only that q ≥ . This substantiallyimproves on previous deterministic algorithms for this problem, the best of which requires q ≥ . , andmatches the natural bound for randomized algorithms obtained by a straightforward application of Markovchain Monte Carlo. In the special case when the graph is also triangle-free, we show that our algorithm appliesunder the condition q ≥ α ∆ + β , where α ≈ . and β = β ( α ) are absolute constants.Our result applies more generally to list colorings, and to the partition function of the anti-ferromagneticPotts model. Our algorithm exploits the so-called “polynomial interpolation” method of Barvinok, identifyinga suitable region of the complex plane in which the Potts model partition function has no zeros. Interestingly,our method for identifying this zero-free region leverages probabilistic and combinatorial ideas that have beenused in the analysis of Markov chains.Some of this work was carried out while the authors were visiting the Simons Institute for the Theory of Computing.Jingcheng Liu, University of California, Berkeley. Email: [email protected] . Research supported by US NSF grant CCF-1815328.Alistair Sinclair, University of California, Berkeley. Email: [email protected] . Research supported by US NSF grantCCF-1815328.Piyush Srivatava, Tata Institute of Fundamental Research. Email: [email protected] . Research supported bya Ramanujan Fellowship of SERB, Indian Department of Science and Technology. a r X i v : . [ c s . CC ] J un . I n t r o d u c t i o n1.1. Background and related work.
Counting colorings of a bounded degree graph is a benchmarkproblem in approximate counting, due both to its importance in combinatorics and statistical physics, aswell as to the fact that it has repeatedly challenged existing algorithmic techniques and stimulated thedevelopment of new ones.Given a finite graph G = ( V, E ) of maximum degree ∆ , and a positive integer q , the goal is to count thenumber of (proper) vertex colorings of G with q colors. It is well known by Brooks’ Theorem [7] that sucha coloring always exists if q ≥ ∆ + 1 . While counting colorings exactly is q ≥ ∆ + 1 .It is known that when q ≤ ∆ , even approximate counting is NP-hard [17].This question has led to numerous algorithmic developments over the past 25 years. The first approachwas via Markov chain Monte Carlo (MCMC), based on the fact that approximate counting can be reduced tosampling a coloring (almost) uniformly at random. Sampling can be achieved by simulating a natural localMarkov chain (or Glauber dynamics) that randomly flips colors on vertices: provided the chain is rapidlymixing, this leads to an efficient algorithm (a fully polynomial randomized approximation scheme , or FPRAS ).Jerrum’s 1995 result [28] that the Glauber dynamics is rapidly mixing for q ≥
2∆ + 1 gave the firstnon-trivial randomized approximation algorithm for colorings and led to a plethora of follow-up work onMCMC (see, e.g., [11, 12, 15, 21–24, 33, 38] and [16] for a survey), focusing on reducing the constant 2 in frontof ∆ . The best constant known for general graphs remains essentially , obtained by Vigoda [38] usinga more sophisticated Markov chain, though this was very recently reduced to − ε for a very small ε by Chen et al. [9]. The constant can be substantially improved if additional restrictions are placed on thegraph: e.g., Dyer et al. [12] achieve roughly q ≥ . provided the girth is at least 6 and the degree isa large enough constant, while Hayes and Vigoda improve this to q ≥ (1 + ε )∆ for girth at least 11 anddegree ∆ = Ω(log n ) , where n is the number of vertices.A significant recent development in approximate counting is the emergence of deterministic approxima-tion algorithms that in some cases match, or even improve upon, the best known MCMC algorithms. Thesealgorithms have made use of one of two main techniques: decay of correlations , which exploits decreasinginfluence of the spins (colors) on distant vertices on the spin at a given vertex; and polynomial interpolation ,which uses the absence of zeros of the partition function in a suitable region of the complex plane. Earlyexamples of the decay of correlations approach include [1, 2, 40], while for early examples of the polynomialinterpolation method, we refer to the monograph of Barvinok [3] (see also, e.g., [4, 13, 25, 27, 30, 34] formore recent examples). Unfortunately, however, in the case of colorings on general bounded degree graphs,these techniques have so far lagged well behind the MCMC algorithms mentioned above. One obstacle togetting correlation decay to work is the lack of a higher-dimensional analog of Weitz’s beautiful algorithmicframework [40], which allows correlation decay to be fully exploited via strong spatial mixing in thecase of spin systems with just two spins (as opposed to the q colors present in coloring). For polynomialinterpolation, the obstacle has been a lack of precise information about the location of the zeros of associatedpartition functions (see below for a definition of the partition function in the context of colorings).So far, the best algorithmic condition for colorings obtained via correlation decay is q ≥ .
58∆ + 1 , dueto Lu and Yin [31], and this remains the best available condition for any deterministic algorithm. Thisimproved on an earlier bound of roughly q ≥ . (proved only for triangle-free graphs), due to Gamarnikand Katz [18]. For the special case ∆ = 3 , Lu et al. [32] give a correlation decay algorithm for counting4-colorings. Furthermore, Gamarnik, Katz and Misra [19] establish the related property of “strong spatialmixing" under the weaker condition q ≥ α ∆ + β for any constant α > α (cid:63) , where α (cid:63) ≈ . is the In this case, the notion of an FPRAS is replaced by that of a fully polynomial time approximation scheme , or
FPTAS . An FPTASfor q -colorings of graphs of maximum degree at most ∆ is an algorithm that given the graph G and an error parameter δ onthe input, produces a (1 ± δ ) -factor multiplicative approximation to the number of q -colorings of G in time poly( | G | , /δ ) (thedegree of the polynomial is allowed to depend upon the constants q and ∆ ). nique solution to xe − /x = 1 and β is a constant depending on α , and under the assumption that G istriangle-free (see also [20, 21] for similar results on restricted classes of graphs). However, as discussedin [19], this strong spatial mixing result unfortunately does not lead to a deterministic algorithm. The newer technique of polynomial interpolation, pioneered by Barvinok [3], has also recently beenbrought to bear on counting colorings. In a recent paper, Bencs et al. [6] use this technique to derive aFPTAS for counting colorings provided q ≥ e ∆ + 1 . This result is of independent interest because it uses adifferent algorithmic approach, and because it establishes a new zero-free region for the associated partitionfunction in the complex plane (see below), but it is weaker than those obtained via correlation decay.In this paper, we push the polynomial interpolation method further and obtain a FPTAS for countingcolorings under the condition q ≥ : Theorem 1.1.
Fix positive integers q and ∆ such that q ≥ . Then there exists a fully polynomial timedeterministic approximation scheme (FPTAS) for counting q -colorings in any graph of maximum degree ∆ . This is the first deterministic algorithm (of any kind) that for all ∆ matches the “natural” bound forMCMC, first obtained by Jerrum [28]. Indeed, q ≥
2∆ + 1 remains the best bound known for rapid mixingof the basic Glauber dynamics that does not require either additional assumptions on the graph or a spectralcomparison with another Markov chain: all the improvements mentioned above require either lower boundson the girth and/or maximum degree, or (in the case of Vigoda’s result [38]) analysis of a more sophisticatedMarkov chain. This is for good reason, since the bound q ≥
2∆ + 1 coincides with the closely relatedDobrushin uniqueness condition from statistical physics [35], which in turn is closely related [39] to thepath coupling method of Bubley and Dyer [8] that provides the simplest currently known proof of the q ≥
2∆ + 1 bound for the Glauber dynamics.We therefore view our result as a promising starting point for deterministic coloring algorithms to finallycompete with their randomized counterparts. In fact, as discussed later in section 1.3, our technique iscapable of directly harnessing strong spatial mixing arguments used in the analysis of Markov chains forcertain classes of graphs. As an example, we can exploit such an argument of Gamarnik, Katz and Misra [19]to improve the bound on q in Theorem 1.1 when the graph is triangle-free, for all but small values of ∆ .(Recall that α (cid:63) ≈ . is the unique positive solution of the equation xe − /x = 1 .) Theorem 1.2.
For every α > α (cid:63) , there exists a β = β ( α ) such that the following is true. For all integers q and ∆ such that q ≥ α ∆ + β , there exists a fully polynomial time deterministic approximation scheme(FPTAS) for counting q -colorings in any triangle-free graph of maximum degree ∆ . We mention also that our technique applies without further effort to the more general setting of list colorings, where each vertex has a list of allowed colors of size q , under the same conditions as above on q .Indeed, our proofs are written to handle this more general situation.In the next subsection we describe our algorithm in more detail.1.2. Our approach.
Let G = ( V, E ) be an n -vertex graph of maximum degree ∆ , and [ q ] := { , . . . , q } aset of colors. Define the polynomial(1) Z G ( w ) := (cid:88) σ : V → [ q ] w |{{ u,v }∈ E : σ ( u )= σ ( v ) }| . Here σ ranges over arbitrary (not necessarily proper) assignments of colors to vertices, and each suchcoloring has a weight w m ( σ ) , where m ( σ ) is the number of monochromatic edges in σ . Note that thenumber of proper q -colorings of G is just Z G (0) .The polynomial Z G ( w ) is the partition function of the Potts model of statistical physics, and implicitlydefines a probability distribution on colorings σ according to their weights in (1). The parameter w measures The strong spatial mixing condition does imply fast mixing of the Glauber dynamics, and hence an FPRAS, but only when thegraph family being considered is “amenable”, i.e., if the size of the (cid:96) -neighborhood of any vertex does not grow exponentially in (cid:96) .This restriction is satisfied by regular lattices, but fails, e.g., for random regular graphs. he strength of nearest-neighbor interactions. The value w = 1 corresponds to the trivial setting wherethere is no constraint on the colors of neighboring vertices, while w = 0 imposes the hard constraint thatno neighboring vertices receive the same color. For intermediate values w ∈ [0 , , neighbors with the samecolor are penalized by a factor of w . Theorems 1.1 and 1.2 are in fact special cases of the following moregeneral theorem. Theorem 1.3.
Suppose that the hypotheses of either Theorem 1.1 or Theorem 1.2 are satisfied, and fix w ∈ [0 , .Then there exists an FPTAS for the partition function Z G ( w ) . Theorem 1.3 of course subsumes Theorems 1.1 and 1.2, but the extension to other values of w is ofindependent interest as the computation of partition functions is a very active area of study in statisticalphysics and combinatorics.To prove Theorem 1.3, we view Z G ( w ) as a polynomial in the complex variable w and identify a regionin the complex plane in which Z G ( w ) is guaranteed to have no zeros. Specifically, we will show that thisholds for the open connected set D ∆ ⊂ C obtained by augmenting the real interval [0 , with a ball ofradius τ ∆ around each point, where τ ∆ is a (small) constant depending only on ∆ . Theorem 1.4.
Fix a positive integer ∆ . Then there exists a τ ∆ > and a region D ∆ of the above formcontaining the interval [0 , such that the following is true. For any graph G of maximum degree ∆ andinteger q satisfying the hypotheses of either Theorem 1.1 or Theorem 1.2, Z G ( w ) (cid:54) = 0 when w ∈ D ∆ . We remark that this theorem is also of independent interest, as the location of zeros of partition functionshas a long and noble history going back to the Lee-Yang theorem of the 1950s [29, 41]. In the case of thePotts model, Sokal [36, 37] proved (in the language of the Tutte polynomial) that the partition functionhas no zeros in the entire unit disk centered at 0, under the strong condition q ≥ . ; the constantwas later improved to 6.907 by Fernández and Procacci [14] (see also [26]). Much more recently, the workof Bencs et al. [6] referred to above gives a zero-free region analogous to that in Theorem 1.4 above, butunder the stronger condition q ≥ e ∆ + 1 . We note also that Barvinok and Soberón [5] (see also [3] for animproved version) established a zero-free region in a disk centered at w = 1 .Theorem 1.4 immediately gives our algorithmic result, Theorem 1.3, by appealing to the recent algorithmicparadigm of Barvinok [3]. The paradigm (see Lemma 2.2.3 of [3]) states that, for a partition function Z ofdegree m , if one can identify a connected, zero-free region D for Z in the complex plane that contains a τ -neighborhood of the interval [0 , , and a point on that interval where the evaluation of Z is easy (inour setting this is the point w = 1 ), then using the first O (cid:0) log (cid:0) me Θ(1 /τ ) /ε (cid:1) (cid:1) coefficients of Z , one canobtain a ± ε multiplicative approximation of Z ( x ) at any point x ∈ D . Barvinok’s framework is based onexploiting the fact that the zero-freeness of Z in D is equivalent to log Z being analytic in D , and thenusing a carefully chosen transformation to deform D into a disk (with the easy point at the center) in orderto perform a convergent Taylor expansion. The coefficients of Z are used to compute the coefficients ofthis Taylor expansion.Barvinok’s framework in general leads to a quasi-polynomial time algorithm as the computation ofthe O (cid:0) log (cid:0) me Θ(1 /τ ) /ε (cid:1) (cid:1) terms of the expansion may take time O (( me Θ(1 /τ ) /ε ) log m ) for the partitionfunctions considered here. However, additional insights provided by Patel and Regts [34] (see, e.g., theproof of Theorem 6.2 in [34]) show how to reduce this computation time to O (( me Θ(1 /τ ) /ε ) log ∆ ) for manymodels on bounded degree graphs of degree at most ∆ , including the Potts model with a bounded numberof colors at each vertex. Hence we obtain an FPTAS. This (by now standard) reduction is the same pathas that followed by Bencs et al. [6, Corollary 1.2]; for completeness, we provide a sketch in Appendix A.We note that for each fixed ∆ and q the running time of our final algorithm is polynomial in n (the sizeof G ) and ε − , as required for an FPTAS. However, as is typical of deterministic algorithms for approximatecounting, the exponent in the polynomial depends on ∆ (through the quantity τ ∆ in Theorem 1.4, which inthe case of bounded list sizes is inverse polynomial in ∆ ). e end this introduction by sketching our approach to proving Theorem 1.4, which is the main contri-bution of the paper.1.3. Technical overview.
The starting point of our proof is a simple geometric observation, versionsof which have been used before for constructing inductive proofs of zero-freeness of partition functions(see, e.g., [3, 6]). Fix a vertex v in the graph G . Given w ∈ C , and a color k ∈ [ q ] , let Z ( k ) v ( w ) denote the restricted partition function in which one only includes those colorings σ in which σ ( v ) = k . Then, since Z G ( w ) = (cid:80) k ∈ [ q ] Z ( k ) v ( w ) , the zero-freeness of Z G will follow if the angles between the complex numbers Z ( k ) v ( w ) , viewed as vectors in R , are all small, and provided that at least one of the Z ( k ) v is non-zero. (Infact, this condition on angles can be relaxed for those Z ( k ) v ( w ) that are sufficiently small in magnitude,and this flexibility is important when w is a complex number close to .) Therefore, one is naturally led toconsider so-called marginal ratios : R ( i,j ) G,v ( w ) := Z ( i ) v ( w ) Z ( j ) v ( w ) . We then require that for any two colors i, j for which Z ( k ) v ( w ) is large enough in magnitude, the ratio R ( i,j ) G,v ( w ) is a complex number with small argument. This is what we prove inductively in sections 4 and 5.The broad contours of our approach as outlined so far are quite similar to some recent work [3, 6].However, it is at the crucial step of how the marginal ratios are analyzed that we depart from these previousresults. Instead of attacking the restricted partition functions or the marginal ratios directly for given w ∈ C , as in these previous works, we crucially exploit the fact that for any ˜ w ∈ [0 , close to the given w , these quantities have natural probabilistic interpretations, and hence can be much better understoodvia probabilistic and combinatorial methods. For instance, when ˜ w ∈ [0 , , the marginal ratio R ( i,j ) G,v ( w ) is in fact a ratio of the marginal probabilities Pr G, ˜ w [ σ ( v ) = i ] and Pr G, ˜ w [ σ ( v ) = j ] , under the naturalprobability distribution on colorings σ . In fact, our analysis cleanly breaks into two separate parts:(1) First, understand the behavior of true marginal probabilities of the form Pr G, ˜ w [ σ ( v ) = i ] for ˜ w ∈ [0 , . This is carried out in section 3.(2) Second, argue that, for complex w ≈ ˜ w , the ratios R ( i,j ) G,v ( w ) remain well-behaved. This is carriedout separately for the two cases when w is close to (in section 4) and when w is bounded awayfrom but still in the vicinity of [0 , (in section 5).A key point in our technical analysis is the notion of “niceness” of vertices, which stipulates that themarginal probability Pr G, ˜ w [ σ ( v ) = i ] should be small enough in terms only of the local neighborhood ofvertex v in G (see Definition 3.1). Note that this condition refers only to real non-negative ˜ w , and henceis amenable to analysis via standard combinatorial tools. Indeed, our proofs that the conditions on q and ∆ in Theorems 1.1 and 1.2 imply this niceness condition are very similar to probabilistic arguments usedby Gamarnik et al. [19] to establish the property of “strong spatial mixing” (in the special case ˜ w = 0 ).We emphasize that this is the only place in our analysis where the lower bounds on q are used. One cantherefore expect that combinatorial and probabilistic ideas used in the analysis of strong spatial mixing andthe Glauber dynamics with smaller number of colors in special classes of graphs can be combined with ouranalysis to obtain deterministic algorithms for those settings, as we have demonstrated in the case of [19].The above ideas are sufficient to understand the real-valued case (part 1 above). For the complex case inpart 2, we start from a recurrence for the marginal ratios R ( i,j ) G,v that is a generalization (to the case w (cid:54) = 0 )of a similar recurrence used by Gamarnik et al. [19] (see Lemma 2.4). The inductive proofs in sections 4and 5 use this recurrence to show that, if ˜ w ∈ [0 , is close to w ∈ C , then all the relevant R ( i,j ) G,v ( w ) remainclose to R ( i,j ) G,v ( ˜ w ) throughout. The actual induction, especially in the case when w is close to , requiresa delicate choice of induction hypotheses (see Lemmas 4.2 and 5.3). The key technical idea is to use the niceness” property of vertices established in part 1 to argue that the two recurrences (real and complex)remain close at every step of the induction. This in turn depends upon a careful application of the meanvalue theorem, separately to the real and imaginary parts (see Lemma 2.5), of a function f κ that arises inthe analysis of the recurrence (see Lemma 2.6).1.4. Comparison with correlation-decay based algorithms.
We conclude this overview with a briefdiscussion of how we are able to obtain a better bound on the number of colors than in correlation decayalgorithms, such as [18, 31] cited earlier. In these algorithms, one first uses recurrences similar to the onementioned above to compute the marginal probabilities, and then appeals to self-reducibility to computethe partition function. Of course, expanding the full tree of computations generated by the recurrence willin general give an exponential time (but exact) algorithm. The core of the analysis of these algorithms is toshow that even if this tree of computations is only expanded to depth about O (log( n/ε )) , and the recurrenceat that point is initialized with arbitrary values, the computation still converges to an ε -approximation of thetrue value. However, the requirement that the analysis be able to deal with arbitrary initializations impliesthat one cannot directly use properties of the actual probability distribution (e.g., the “niceness” propertyalluded to above); indeed, this issue is also pointed out by Gamarnik et al. [19]. In contrast, our analysis doesnot truncate the recurrence, and thus only has to handle initializations that make sense in the context ofthe graph being considered. Moreover, the exponential size of the recursion tree is no longer a barrier since,in contrast to correlation decay algorithms, we are using the tree only as a tool to establish zero-freeness;the algorithm itself follows from Barvinok’s polynomial interpolation paradigm. Our approach suggeststhat this paradigm can be viewed as a method for using (complex-valued generalizations of) strong spatialmixing results to obtain deterministic algorithms.2 . P r e l i m i n a r i e s2.1. Colorings and the Potts model.
Throughout, we assume that the graphs that we consider areaugmented with a list of colors for every vertex. Formally, a graph is a triple G = ( V, E, L ) , where V isthe vertex set, E is the edge set, and L : V → N specifies a list of colors for every vertex. The partitionfunction as defined in the introduction generalizes naturally to this setting: the sum is now over all thosecolorings σ which satisfy σ ( v ) ∈ L ( v ) .We also allow graphs to contain pinned vertices: a vertex v is said to be pinned to a color c if only thosecolorings of G are allowed in which v has color v . Suppose that a vertex v of degree d v in a graph G is pinned to a color c , and consider the graph G (cid:48) obtained by replacing v with d v copies of itself, eachof which is pinned to c and connected to exactly one of the original neighbors of v in G . It is clear that Z G (cid:48) ( w ) = Z G ( w ) for all w . We will therefore assume that all pinned vertices in our graphs G have degreeexactly one. The size of graph, denoted as | G | , is defined to be the number of unpinned vertices. It is worthnoting that the above operation of duplicating pinned vertices does not change the size of the graph.Let G be a graph and v an unpinned vertex in G . A color c in the list of v is said to be good for v if forevery pinned neighbor u of v is pinned to a color different from c . The set of good colors for a vertex v ingraph G is denoted Γ G,v . We sometimes omit the graph G and write Γ v when G is clear from the context.A color c that is not in Γ v is called bad for v . Further, given a graph G with possibly pinned vertices, we saythat the graph is unconflicted if no two neighboring vertices in G are pinned to the same color. Note thatsince all pinned vertices have degree exactly one, each conflicted graph is the direct sum of an unconflictedgraph and a collection of disjoint, conflicted edges.We will assume throughout that all unconflicted graphs G we consider have at least one proper coloring:this will be guaranteed in our applications since we will always have | L ( u ) | ≥ deg G ( u ) + 1 for everyunpinned vertex u in G . Definition 2.1.
For a graph G , a vertex v and a color i ∈ L ( v ) , the restricted partition function Z ( i ) G,v ( w ) isthe partition function restricted to colorings in which the vertex v receives color i . efinition 2.2. Let ω be a formal variable. For any G , a vertex v and colors i, j ∈ L ( v ) , we definethe marginal ratio of color i to color j as R ( i,j ) G,v ( ω ) := Z ( i ) G,v ( ω ) Z ( j ) G,v ( ω ) . Similarly we also define formally thecorresponding pseudo marginal probability as P G,ω [ c ( v ) = i ] := Z ( i ) G,v ( ω ) Z G ( ω ) . Remark 1.
Note that when a numerical value w ∈ C is substituted in place of ω in the above formaldefinition, R ( i,j ) G,v ( w ) is numerically well-defined as long as Z ( j ) G,v ( w ) (cid:54) = 0 , and P G,w [ c ( v ) = i ] is numericallywell-defined as long as Z G ( w ) (cid:54) = 0 . In the proof of the main theorem in sections 4 and 5, we will ensurethat the above definitions are numerically instantiated only in cases where the corresponding conditions forsuch an instantiation to be well-defined, as stated above, are satisfied. For instance, when w ∈ [0 , , this isthe case for the first definition when either (i) w (cid:54) = 0 ; or (ii) w = 0 , but G is unconflicted and j ∈ Γ G,v ;while for the second definition, this is the case when either (i) w (cid:54) = 0 ; or (ii) w = 0 , but G is unconflicted. Remark 2.
Note also that when w ∈ [0 , , the pseudo probabilities, if well-defined, are actual marginalprobabilities. In this case, we will also write P G,w [ c ( v ) = i ] as Pr G,w [ c ( v ) = i ] . For arbitrary complex w ,this interpretation as probabilities is of course not valid (since P G,w [ c ( v ) = i ] can be non-real), but providedthat Z G ( w ) (cid:54) = 0 it is still true that (cid:80) i ∈ L ( v ) P G,w [ c ( v ) = i ] = Z G ( w ) (cid:80) i ∈ L ( v ) Z ( i ) G,v ( w ) = Z G ( w ) Z G ( w ) = 1 . Wealso note that if v is pinned to color k , then P G,w [ c ( v ) = i ] is when k = i and when k (cid:54) = i . Notation . For the case w = 0 we will sometimes shorten the notations P G, [ c ( v ) = i ] and Pr G, [ c ( v ) = i ] to P G [ c ( v ) = i ] and Pr G [ c ( v ) = i ] respectively. Definition 2.3 ( The graphs G ( i,j ) k ) . Given a graph G and a vertex u in G , let v , · · · , v deg G ( u ) be theneighbors of u . We define G ( i,j ) k (the vertex u will be understood from the context) to be the graph obtainedfrom G as follows: • first we replace vertex u with u , · · · , u deg G ( u ) , and connect u to v , u to v , and so on; • next we pin vertices u , · · · , u k − to color i , and vertices u k +1 , · · · , u deg G ( u ) to color j ; • finally we remove the vertex u k .Note that the graph G ( i,j ) k has one fewer unpinned vertex than G .We now derive a recurrence relation between the marginal ratios of the graph G and pseudo marginalprobabilities of the graphs G ( i,j ) k . This is an extension to the Potts model of a similar recurrence relationderived by Gamarnik, Katz and Mishra [19] for the special case of colorings (that is, w = 0 ). Lemma 2.4.
Let ω be a formal variable. For a graph G , a vertex u and colors i, j ∈ L ( u ) , we have R ( i,j ) G,u ( ω ) = (cid:81) deg G ( u ) k =1 (cid:16) − γ · P G ( i,j ) k ,ω [ c ( v k ) = i ] (cid:17)(cid:81) deg G ( u ) k =1 (cid:16) − γ · P G ( i,j ) k ,ω [ c ( v k ) = j ] (cid:17) , where we define γ := 1 − ω . In particular, when a numerical value w ∈ C is substituted in place of ω , the aboverecurrence is valid as long as the quantities Z G ( i,j ) k ( w ) and − γ · P G ( i,j ) k ,w [ c ( v k ) = j ] for ≤ k ≤ deg G ( u ) are all non-zero.Proof. For ≤ k ≤ deg G ( u ) , let H k be the graph obtained from G as follows: • first we replace vertex u with u , · · · , u deg G ( u ) , and connect u to v , u to v , and so on; • we then pin vertices u , · · · , u k to color i , and vertices u k +1 , · · · , u deg G ( u ) to color j . ote that H k is the same as G ( i,j ) k , except that the last step of the construction of G ( i,j ) k is skipped, i.e, thevertex u k is not removed, and, further, u k is pinned to color i . We can now write R ( i,j ) G,u ( ω ) = Z ( i ) G,u ( ω ) Z ( j ) G,u ( ω ) = Z H deg G ( u ) ( ω ) Z H ( ω ) = deg G ( u ) (cid:89) k =1 Z H k ( ω ) Z H k − ( ω ) . Next, for ≤ k ≤ deg G ( u ) , let Y k := Z G ( i,j ) k ( ω ) and Y ( i ) k := Z ( i ) G ( i,j ) k ,v k ( ω ) . We observe that P G ( i,j ) k ,ω [ c ( v k ) = i ] = Y ( i ) k Y k ,Z H k ( ω ) = Y k − (1 − ω ) · Y ( i ) k ,Z H k − ( ω ) = Y k − (1 − ω ) · Y ( j ) k . Therefore we have R ( i,j ) G,u ( ω ) = deg G ( u ) (cid:89) k =1 Y k − (1 − ω ) · Y ( i ) k Y k − (1 − ω ) · Y ( j ) k = (cid:81) deg G ( u ) k =1 (cid:16) − γ · P G ( i,j ) k ,ω [ c ( v k ) = i ] (cid:17)(cid:81) deg G ( u ) k =1 (cid:16) − γ · P G ( i,j ) k ,ω [ c ( v k ) = j ] (cid:17) , where γ = 1 − ω . The claim about the validity of the recurrence on numerical substitution then followsfrom the conditions outlined in Definition 2.2. (cid:3) Complex analysis.
In this subsection we collect some tools and observations from complex analysis.Throughout this paper, we use ι to denote the imaginary unit √− , in order to avoid confusion withthe symbol “ i ” used for other purposes. For a complex number z = a + ιb with a, b ∈ R , we denote itsreal part a as (cid:60) z , its imaginary part b as (cid:61) z , its length √ a + b as | z | , and, when z (cid:54) = 0 , its argument sin − (cid:16) b | z | (cid:17) ∈ ( − π, π ] as arg z . We also generalize the notation [ x, y ] used for closed real intervals to thecase when x, y ∈ C , and use it to denote the closed straight line segment joining x and y .We start with a consequence of the mean value theorem for complex functions, specifically tailored toour application. Let D be any domain in C with the following properties. • For any z ∈ D , (cid:60) z ∈ D . • For any z , z ∈ D , there exists a point z ∈ D such that one of the numbers z − z , z − z haszero real part while the other has zero imaginary part. • If z , z ∈ D are such that either (cid:61) z = (cid:61) z or (cid:60) z = (cid:60) z , then the segment [ z , z ] lies in D .We remark that a rectangular region symmetric about the real axis will satisfy all the above properties. Lemma 2.5 ( Mean value theorem for complex functions ) . Let f be a holomorphic function on D suchthat for z ∈ D , (cid:61) f ( z ) has the same sign as (cid:61) z . Suppose further that there exist positive constants ρ I and ρ R such that • for all z ∈ D , |(cid:61) f (cid:48) ( z ) | ≤ ρ I ; • for all z ∈ D , (cid:60) f (cid:48) ( z ) ∈ [0 , ρ R ] .Then for any z , z ∈ D , there exists C z ,z ∈ [0 , ρ R ] such that |(cid:60) ( f ( z ) − f ( z )) − C z ,z · (cid:60) ( z − z ) | ≤ ρ I · |(cid:61) ( z − z ) | , and |(cid:61) ( f ( z ) − f ( z )) | ≤ ρ R · (cid:40) |(cid:61) ( z − z ) | , when ( (cid:61) z ) · ( (cid:61) z ) ≤ . max {|(cid:61) z | , |(cid:61) z | } otherwise. ue to space considerations, we defer the proof to Appendix B.We will apply the above lemma to the function(2) f κ ( x ) := − ln(1 − κe x ) , which, as we shall see later, will play a central role in our proofs. (We note that here, and also later inthe paper, we use ln to denote the principal branch of the complex logarithm; i.e., if z = re ιθ with r > and θ ∈ ( − π, π ) , then ln z = ln r + ιθ .) Below we verify that such an application is valid, and record theconsequences. Lemma 2.6.
Consider the domain D given by D := { z | (cid:60) z ∈ ( −∞ , − ζ ) and |(cid:61) z | < τ } , where τ < / and ζ are positive real numbers such that τ + e − ζ < . Suppose κ ∈ [0 , and consider thefunction f κ as defined in eq. (2) . Then,(1) The function f κ and the domain D satisfy the hypotheses of Lemma 2.5, if ρ R and ρ I in the statementof the theorem are taken to be e − ζ − e − ζ and τ · e − ζ ( − e − ζ ) , respectively.(2) If ε > and κ (cid:48) are such that | κ (cid:48) − κ | < ε and (1 + ε ) < e ζ , then for any z ∈ D , | f κ (cid:48) ( z ) − f κ ( z ) | ≤ εe ζ − − ε . In particular, we note that the domain D is indeed rectangular and symmetric about the real axis. Due tospace considerations, we defer the proof to Appendix C.We will also need the following simple geometric lemma, versions of which have been used in the workof Barvinok [3] and also Bencs et al. [6]; for completeness we provide the proof in Appendix D. Lemma 2.7.
Let z , z , . . . , z n be complex numbers such that the angle between any two non-zero z i is atmost α ∈ [0 , π/ . Then | (cid:80) ni =1 z i | ≥ cos( α/ (cid:80) ni =1 | z i | . w ∈ [0 , is real (and hence, γ = 1 − w ∈ [0 , ).We remark that in all graphs G appearing in our analysis, we will be able to assume that for any unpinnedvertex u in G , | L ( u ) | ≥ deg G ( u ) + 1 . Thus, Z G ( w ) (cid:54) = 0 whenever either (i) w ∈ (0 , ; or (ii) w = 0 ,but G is unconflicted. As discussed in the previous section, this implies that the marginal ratios and thepseudo marginal probabilities are well-defined, and, further, the latter are actual probabilities. Moreover, if G is not connected, and G (cid:48) is the connected component containing u , then we have R ( i,j ) G,u ( w ) = R ( i,j ) G (cid:48) ,u ( w ) and P G,w [ c ( u ) = i ] = P G (cid:48) ,w [ c ( u ) = i ] . Thus without loss of generality, we will only consider connectedgraphs in this section.We now formally state the conditions on the list sizes under which our main theorem holds. Condition 1 ( Large lists ) . The graph G satisfies at least one of the following two conditions.(1) | L ( v ) | ≥ max { , · deg G ( v ) } for each unpinned vertex v in G .(2) The graph G is triangle-free and further, for each vertex v of G , | L ( v ) | ≥ α · deg G ( v ) + β, where α is any fixed constant larger than the unique positive solution α (cid:63) of the equation xe − x = 1 and β = β ( α ) ≥ α is a constant chosen so that α · e − α (1+ β ) ≥ . We note that α (cid:63) lies in the interval [1 . , . , and β as chosen above is at least / . emark 3. Note that the condition | L ( v ) | ≥ imposed in case 1 above is without loss of generality, sinceany vertex with | L ( v ) | = 1 can be removed from G after removing the unique color in its list from the listsof its neighbors, without changing the number of colorings of G .As stated in the introduction, an important element of our analysis is going to be the fact that underCondition 1, one can show that certain vertices are “nice” in the sense of the following definition. Weemphasize that Condition 1 is ancillary to our main technical development: any condition under which theprobability bounds imposed in the following definition can be proved (as is done in Lemma 3.2 below) willbe sufficient for the analysis. Definition 3.1.
Given a graph G and an unpinned vertex u in G , let d be the number of unpinned neighborsof u . We say the vertex u is nice in G if for any w ∈ [0 , and any color i ∈ L ( u ) , Pr G,w [ c ( u ) = i ] ≤ d +2 . Remark 4.
We adopt the convention that if G is a conflicted graph (so that it has no proper colorings)and w = 0 , then Pr G,w [ c ( u ) = i ] = 0 for every color i and every unpinned vertex u in G . This is justto simplify the presentation in this section by avoiding the need to explicitly exclude this case from thelemmas below. In the proof of our main result in sections 4 and 5, we will never consider conflicted graphsin a situation where w could be , so that this convention will then be rendered moot. Lemma 3.2. If G satisfies Condition 1 then for any vertex u in G , and any unpinned neighbor v k of u , wehave that v k is nice in G ( i,j ) k . We prove this lemma separately for each of the two cases in Condition 1.3.1.
Analysis for case 1 of Condition 1.Lemma 3.3.
Let G be a graph that satisfies case 1 of Condition 1. Then for any unpinned vertex u in G , andany unpinned neighbor v k of u , we have that v k is nice in G ( i,j ) k .Proof. For ease of notation, we denote G ( i,j ) k by H and v k by v . Since G satisfies case 1 of Condition 1, and deg H ( v ) = deg G ( v k ) − (since the neighbor u of v k in G is dropped in the construction of H = G ( i,j ) k ),we have | L H ( v ) | = | L G ( v k ) | ≥ G ( v k ) ≥ · deg H ( v ) + 2 . Consider any valid coloring σ (cid:48) of the neighbors of v in H . For k ∈ L H ( v ) , let n k denote the number ofneighbors of v that are colored k in σ (cid:48) . Then for any w ∈ [0 , and i ∈ L H ( v ) , Pr H,w (cid:2) c ( v ) = i | σ (cid:48) (cid:3) = w n i (cid:80) j ∈ L H ( v ) w n j ≤ | L H ( v ) | − deg H ( v ) , since at most deg H ( v ) of the n j can be positive. Note in particular that if i is not a good color for v in H ,then the probability is . Since this holds for any coloring σ (cid:48) , we have Pr H,w [ c ( v ) = i ] ≤ | L H ( v ) |− deg H ( v ) . Now, let d be the number of unpinned neighbors of v in H . Noting that deg H ( v ) ≥ d , and recalling theobservation above that | L H ( v ) | ≥ H ( v ) + 2 , we thus have Pr G ( i,j ) k ,w [ c ( v k ) = i ] = Pr H,w [ c ( v ) = i ] ≤ | L H ( v ) | − deg H ( v ) ≤ d + 2 . Thus v k is nice in G ( i,j ) k . (cid:3) Here, we say that a coloring σ is valid if the color σ assigns to any vertex v is from L ( v ) , and further, in case w = 0 , no twoneighbors are assigned the same color by σ . .2. Analysis for case 2 of Condition 1.
Notice that if G satisfies case 2 of Condition 1, then so does G ( i,j ) k . Thus in order to show that v k is nice in G ( i,j ) k , it suffices to show the following more general fact. Lemma 3.4.
Let G be any graph that satisfies case 2 of Condition 1, and let u be any unpinned vertex in G ,then u is nice in G . The proof of this lemma is almost identical to arguments that appear in the work of Gamarnik, Katz andMisra [19] on strong spatial mixing; hence we defer the proof to Appendix E.The proof of Lemma 3.2 is immediate from Lemmas 3.3 and 3.4.
Proof of Lemma 3.2. If G satisfies case 1 of Condition 1 then we apply Lemma 3.3. If G satisfies case 2 ofCondition 1 then we apply Lemma 3.4 after noting that if G satisfies case 2 of Condition 1, then so does G ( i,j ) k , and further that, as assumed in the hypothesis of Lemma 3.2, v k is unpinned in G ( i,j ) k . (cid:3) We conclude this section by noting that, the niceness condition can be strengthened in the case when allthe list sizes are uniformly large (e.g., as in the case of q -colorings). Remark 5.
In Condition 1, if we replace the degree of a vertex by the maximum degree ∆ (e.g., in case 1of the condition, if we assume | L ( v ) | ≥ , instead of G ( v ) , for each v ), then for every vertex v inthe graph G , it holds that Pr G,w [ c ( v ) = i ] < min (cid:8) , (cid:9) .To see this, notice that the same calculation as in the proof of Lemma 3.3 above gives Pr G,w [ c ( v ) = i ] ≤ | L ( v ) |− ∆ ≤ α − β ≤ α − < . We will refer to this stronger condition on list sizes (which holds,in particular, when one is considering the case of q -colorings), as the uniformly large list size condition.4 . Z e r o - f r e e r e g i o n f o r s m a l l | w | As explained in the introduction, all our algorithmic results follow from Theorem 1.4, which establishesa zero-free region for the partition function Z G ( w ) around the interval [0 , in the complex plane. Wesplit the proof of Theorem 1.4 into two parts: in this section, we establish the existence of a zero-free diskaround the endpoint w = 0 (see Theorem 4.1): this is the most delicate case because w = 0 corresponds toproper colorings. Then in section 5 (see Theorem 5.1) we derive a zero-free region around the remainder ofthe interval, using a similar but less delicate approach. Taken together, Theorems 4.1 and 5.1 immediatelyimply Theorem 1.4, so this will conclude our analysis. Theorem 4.1.
Fix a positive integer ∆ . There exists a ν w = ν w (∆) such that the following is true. Let G be agraph of maximum degree ∆ satisfying Condition 1, and having no pinned vertices. Then, Z G ( w ) (cid:54) = 0 for any w satisfying | w | ≤ ν w . In the proof, we will encounter several constants which we now fix. Given the degree bound ∆ ≥ , wedefine(3) ε R := 0 . , ε I := ε R · . , and ε w := ε I · . . We will then see that the quantity ν w in the statement of the theorem can be chosen to be . ε w / ∆ . (Infact, we will show that if one has the slightly stronger assumption of uniformly large list sizes consideredin Remark 5, then ν w can be chosen to be ε w / (300∆) ).Throughout the rest of this section, we fix ∆ to be the maximum degree of the graphs, and let ε w , ε I , ε R be as above.We now briefly outline our strategy for the proof. Recall that, for a vertex u and colors i, j , the marginalratio is given by R ( i,j ) G,u ( w ) = Z ( i ) G,u ( w ) Z ( j ) G,u ( w ) . When G is an unconflicted graph, R ( i,j ) G,u (0) is always a well-definednon-negative real number. Intuitively, we would like to show that R ( i,j ) G,u ( w ) ≈ R ( i,j ) G,u (0) , independent ofthe size of G , when w ∈ C is close to . Given such an approximation one can use a simple geometric rgument (see Consequence 4.3) to conclude that the partition function does not vanish for such w . Inorder to prove the above approximate equality inductively for a given graph G , we take an approach thatexploits the properties of the “real” case (i.e., of R ( i,j ) G,u (0) ) and then uses the notion of “niceness” of certainvertices described earlier to control the accumulation of errors. To this end, we will prove the followinglemma via induction on the number of unpinned vertices in G . Theorem 4.1 will follow almost immediatelyfrom the lemma; see the end of this section for the details. Lemma 4.2.
Let G be an unconflicted graph of maximum degree ∆ satisfying Condition 1, and u be anyunpinned vertex in G . Then, the following are true (with ε w , ε I , and ε I as defined in eq. (3) ):(1) For i ∈ Γ u , (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) > .(2) For i, j ∈ Γ u , if u has all neighbors pinned, then R ( i,j ) G,u ( w ) = R ( i,j ) G,u (0) = 1 .(3) For i, j ∈ Γ u , if u has d ≥ unpinned neighbors, then d (cid:12)(cid:12)(cid:12) (cid:60) ln R ( i,j ) G,u ( w ) − (cid:60) ln R ( i,j ) G,u (0) (cid:12)(cid:12)(cid:12) < ε R . (4) For any i, j ∈ Γ u , if u has d ≥ unpinned neighbors, we have d (cid:12)(cid:12)(cid:12) (cid:61) ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) < ε I .(5) For any i (cid:54)∈ Γ u , j ∈ Γ u , then (cid:12)(cid:12)(cid:12) R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ ε w . We will refer to items 1 to 5 as “items of the induction hypothesis”. The rest of this section is devoted tothe proof of this lemma via induction on the number of unpinned vertices in G .We begin by verifying that the induction hypothesis holds in the base case when u is the only unpinnedvertex in an unconflicted graph G . In this case, items 3 and 4 are vacuously true since u has no unpinnedneighbors. Since all neighbors of u in G are pinned, the fact that all pinned vertices have degree at most oneimplies that G can be decomposed into two disjoint components G and G , where G consists of u andits pinned neighbors, while G consists of a disjoint union of unconflicted edges (since G is unconflicted).Now, since G and G are disjoint components, we have Z ( i ) G,u ( w ) = Z G ( w ) = 1 for all i ∈ Γ G,u and all w ∈ C . This proves items 1 and 2. Similarly, when i (cid:54)∈ Γ G,u , we have Z ( i ) G,u ( w ) = w n i , where n i ≥ is thenumber of neighbors of u pinned to color i . This gives (cid:12)(cid:12)(cid:12) R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ | w | n i ≤ ε w , since | w | ≤ ε w ≤ , and proves item 5.We now derive some consequences of the above induction hypothesis that will be helpful in carryingout the induction. Throughout, we assume that G is an unconflicted graph satisfying Condition 1. Consequence 4.3. If | L ( u ) | ≥ deg G ( u ) + 1 then | Z G ( w ) | ≥ . i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,v ( w ) (cid:12)(cid:12)(cid:12) > . Proof.
Note that Z G ( w ) = (cid:80) i ∈ L ( u ) Z ( i ) G,u ( w ) . From item 4, we see that the angle between the complexnumbers Z ( i ) G,u ( w ) and Z ( j ) G,u ( w ) , when i, j ∈ Γ u , is at most dε I . Applying Lemma 2.7 to the terms orresponding to the good colors and item 5 to the terms corresponding to the bad colors, we then have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i ∈ L ( u ) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ (cid:18) | Γ u | cos dε I − | L ( u ) \ Γ u | ε w (cid:19) min i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≥ (cid:18) ( | L ( u ) | − deg G ( u )) cos dε I − | deg G ( u ) | ε w (cid:19) min i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≥ (cid:18) cos dε I − | deg G ( u ) | ε w (cid:19) min i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) , where we use the fact that | L ( u ) \ Γ u | ≤ deg G ( u ) in the second inequality, and | L ( u ) | ≥ deg G ( u ) +1 in the last inequality. Since dε I ≤ . and ε w ≤ . / ∆ , we then have (cid:12)(cid:12)(cid:12)(cid:80) i ∈ L ( u ) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≥ . i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,v ( w ) (cid:12)(cid:12)(cid:12) , which in turn is positive from item 1. (cid:3) Consequence 4.4.
The pseudo-probabilities approximate the real probabilities in the following sense:(1) for any i (cid:54)∈ Γ u , |P G,w [ c ( u ) = i ] | ≤ . ε w .(2) for any j ∈ Γ u , (cid:12)(cid:12)(cid:12)(cid:12) (cid:61) ln P G,w [ c ( u ) = j ] P G [ c ( u ) = j ] (cid:12)(cid:12)(cid:12)(cid:12) = |(cid:61) ln P G,w [ c ( u ) = j ] | ≤ dε I + 2∆ ε w , and (cid:12)(cid:12)(cid:12)(cid:12) (cid:60) ln P G,w [ c ( u ) = j ] P G [ c ( u ) = j ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ dε R + dε I + 2∆ ε w , where d is the number of unpinned neighbors of u in G .Proof. For part (1), by Consequence 4.3 we have |P G,w [ c ( u ) = i ] | = (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) | Z G ( w ) | ≤ (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) . j ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ . ε w , where the last inequality follows from induction hypothesis item 5.For part (2), by items 2 to 4 of the induction hypothesis, there exist complex numbers ξ i (for all i ∈ Γ u )satisfying |(cid:60) ξ i | ≤ dε R and |(cid:61) ξ i | ≤ dε I such that P G,w [ c ( u ) = j ] = (cid:88) i ∈ L ( u ) Z ( i ) G,u ( w ) Z ( j ) G,u ( w )= (cid:88) i ∈ Γ u Z ( i ) G,u (0) Z ( j ) G,u (0) e ξ i (cid:124) (cid:123)(cid:122) (cid:125) := A + (cid:88) i ∈ L ( u ) \ Γ u Z ( i ) G,u ( w ) Z ( j ) G,u ( w ) (cid:124) (cid:123)(cid:122) (cid:125) := B . Next we show that A ≈ P G [ c ( u )= j ] and B is negligible. From item 5 of the induction hypothesis we have(4) P G [ c ( u ) = j ] · | B | ≤ ∆ ε w . Now, note that (cid:80) i ∈ Γ u Z ( i ) G,u (0) Z ( j ) G,u (0) = P G [ c ( u )= j ] . Further, when ε I ≤ . / ∆ , we also have (5) (cid:60) e ξ i ∈ (exp( − dε R ) − d ε I , exp( dε R )) , and | arg e ξ i | ≤ dε I . Here, we also use the elementary facts that if z is a complex number satisfying (cid:60) z = r ∈ [ − , and |(cid:61) z | = θ ≤ . then | arg e z | = |(cid:61) z | = θ , and e r ≥ (cid:60) e z = e r cos θ = exp( r + log cos θ ) ≥ exp( r − θ ) . he above will therefore be true also for any convex combination of the e ξ i . Noting that P G [ c ( u ) = j ] · A isjust such a convex combination (as the coefficients of the e ξ i are non-negative reals summing to ), we have P G [ c ( u ) = j ] · (cid:60) A ∈ (exp( − dε R ) − d ε I , exp( dε R )) , and(6) | arg( P G [ c ( u ) = j ] · A ) | ≤ dε I . (7)Together, eqs. (4), (6) and (7) imply that if C := P G [ c ( u )= j ] P G,w [ c ( u )= j ] then (using the values of ε R , ε I , and ε w ) (cid:60) C ∈ (cid:0) exp( − dε R ) − d ε I − ∆ ε w , exp( dε R ) + ∆ ε w (cid:1) , and arg C ∈ ( − dε I − ε w , dε I + 2∆ ε w ) . Thus, since ε I , ε R are small enough and ε w ≤ .
01 min { ε I , ε R } , we have |(cid:60) ln C | ≤ dε R + dε I + 2∆ ε w , and |(cid:61) ln C | ≤ dε I + 2∆ ε w . (cid:3) In the next consequence, we show that the error contracts during the induction. We first set up somenotation. For a graph G , a vertex u , and a color i ∈ Γ u , we let a ( i ) G,u ( w ) = ln P G,w [ c ( u ) = i ] . We also recallthe definition of the function f γ ( x ) := − ln(1 − γe x ) from eq. (2). Consequence 4.5.
There exists a positive constant η ∈ [0 . , so that the following is true. Let d be thenumber of unpinned neighbors of u . Assume further that u is nice in G . Then, for any colors i, j ∈ Γ u , thereexists a real constant c = c G,u,i ∈ [0 , d + η ] such that (cid:12)(cid:12)(cid:12) (cid:60) f γ ( a ( i ) G,u ( w )) − f ( a ( i ) G,u (0)) − c · (cid:60) (cid:16) a ( i ) G,u ( w ) − a ( i ) G,u (0) (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε I + ε w . (8) (cid:12)(cid:12)(cid:12) (cid:61) f γ ( a ( i ) G,u ( w )) − f γ ( a ( j ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ d + η · ( dε I + 4∆ ε w ) + 2 ε w . (9) (cid:12)(cid:12)(cid:12) (cid:61) f γ ( a ( i ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ d + η · ( dε I + 4∆ ε w ) + ε w . (10) Proof.
Since u is nice in G , the bound P G, [ c ( u ) = k ] ≤ d +2 (for any k ∈ Γ G,u ) applies. Combining themwith Consequence 4.4 we see that a ( i ) G,u ( w ) , a ( i ) G,u (0) , a ( j ) G,u ( w ) , a ( j ) G,u (0) lie in a domain D as described inLemma 2.6 (with the parameter κ therein set to ), with the parameters ζ and τ in that observation chosenas(11) ζ = ln( d + 2) − dε R − dε I − ε w and τ = dε I + 2∆ ε w . Here, for the bound on ζ , we use the fact that for j ∈ Γ G,u , P G [ c ( u ) = j ] ≤ d +2 , which is due to u beingnice in G .The bounds on ε w , ε I and ε R now imply e ζ ≥ ( d + 2) (cid:0) − . (cid:1) ≥ d + 1 . , and also that τ ≤ . / ∆ .Thus, the conditions required on ζ and τ in Lemma 2.6 (i.e. that τ < / and τ + e − ζ < ) are satisfied.Further, ρ R and ρ I as set in the observation satisfy ρ R ≤ d + η , where η can be taken to be . , and ρ I < ε I . Using Lemma 2.5 followed by the value of ε w , we then have(12) (cid:12)(cid:12)(cid:12) (cid:60) f ( a ( i ) G,u ( w )) − f ( a ( i ) G,u (0)) − c · (cid:60) (cid:16) a ( i ) G,u ( w ) − a ( i ) G,u (0) (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε I ( dε I + 2∆ ε w ) ≤ dε I ≤ ε I , for an appropriate positive c ≤ / ( d + η ) . This is almost eq. (8), whose difference will be handled later. Here, for the second inclusion, we use the following elementary computation. Let z, s be complex numbers such that (cid:60) z = r ∈ [0 . , . , | arg z | = θ ≤ . and | s | = q ≤ . . Then, we have (cid:60) ( z + s ) ≥ r − q and |(cid:61) ( z + s ) | ≤ rθ + q . Thus, | arg( z + s ) | ≤ |(cid:61) ( z + s ) ||(cid:60) ( z + s ) | ≤ rθ + qr − q = θ + q · θr − q ≤ θ + 2 q . Here we use the elementary fact that if z is a complex number satisfying (cid:60) z = r ∈ [0 , , . and | arg z | = θ ≤ . then (cid:60) log z = log | z | and (cid:61) log z = arg z . Further, for such θ , we also have log r ≤ (cid:60) log z ≤ log r + log sec θ ≤ log r + θ . imilarly, applying Lemma 2.5 to the imaginary part we have(13) (cid:12)(cid:12)(cid:12) (cid:61) f ( a ( i ) G,u ( w )) − f ( a ( j ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ ρ R · max (cid:110)(cid:12)(cid:12)(cid:12) (cid:61) (cid:16) a ( i ) G,u ( w ) − a ( j ) G,u ( w ) (cid:17)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) (cid:61) a ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) (cid:61) a ( j ) G,u ( w ) (cid:12)(cid:12)(cid:12)(cid:111) , where, as noted above, ρ R ≤ d + η . Now, note that the first term in the above maximum is less than dε I by item 4 of the induction hypothesis, while the other two terms are at most dε I + 2∆ ε w from item 2 ofConsequence 4.4. This is almost the bound in eq. (9), whose difference will be handled later.To prove the bound in eq. (10), we first apply the imaginary part of Lemma 2.5 along with the fact that (cid:61) a ( i ) G,u (0) = 0 to get(14) (cid:12)(cid:12)(cid:12) (cid:61) f ( a ( i ) G,u ( w )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) (cid:61) f ( a ( i ) G,u ( w )) − f ( a ( i ) G,u (0)) (cid:12)(cid:12)(cid:12) ≤ ρ R · (cid:12)(cid:12)(cid:12) (cid:61) (cid:16) a ( i ) G,u ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d + η ( dε I + ∆ ε w ) . Finally, we use item 2 of Lemma 2.6 (with the parameter κ (cid:48) therein set to γ ) to conclude the proofsof eqs. (8) to (10). To this end, we note that γ satisfies | γ − | ≤ ε w , so that the condition (1 + ε w ) < e ζ required for item 2 to apply is satisfied. Thus we see that for any z ∈ D , | f γ ( z ) − f ( z ) | ≤ ε w , so that the quantities |(cid:60) f γ ( a ( i ) G,u ( w )) − (cid:60) f ( a ( i ) G,u ( w )) | , |(cid:61) f γ ( a ( i ) G,u ( w )) − (cid:61) f ( a ( i ) G,u ( w )) | , |(cid:61) f γ ( a ( j ) G,u ( w )) −(cid:61) f ( a ( j ) G,u ( w )) | , and |(cid:61) f γ ( a ( j ) G,u ( w )) − (cid:61) f ( a ( j ) G,u ( w )) | are all at most ε w . The desired bounds of eqs. (8)to (10) now follow from the triangle inequality and the bounds in eqs. (12) to (14). (cid:3) We set up some further notation for the next consequence. For a color i ∈ L ( u ) \ Γ u we let b ( i ) G,u ( w ) = P G,w [ c ( u ) = i ] . We then consider the function g γ ( x ) := − ln(1 − γx ) . Consequence 4.6.
For every color i (cid:54)∈ Γ u , (cid:12)(cid:12)(cid:12) g γ ( b ( i ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ ε w . Proof.
Item 1 of Consequence 4.4 implies that (cid:12)(cid:12)(cid:12) b ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ . ε w . Thus, recalling that | γ − | ≤ ε w , we getthat for all ε w < . , (cid:12)(cid:12)(cid:12) g γ ( b ( i ) G,u ( w )) (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) ln(1 − γb ( i ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ ε w . (cid:3) Inductive proof of Lemma 4.2.
We are now ready to see the induction step in the proof of Lemma 4.2;recall that the base case was already established following the statement of the lemma. Let G be anyunconflicted graph which satisfies Condition 1 and had at least two unpinned vertices (the base case when | G | = 1 was already handled above). We first prove induction item 1 for any vertex u ∈ G . Consider thegraph G (cid:48) obtained from G by pinning vertex u to color i . Note that by the definition of the pinning operation, Z ( i ) G,u ( w ) = Z G (cid:48) ( w ) , and when i ∈ Γ G,u , the graph G (cid:48) is also unconflicted and satisfies Condition 1, andhas one fewer unpinned vertex than G . Thus, from Consequence 4.3 of the induction hypothesis applied to G (cid:48) , we have that (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) = | Z G (cid:48) ( w ) | > .We now consider item 2. When all neighbors of u in G are pinned, the fact that all pinned vertices havedegree at most one implies that G can be decomposed into two disjoint components G and G , where G consists of u and its pinned neighbors, while G is also unconflicted (when G is unconflicted) and has onefewer unpinned vertex than G . Now, since G and G are disjoint components, we have Z ( k ) G,u ( x ) = Z G ( x ) for all k ∈ Γ G,u and all x ∈ C . Further, from Consequence 4.3 of the induction hypothesis applied to G , we also have that Z G ( w ) and Z G (0) are both non-zero. It therefore follows that when i, j ∈ Γ G,u , R ( i,j ) G,u ( w ) = R ( i,j ) G,u (0) = 1 . e now consider items 3 and 4. Recall that by Lemma 2.4, we have(15) R ( i,j ) G,u ( w ) = deg G ( u ) (cid:89) k =1 (cid:16) − γ P G ( i,j ) k ,w [ c ( v k ) = i ] (cid:17)(cid:16) − γ P G ( i,j ) k ,w [ c ( v k ) = j ] (cid:17) . For simplicity we write G k := G ( i,j ) k . Note that when i, j ∈ Γ G,u , and G is unconflicted, so are the G k .Further, each G k has exactly one fewer unpinned vertex than G , so that the induction hypothesis applies toeach G k . Note also that when i, j ∈ Γ G,u , we can restrict the product above to the d unpinned neighborsof u , since for such i, j , the contribution of the factor corresponding to a pinned neighbor is , irrespectiveof the value of w . Without loss of generality, we relabel these unpinned neighbors as v , v , . . . , v d .Now, as before, for s ∈ Γ G k ,v k we define a ( s ) G k ,v k ( w ) := ln P G k ,w [ c ( v k ) = s ] ; while for t ∈ L ( v k ) \ Γ G k ,v k we let b ( t ) G k ,v k ( w ) := P G k ,w [ c ( v k ) = t ] . For a graph G , a vertex u and a color s , we let B G,u ( s ) be the set ofthose neighbors of u for which s is a bad color in G \ { u } . For simplicity we will also write B ( s ) := B G,u ( s ) when it is clear from the context. As before, we have γ = 1 − w , f γ ( x ) = − ln(1 − γe x ) , g γ ( x ) = − ln(1 − γx ) . From the above recurrence, we then have, − ln R ( i,j ) G,u ( w ) = (cid:88) v k ∈ B ( i ) ∩ B ( j ) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) + (cid:88) v k ∈ B ( i ) ∩ B ( j ) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − (cid:88) v k ∈ B ( i ) ∩ B ( j ) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − (cid:88) v k ∈ B ( i ) ∩ B ( j ) g γ (cid:16) b ( j ) G k ,v k ( w ) (cid:17) + (cid:88) v k ∈ B ( i ) ∩ B ( j ) g γ (cid:16) b ( i ) G k ,v k ( w ) (cid:17) + (cid:88) v k ∈ B ( i ) ∩ B ( j ) g γ (cid:16) b ( i ) G k ,v k ( w ) (cid:17) − g γ (cid:16) b ( j ) G k ,v k ( w ) (cid:17) . (16)Note that the same recurrence also applies when w is replaced by (and hence γ by ), except in that casethe last three sums are (as, when i is bad for v k in G k , we have b ( i ) G k ,v k (0) := Pr G k [ c ( v k ) = i ] = 0 ): − ln R ( i,j ) G,u (0) = (cid:88) v k ∈ B ( i ) ∩ B ( j ) f (cid:16) a ( i ) G k ,v k (0) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17) + (cid:88) v k ∈ B ( i ) ∩ B ( j ) f (cid:16) a ( i ) G k ,v k (0) (cid:17) − (cid:88) v k ∈ B ( i ) ∩ B ( j ) f (cid:16) a ( j ) G k ,v k (0) (cid:17) . (17)Further, by Consequence 4.6 of the induction hypothesis applied to the graph G k at a vertex v k ∈ B ( i ) (respectively, v k ∈ B ( j ) ) we see that (cid:12)(cid:12)(cid:12) g γ (cid:16) b ( i ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε w (respectively, g γ (cid:16) b ( j ) G k ,v k ( w ) (cid:17) ≤ ε w ). Thus, pplying the triangle inequality to the real part of the difference of the two recurrences, we get d (cid:12)(cid:12)(cid:12) (cid:60) ln R ( i,j ) G,u (0) − ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ ε w + max (cid:40) max v k ∈ B ( i ) ∩ B ( j ) (cid:110)(cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12)(cid:111) , max v k ∈ B ( i ) ∩ B ( j ) (cid:110)(cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17)(cid:12)(cid:12)(cid:12)(cid:111) , max v k ∈ B ( j ) ∩ B ( i ) (cid:110)(cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17)(cid:12)(cid:12)(cid:12)(cid:111)(cid:41) . (18) Notation . In what follows, we will always denote by d k the number of unpinned neighbors of v k in G k .Before proceeding with the analysis, we note that the graphs G k are unconflicted and satisfy Condition 1,and further that v k is nice in G k (this last fact follows from Lemma 3.2 and the fact that G satisfiesCondition 1). Thus, the preconditions of Consequence 4.5 apply to the vertex v k in graph G k . We nowproceed with the analysis.We first consider v k ∈ B ( i ) ∩ B ( j ) . Note that this implies that i ∈ Γ G k ,v k . Thus, the conditions ofConsequence 4.5 of the induction hypothesis instantiated on G k apply to v k with color i , and we thus havefrom eq. (8) that (cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k + η (cid:12)(cid:12)(cid:12) (cid:60) a ( i ) G k ,v k ( w ) − a ( i ) G k ,v k (0) (cid:12)(cid:12)(cid:12) + ε I + ε w , where d k is the number of unpinned neighbors of v k and η ∈ [0 . , is as in the statement of Conse-quence 4.5. Applying item 2 of Consequence 4.4 (which, again, is applicable because i ∈ Γ G k ,v k ), we thenhave (cid:12)(cid:12)(cid:12) (cid:60) a ( i ) G k ,v k ( w ) − a ( i ) G k ,v k (0) (cid:12)(cid:12)(cid:12) ≤ d k ( ε R + ε I ) + 2∆ ε w , so that(19) (cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε R + 2 ε I + 3∆ ε w . By interchanging the roles of i and j in the above argument, we see that, for v k ∈ B ( j ) ∩ B ( i ) (20) (cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε R + 2 ε I + 3∆ ε w . We now consider v k ∈ B ( i ) ∩ B ( j ) . Note that both i and j are good for v k in G k , so that (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) ≤ max i (cid:48) ,j (cid:48) ∈ Γ Gk,vk (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i (cid:48) ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i (cid:48) ) G k ,v k (0) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j (cid:48) ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j (cid:48) ) G k ,v k (0) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) , Now, for any color s ∈ Γ G k ,v k , Consequence 4.5 of the induction hypothesis instantiated on G k and appliedto v k and s shows that there exists a C s = C s,v k ,G k ∈ [0 , / ( d k + η )] such that (cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( s ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( s ) G k ,v k (0) (cid:17) − C s ( (cid:60) a ( s ) G k ,v k ( w ) − a ( s ) G k ,v k (0)) (cid:12)(cid:12)(cid:12) ≤ ε I + ε w . ubstituting this in the previous display shows that (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) ≤ max i (cid:48) ,j (cid:48) ∈ Γ Gk,vk (cid:12)(cid:12)(cid:12) C i (cid:48) ( (cid:60) a ( i (cid:48) ) G k ,v k ( w ) − a ( i (cid:48) ) G k ,v k (0)) − C j (cid:48) ( (cid:60) a ( j (cid:48) ) G k ,v k ( w ) − a ( j (cid:48) ) G k ,v k (0)) (cid:12)(cid:12)(cid:12) + 2 ε I + 2 ε w = 2 ε I + 2 ε w + max i (cid:48) ,j (cid:48) ∈ Γ Gk,vk (cid:12)(cid:12) C i (cid:48) (cid:60) ξ i (cid:48) − C j (cid:48) (cid:60) ξ j (cid:48) (cid:12)(cid:12) , = 2 ε I + 2 ε w + C s (cid:60) ξ s − C t (cid:60) ξ t , (21)where ξ l := a ( l ) G k ,v k ( w ) − a ( l ) G k ,v k (0) for l ∈ Γ G k ,v k , and s and t are given by s := arg max i (cid:48) ∈ Γ Gk,vk C i (cid:48) (cid:60) ξ i (cid:48) and t := arg min i (cid:48) ∈ Γ Gk,vk C i (cid:48) (cid:60) ξ i (cid:48) . We now have the following two cases:
Case 1: ( (cid:60) ξ s ) · ( (cid:60) ξ t ) ≤ . Recall that C s , C t are non-negative and lie in [0 , / ( d k + η )] . Thus, in this case,we must have (cid:60) ξ s ≥ and (cid:60) ξ t ≤ , so that(22) C s (cid:60) ξ s − C t (cid:60) ξ t = C s (cid:60) ξ s + C t |(cid:60) ξ t | ≤ d k + η ( (cid:60) ξ s + |(cid:60) ξ t | ) = 1 d k + η |(cid:60) ξ s − (cid:60) ξ t | . Now, note that (cid:60) ξ s − (cid:60) ξ t = (cid:60) ln P G k ,w [ c ( v k ) = s ] P G k [ c ( v k ) = s ] − (cid:60) ln P G k ,w [ c ( v k ) = t ] P G k [ c ( v k ) = t ]= (cid:60) ln P G k ,w [ c ( v k ) = s ] P G k ,w [ c ( v k ) = t ] − (cid:60) ln P G k [ c ( v k ) = s ] P G k [ c ( v k ) = t ]= (cid:60) ln R ( s,t ) G k ,v k ( w ) − ln R ( s,t ) G k ,v k (0) . Note that all the logarithms in the above are well defined from Consequence 4.4 of the induction hypothesisapplied to G k and v k (as s, t ∈ Γ G k ,v k ) . Further from items 2 and 3 of the induction hypothesis, the lastterm is at most d k ε R in absolute value. Substituting this in eq. (22), we get(23) C s (cid:60) ξ s − C t (cid:60) ξ t ≤ d k d k + η ε R . This concludes the analysis of Case 1.
Case 2: (cid:60) ξ i (cid:48) for i (cid:48) ∈ Γ G k ,v k all have the same sign. Suppose first that (cid:60) ξ i (cid:48) ≥ for all i (cid:48) ∈ Γ G k ,v k . Then,we have(24) ≤ C s (cid:60) ξ s − C t (cid:60) ξ t ≤ d k + η (cid:60) ξ s ≤ d k d k + η ε R + ε I + 4∆ ε w , where the last inequality follows from item 2 of Consequence 4.5 of the induction hypothesis applied to G k at vertex v k with color s , which states that |(cid:60) ξ s | ≤ d k ( ε R + ε I ) + 4∆ ε w . Similarly, when (cid:60) ξ i (cid:48) ≤ for all i (cid:48) ∈ Γ G k ,v k , we have ≤ C s (cid:60) ξ s − C t (cid:60) ξ t = C t |(cid:60) ξ t | − C s |(cid:60) ξ s |≤ d k + η |(cid:60) ξ t | ≤ d k d k + η ε R + ε I + 4∆ ε w , (25)where the last inequality follows from item 2 of Consequence 4.5 of the induction hypothesis applied to G k at vertex v k with color t , which states that |(cid:60) ξ t | ≤ d k ( ε R + ε I ) + 4∆ ε w . This concludes the analysis ofCase 2. ow, substituting eqs. (23) to (25) into eq. (21), we get(26) (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( i ) G k ,v k (0) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f (cid:16) a ( j ) G k ,v k (0) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε R + 3 ε I + 5∆ ε w . Substituting eqs. (19), (20) and (26) into eq. (18), we get d (cid:12)(cid:12)(cid:12) (cid:60) ln R ( i,j ) G,u ( w ) − ln R ( i,j ) G,u (0) (cid:12)(cid:12)(cid:12) ≤ d k d k + η ε R + 3 ε I + 7∆ ε w < ε R , where the last inequality follows since ηε R > (∆ + 1)(3 ε I + 7∆ ε w ) (recalling that ≤ d k ≤ ∆ and η ∈ [0 . , ). This verifies item 3 of the induction hypothesis.For item 4, we consider the imaginary part of eq. (16). As in the derivation of eq. (18), we use the factthat the induction hypothesis applied to the graph G k at the vertex v k ∈ B ( i ) (respectively, v k ∈ B ( j ) )implies that (cid:12)(cid:12)(cid:12) g γ (cid:16) b ( i ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε w (respectively, g γ (cid:16) b ( j ) G k ,v k ( w ) (cid:17) ≤ ε w ). This yields d (cid:12)(cid:12)(cid:12) (cid:61) ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ ε w + max (cid:40) max v k ∈ B ( i ) ∩ B ( j ) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − (cid:61) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) max v k ∈ B ( i ) ∩ B ( j ) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) , max v k ∈ B ( j ) ∩ B ( i ) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12)(cid:41) . (27)We first consider v k ∈ B ( i ) ∩ B ( j ) . Applying eq. (9) of Consequence 4.5 of the induction hypothesis to thegraph G k at vertex v k with colors i, j ∈ Γ G k ,v k gives(28) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − (cid:61) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε I + 6∆ ε w . Now consider v k ∈ B ( i ) ∩ B ( j ) . For this case, eq. (10) of Consequence 4.5 of the induction hypothesisapplied to G k at vertex v k with color i ∈ Γ G k ,v k gives(29) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε I + 5∆ ε w . Similarly, for v k ∈ B ( j ) ∩ B ( i ) . For this case, eq. (10) of Consequence 4.5 of the induction hypothesisapplied to G k at vertex v k with color j ∈ Γ G k ,v k gives(30) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε I + 5∆ ε w . Substituting eqs. (28) to (30) into eq. (27) we then have d (cid:12)(cid:12)(cid:12) (cid:61) ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ d k d k + η ε I + 8∆ ε w < ε I , where the last inequality holds since ηε I > ε w (recalling that ≤ d k ≤ ∆ and η ∈ [0 . , ).This completes the proof of item item 4 of the induction hypothesis.Finally, we prove item 5. Since i (cid:54)∈ Γ u , there exist n i > neighbors of u that are pinned to color i . Let H be the graph obtained by removing these neighbors of u from G . Then, H is an unconflicted graph withthe same number of unpinned vertices as G which also satisfies i, j ∈ Γ H,u ; we can therefore apply thealready proved items 1 to 3 to H to conclude that(31) (cid:12)(cid:12)(cid:12) R ( i,j ) H ( w ) (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) R ( i,j ) H (0) (cid:12)(cid:12)(cid:12) exp( dε R ) . ow, since i, j ∈ Γ H,u , we can apply the recurrence of Lemma 2.4 in the same way as in the derivation ofeq. (15) above to get(32) R ( i,j ) H,u ( w ) = deg H ( u ) (cid:89) k =1 (cid:16) − P H ( i,j ) k [ c ( v k ) = i ] (cid:17)(cid:16) − P H ( i,j ) k [ c ( v k ) = j ] (cid:17) , where, for the reasons described in the discussion following eq. (15), the product can be restricted tounpinned neighbors of u in H . Renaming these unpinned neighbors as v , v , . . . , v d , we then have(33) ≤ R ( i,j ) H (0) = d (cid:89) k =1 (1 − P H k [ c ( v k ) = i ] )(1 − P H k [ c ( v k ) = j ] ) , where as before, H k := H ( i,j ) k . Now, since G satisfies Condition 1, so does H . Thus, for ≤ k ≤ d , v k isnice in H k (Lemma 3.2), and hence, P H k [ c ( v k ) = j ] ≤ d k +2 for ≤ k ≤ d , where d k ≥ is the number ofunpinned neighbors of v k in H k . We then have ≤ R ( i,j ) H (0) = d (cid:89) k =1 (1 − P H k [ c ( v k ) = i ] )(1 − P H k [ c ( v k ) = j ] ) ≤ d (cid:89) k =1 − d k +2 = d (cid:89) k =1 d k + 2 d k + 1 ≤ ∆ . (As an aside, we note that one could get a better bound under the slightly stronger assumption of uniformlylarge list sizes considered in Remark 5. Under the conditions of that remark, we have P H k [ c ( v k ) = j ] < min (cid:8) , (cid:9) , so that the above upper bound can be improved to R ( i,j ) H (0) ≤ e for ∆ > .)Combining the estimate with eq. (31), we get (cid:12)(cid:12)(cid:12) R ( i,j ) H ( w ) (cid:12)(cid:12)(cid:12) ≤ · ∆ since dε R ≤ / . Now note that since j ∈ Γ G,u , Z ( i ) G,u ( w ) = w n i Z ( i ) H,u ( w ) , and Z ( j ) G,u ( w ) = Z ( j ) H,u ( w ) , so that (cid:12)(cid:12)(cid:12) R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) = | w | n i (cid:12)(cid:12)(cid:12) R ( i,j ) H,u ( w ) (cid:12)(cid:12)(cid:12) ≤ · ∆ · | w | n i . The latter is at most ε w whenever | w | ≤ . ε w / ∆ .This proves item 5, and also completes the inductive proof of Lemma 4.2. (Note also that using the strongerupper bound above under the condition of uniformly large list sizes, we can in fact relax the requirementfurther to | w | ≤ ε w / (300∆) .) (cid:3) We conclude this section by using Lemma 4.2 to prove Theorem 4.1.
Proof of Theorem 4.1.
Let G be a graph satisfying Condition 1. Since G has no pinned vertices, G isunconflicted. Let u be an unpinned vertex in G . By Consequence 4.3 of the induction hypothesis (which weproved in Lemma 4.2), we then have Z w ( G ) (cid:54) = 0 provided ν w ≤ . ε w / ∆ .Furthermore, as discussed above, under a slightly stronger assumption of uniformly large list sizesconsidered in Remark 5, ν w can be chosen to be ε w / (300∆) . (cid:3) (0 , In this section, we consider the case of w close to [0 , but bounded away from . In particular, we provethe following theorem, which complements Theorem 4.1. Theorem 5.1.
Fix a positive integer ∆ and let ν w = ν w (∆) be as in Theorem 4.1. Then, for any w satisfying (34) (cid:60) w ∈ [ ν w / , ν w / and |(cid:61) w | ≤ ν w / , and any graph G satisfying Condition 1, we have Z G ( w ) (cid:54) = 0 . (Here, we recall that as described in the discussion following Theorem 4.1, ν w can be chosen to be ε w / (300∆) when the uniformly large list size condition of Remark 5 is satisfied. However, as in thattheorem, in the case of general list coloring, one chooses ν w = 0 . ε w / ∆ .) or w as in eq. (34), we define ˜ w to be the point on the interval [0 , which is closest to w . Thus ˜ w := (cid:40) (cid:60) w when (cid:60) w ∈ [ ν w / , ; when (cid:60) w ∈ (1 , ν w / .We also define, in analogy with the last section, γ := 1 − w and ˜ γ := 1 − ˜ w . We record a few properties ofthese quantities in the following observation. Observation 5.2.
With w, γ, ˜ w and ˜ γ as above, we have(1) ≤ ˜ γ, | γ | < .(2) | ln w − ln ˜ w | ≤ ν w .Proof. We have ˜ γ ∈ [0 , − ν w / , (cid:60) γ ∈ [ − ν / , − ν w / and |(cid:61) γ | ≤ ν w / . Since ν w ≤ . , thesebounds taken together imply item 1. We also have ≤ ˜ w ≤ | w | ≤ ˜ w + ν w / and ˜ w ≥ ν w / . Thus ≤ (cid:60) (ln w − ln ˜ w ) = ln | w | ˜ w ≤ ln (cid:18) ν w w (cid:19) ≤ ν w . Similarly, (cid:61) (ln w − ln ˜ w ) = (cid:61) ln w = arg w , so that |(cid:61) (ln w − ln ˜ w ) | ≤ | arg w | ≤ |(cid:61) w |(cid:60) w ≤ ν w . Together, the above two bounds imply item 2. (cid:3)
In analogous fashion to the proof of Theorem 4.1, we would like to show that R ( i,j ) G,u ( w ) ≈ R ( i,j ) G,u ( ˜ w ) independent of the size of G . (Note that for positive ˜ w , R ( i,j ) G,u ( ˜ w ) is a well defined positive real numberfor any graph.) To this end, we will prove the following analog of Lemma 4.2 for any graph G satisfyingCondition 1 and any vertex u in G , via an induction on the number of unpinned vertices in G . The inductionis very similar in structure to that used in the proof of Lemma 4.2, except that the fact that w has strictlypositive real part allows us to simplify several aspects of the proof. In particular, we do not need to considergood and bad colors separately, and do not require the underlying graphs to be unconflicted.As in the previous section, we assume that all graphs in this section have maximum degree at most ∆ ≥ , and define the quantities ε w , ε R , ε I in terms of ∆ using eq. (3). Lemma 5.3.
Let G be a graph of maximum degree ∆ satisfying Condition 1 and let u be any unpinned vertexin G . Then, the following are true (here, ε w , ε I , ε R are as defined in eq. (3) ):(1) For i ∈ L ( u ) , (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) > .(2) For i, j ∈ L ( u ) , if u has all neighbors pinned, then | ln R ( i,j ) G,u ( w ) − ln R ( i,j ) G,u ( ˜ w ) | < ε w .(3) For i, j ∈ L ( u ) , if u has d ≥ unpinned neighbors, then d (cid:12)(cid:12)(cid:12) (cid:60) ln R ( i,j ) G,u ( w ) − (cid:60) ln R ( i,j ) G,u ( ˜ w ) (cid:12)(cid:12)(cid:12) < ε R . (4) For any i, j ∈ L ( u ) , if u has d ≥ unpinned neighbors, then d (cid:12)(cid:12)(cid:12) (cid:61) ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) < ε I . We will refer to items 1 to 4 as “items of the induction hypothesis”. The rest of this section is devoted tothe proof of this lemma via an induction on the number of unpinned vertices in G .We begin by verifying that the induction hypothesis holds in the base case when u is the only unpinnedvertex in a graph G . In this case, items 3 and 4 are vacuously true since u has no unpinned neighbors. Sinceall neighbors of u in G are pinned, the fact that all pinned vertices have degree at most one implies that G can be decomposed into two disjoint components G and G , where G consists of u and its pinnedneighbors, while G consists of a disjoint union of edges with pinned end-points. Let m be the number ofconflicted edges on G , and let n k denote the number of neighbors of u pinned to color k . We then have ( k ) G,u ( x ) = x n k Z G ( x ) = x n k + m for all x ∈ C . This already proves item 1 since w, ˜ w (cid:54) = 0 . Item 2 followsvia the following computation (which uses item 2 of Observation 5.2): | ln R ( i,j ) G,u ( w ) − ln R ( i,j ) G,u ( ˜ w ) | = | n i − n j | · | ln w − ln ˜ w | ≤ ∆ ν w < ε w . We now derive some consequences of the above induction hypothesis that will be helpful in carrying outthe induction.
Consequence 5.4. If | L ( u ) | ≥ , then | Z G ( w ) | > . Proof.
Note that Z G ( w ) = (cid:80) i ∈ L ( u ) Z ( i ) G,u ( w ) . From item 4, we see that the angle between the complexnumbers Z ( i ) G,u ( w ) and Z ( j ) G,u ( w ) , for all i, j ∈ L ( u ) , is at most dε I . Applying Lemma 2.7 we then have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) i ∈ L ( u ) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ | L ( u ) | cos dε I · min i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≥ . i ∈ Γ u (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) , when | L ( u ) | ≥ and dε I ≤ . . This last quantity is positive from item 1. (cid:3) Consequence 5.5.
For all ε R , ε I , ε w small enough such that ε I ≤ ε R and ε w ≤ . ε I , the pseudo-probabilities approximate the real probabilities in the following sense: for any j ∈ L ( u ) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:61) ln P G,w [ c ( u ) = j ] P G, ˜ w [ c ( u ) = j ] (cid:12)(cid:12)(cid:12)(cid:12) = |(cid:61) ln P G,w [ c ( u ) = j ] | ≤ dε I + 2∆ ε w , and (cid:12)(cid:12)(cid:12)(cid:12) (cid:60) ln P G,w [ c ( u ) = j ] P G, ˜ w [ c ( u ) = j ] (cid:12)(cid:12)(cid:12)(cid:12) ≤ dε R + dε I + 2∆ ε w , where d is the number of unpinned neighbors of u in G .Proof. Using items 2 to 4 of the induction hypothesis, there exist complex numbers ξ i (for all i ∈ Γ u )satisfying |(cid:60) ξ i | ≤ dε R + ε w and |(cid:61) ξ i | ≤ dε I + ε w such that(35) P G, ˜ w [ c ( u ) = j ] P G,w [ c ( u ) = j ] = P G, ˜ w [ c ( u ) = j ] (cid:88) i ∈ L ( u ) Z ( i ) G,u ( w ) Z ( j ) G,u ( w ) = P G, ˜ w [ c ( u ) = j ] (cid:88) i ∈ L ( u ) Z ( i ) G,u ( ˜ w ) Z ( j ) G,u ( ˜ w ) e ξ i Now, note that (cid:80) i ∈ L ( u ) Z ( i ) G,u ( ˜ w ) Z ( j ) G,u ( ˜ w ) = P G, ˜ w [ c ( u )= j ] , so that the sum above is a convex combination of the exp( ξ i ) . From the bounds on the real and imaginary parts of the ξ i quoted above, we also have (when ε I , ε w ≤ . / ∆ ) (36) (cid:60) e ξ i ∈ (exp( − dε R − ε w ) − ( dε I + ε w ) , exp( dε R + ε w )) , and | arg e ξ i | ≤ dε I + ε w . The above will therefore be true also for any convex combination of the e ξ i , in particular the one in eq. (35).We therefore have for C := P G, ˜ w [ c ( u )= j ] P G,w [ c ( u )= j ] (cid:60) C ∈ (exp( − dε R − ε w ) − ( dε I + ε w ) , exp( dε R + ε w )) , and(37) | arg C | ≤ dε I + ε w . (38)Now recall that for | θ | ≤ π/ , we have − θ ≤ ln cos θ ≤ − θ / . Thus, using the values of ε w , ε I and ε R ,we have |(cid:60) ln C | ≤ dε R + dε I + 2∆ ε w , and |(cid:61) ln C | ≤ dε I + ε w . (cid:3) Here, we also use the elementary facts that if z is a complex number satisfying (cid:60) z = r ∈ [ − , and |(cid:61) z | = θ ≤ . then | arg e z | = |(cid:61) z | = θ , and e r ≥ (cid:60) e z = e r cos θ = exp( r + log cos θ ) ≥ exp( r − θ ) . s before we define a ( i ) G,u ( w ) = ln P G,w [ c ( u ) = i ] , and recall the definition of the function f γ ( x ) := − ln(1 − γe x ) . Consequence 5.6.
There exists a positive constant η ∈ [0 . , so that the following is true. Let d be thenumber of unpinned neighbors of u . Assume further that the vertex u is nice in G . Then, for any colors i, j ∈ L ( u ) , there exist a real constant c = c G,u,i ∈ [0 , d + η ] such that (cid:12)(cid:12)(cid:12) (cid:60) f γ ( a ( i ) G,u ( w )) − f ˜ γ ( a ( i ) G,u ( ˜ w )) − c · (cid:60) (cid:16) a ( i ) G,u ( w ) − a ( i ) G,u ( ˜ w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε I + ε w . (39) (cid:12)(cid:12)(cid:12) (cid:61) f γ ( a ( i ) G,u ( w )) − f γ ( a ( j ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ d + η · ( dε I + 4∆ ε w ) + 2 ε w . (40) Proof.
Since u is nice in G , the bound P G, ˜ w [ c ( u ) = k ] ≤ d +2 (for any k ∈ L ( u ) ) applies. Combining themwith Consequence 5.5 we see that a ( i ) G,u ( w ) , a ( i ) G,u ( ˜ w ) , a ( j ) G,u ( w ) , a ( j ) G,u ( ˜ w ) lie in a domain D as described inLemma 2.6, with the parameters ζ and τ in that lemma chosen as(41) ζ = ln( d + 2) − dε R − dε I − ε w and τ = dε I + 2∆ ε w . Here, for the bound on ζ , we use the fact that for k ∈ L ( u ) , P G, ˜ w [ c ( u ) = k ] ≤ d +2 , since u is nice in G .As in the proof of Consequence 4.5, we use the values of ε w , ε I , ε R to verify that the condition τ < / and τ + e − ζ < are satisfied, so that item 1 of Lemma 2.6 applies (with the parameter κ therein set to ˜ γ )and further that ρ R and ρ I as set there satisfy ρ R ≤ d + η and ρ I < ε I , with η = 0 . . Using Lemma 2.5followed by the bound on ε w , we then have(42) (cid:12)(cid:12)(cid:12) (cid:60) f ˜ γ ( a ( i ) G,u ( w )) − f ˜ γ ( a ( i ) G,u ( ˜ w )) − c · (cid:60) (cid:16) a ( i ) G,u ( w ) − a ( i ) G,u ( ˜ w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ ε I ( dε I + 2∆ ε w ) ≤ dε I ≤ ε I , for an appropriate positive c ≤ / ( d + η ) . This is almost eq. (39), whose difference will be handled later.Similarly, applying Lemma 2.5 to the imaginary part we have(43) (cid:12)(cid:12)(cid:12) (cid:61) f ˜ γ ( a ( i ) G,u ( w )) − f ˜ γ ( a ( j ) G,u ( w )) (cid:12)(cid:12)(cid:12) ≤ ρ R · max (cid:110)(cid:12)(cid:12)(cid:12) (cid:61) (cid:16) a ( i ) G,u ( w ) − a ( j ) G,u ( w ) (cid:17)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) (cid:61) a ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12) (cid:61) a ( j ) G,u ( w ) (cid:12)(cid:12)(cid:12)(cid:111) , where, as noted above, ρ R ≤ d + η . Now, note that the first term in the above maximum is less than dε I + ε w by items 2 and 4 of the induction hypothesis, while the other two are at most dε I + 2∆ ε w from item 2 ofConsequence 5.5.Finally, we use item 2 of Lemma 2.6 with the parameter κ (cid:48) therein set to γ . To this end, we note that | γ − ˜ γ | ≤ ε w , and that with the fixed values of ε w , ε R , and ε I , the condition (1 + ε w ) < e ζ is satisfied, sothat the item applies. Using the item, we then see that for any z ∈ D , | f γ ( z ) − f ˜ γ ( z ) | ≤ ε w . Thus, the quantities |(cid:60) f γ ( a ( i ) G,u ( w )) − (cid:60) f ˜ γ ( a ( i ) G,u ( w )) | , |(cid:61) f γ ( a ( i ) G,u ( w )) − (cid:61) f ˜ γ ( a ( i ) G,u ( w )) | , |(cid:61) f γ ( a ( j ) G,u ( w )) −(cid:61) f ˜ γ ( a ( j ) G,u ( w )) | , and |(cid:61) f γ ( a ( j ) G,u ( w )) − (cid:61) f ˜ γ ( a ( j ) G,u ( w )) | are all at most ε w . The desired bounds now followfrom the triangle inequality and the bounds in eqs. (42) and (43). (cid:3) Inductive proof of Lemma 5.3.
We are now ready to see the inductive proof of Lemma 5.3; recall thatthe base case was already established following the statement of the lemma. Let G be any graph whichsatisfies Condition 1 and had at least two unpinned vertices (the base case when | G | = 1 was alreadyhandled above). We first prove induction item 1 for any vertex u in G . Consider the graph G (cid:48) obtained from G by pinning vertex u to color i . Note that by the definition of the pinning operation, Z iG,u ( w ) = Z G (cid:48) ( w ) . urther, the graph G (cid:48) also satisfies Condition 1, and has one fewer unpinned vertex than G . Thus, fromConsequence 5.4 of the induction hypothesis applied to G (cid:48) , we have that (cid:12)(cid:12)(cid:12) Z ( i ) G,u ( w ) (cid:12)(cid:12)(cid:12) = | Z G (cid:48) ( w ) | > .We now consider item 2. When all neighbors of u in G are pinned, the fact that all pinned vertices havedegree at most one implies that G can be decomposed into two disjoint components G and G , where G consists of u and its pinned neighbors, while G has one fewer unpinned vertex than G . Let n k bethe number of neighbors of u pinned to color k . Now, since G and G are disjoint components, we have Z ( k ) G,u ( x ) = x n k Z G ( x ) for all k ∈ L ( u ) and all x ∈ C . Further, from Consequence 5.4 of the inductionhypothesis applied to G , we also have that Z G ( w ) and Z G ( ˜ w ) are both non-zero. It therefore followsthat | ln R ( i,j ) G,u ( w ) − ln R ( i,j ) G,u ( ˜ w ) | = | n i − n j | · | ln w − ln ˜ w | ≤ ∆ ν w < ε w . We now consider items 3 and 4. Recall that by Lemma 2.4, we have R ( i,j ) G,u ( w ) = deg G ( u ) (cid:89) k =1 (cid:16) − γ P G ( i,j ) k ,w [ c ( v k ) = i ] (cid:17)(cid:16) − γ P G ( i,j ) k ,w [ c ( v k ) = j ] (cid:17) . As before, for simplicity we write G k := G ( i,j ) k . Note that each G k has exactly one fewer unpinned vertexthan G , so that the induction hypothesis applies to each G k . Without loss of generality, we relabel theunpinned neighbors of u as v , v , . . . , v d . Let n k be the number of neighbors of u pinned to color k .Recalling that − γ = w , we can then simplify the above recurrence to R ( i,j ) G,u ( w ) = w n i − n j d (cid:89) k =1 (cid:16) − γ P G ( i,j ) k ,w [ c ( v k ) = i ] (cid:17)(cid:16) − γ P G ( i,j ) k ,w [ c ( v k ) = j ] (cid:17) . Now, as before, for s ∈ L ( v k ) we define a ( s ) G k ,v k ( w ) := ln P G k ,w [ c ( v k ) = s ] . From the above recurrence, wethen have, − ln R ( i,j ) G,u ( w ) = ( n i − n j ) ln w + d (cid:88) k =1 f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) . (44)Note that the same recurrence also applies when w is replaced by ˜ w (and hence γ by ˜ γ ): − ln R ( i,j ) G,u ( ˜ w ) = ( n i − n j ) ln ˜ w + d (cid:88) k =1 f ˜ γ (cid:16) a ( i ) G k ,v k ( ˜ w ) (cid:17) − f ˜ γ (cid:16) a ( j ) G k ,v k ( ˜ w ) (cid:17) . (45)(Recall that since (cid:60) w, ˜ w > , ln w and ln ˜ w are well defined).Using item 2 of Observation 5.2, | n i − n j | ≤ ∆ , and the fact that ∆ ν w ≤ ε w , we have | n i − n j | | ln w − ln ˜ w | ≤ ε w . Applying the triangle inequality to the real part of the difference of the two recurrences, we therefore get(46) d (cid:12)(cid:12)(cid:12) (cid:60) ln R ( i,j ) G,u ( w ) − ln R ( i,j ) G,u ( ˜ w ) (cid:12)(cid:12)(cid:12) ≤ ε w + max ≤ i ≤ d (cid:110)(cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( i ) G k ,v k ( ˜ w ) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( j ) G k ,v k ( ˜ w ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) (cid:111) . otation . In what follows, we will always denote by d k the number of unpinned neighbors of v k in G k .Before proceeding with the analysis, we note that the graphs G k satisfy Condition 1, and further that v k is nice in G k (the latter fact follows from Lemma 3.2 and the fact that G has Condition 1). Thus, thepreconditions of Consequence 5.6 applies to the vertex v k in graph G k . We now proceed with the analysis.We begin by noting that (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( i ) G k ,v k ( ˜ w ) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( j ) G k ,v k ( ˜ w ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) ≤ max i (cid:48) ,j (cid:48) ∈ L ( v k ) (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i (cid:48) ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( i (cid:48) ) G k ,v k ( ˜ w ) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j (cid:48) ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( j (cid:48) ) G k ,v k ( ˜ w ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) . On the other hand, for any color s ∈ L ( v k ) , Consequence 5.6 of the induction hypothesis instantiated on G k and applied to v k and s shows that there exists a C s = C s,v k ,G k ∈ [0 , / ( d k + η )] such that (cid:12)(cid:12)(cid:12) (cid:60) f γ (cid:16) a ( s ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( s ) G k ,v k ( ˜ w ) (cid:17) − C s ( (cid:60) a ( s ) G k ,v k ( w ) − a ( s ) G k ,v k ( ˜ w )) (cid:12)(cid:12)(cid:12) ≤ ε I + ε w . Substituting this in the previous display shows that (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( i ) G k ,v k ( ˜ w ) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( j ) G k ,v k ( ˜ w ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) ≤ max i (cid:48) ,j (cid:48) ∈ L ( v k ) (cid:12)(cid:12)(cid:12) C i (cid:48) ( (cid:60) a ( i (cid:48) ) G k ,v k ( w ) − a ( i (cid:48) ) G k ,v k ( ˜ w )) − C j (cid:48) ( (cid:60) a ( j (cid:48) ) G k ,v k ( w ) − a ( j (cid:48) ) G k ,v k ( ˜ w )) (cid:12)(cid:12)(cid:12) + 2 ε I + 2 ε w = 2 ε I + 2 ε w + max i (cid:48) ,j (cid:48) ∈ L ( v k ) (cid:12)(cid:12) C i (cid:48) (cid:60) ξ i (cid:48) − C j (cid:48) (cid:60) ξ j (cid:48) (cid:12)(cid:12) , = 2 ε I + 2 ε w + C s (cid:60) ξ s − C t (cid:60) ξ t , (47)where ξ l := a ( l ) G k ,v k ( w ) − a ( l ) G k ,v k ( ˜ w ) for l ∈ Γ G k ,v k , and s and t are given by s := arg max i (cid:48) ∈ L ( v k ) C i (cid:48) (cid:60) ξ i (cid:48) and t := arg min i (cid:48) ∈ L ( v k ) C i (cid:48) (cid:60) ξ i (cid:48) . We now have the following two cases:
Case 1: ( (cid:60) ξ s ) · ( (cid:60) ξ t ) ≤ . Recall that C s , C t are non-negative and lie in [0 , / ( d k + η )] . Thus, in this case,we must have (cid:60) ξ s ≥ and (cid:60) ξ t ≤ , so that(48) C s (cid:60) ξ s − C t (cid:60) ξ t = C s (cid:60) ξ s + C t |(cid:60) ξ t | ≤ d k + η ( (cid:60) ξ s + |(cid:60) ξ t | ) = 1 d k + η |(cid:60) ξ s − (cid:60) ξ t | . Now, note that (cid:60) ξ s − (cid:60) ξ t = (cid:60) ln P G k ,w [ c ( v k ) = s ] P G k , ˜ w [ c ( v k ) = s ] − (cid:60) ln P G k ,w [ c ( v k ) = t ] P G k , ˜ w [ c ( v k ) = t ]= (cid:60) ln P G k ,w [ c ( v k ) = s ] P G k ,w [ c ( v k ) = t ] − (cid:60) ln P G k , ˜ w [ c ( v k ) = s ] P G k , ˜ w [ c ( v k ) = t ]= (cid:60) ln R ( s,t ) G k ,v k ( w ) − ln R ( s,t ) G k ,v k ( ˜ w ) . Note that all the logarithms in the above are well defined from Consequence 5.5 of the induction hypothesisapplied to G k and v k . Further, from items 2 and 3 of the induction hypothesis, the last term is at most d k ε R + ε w in absolute value. Substituting this in eq. (48), we get(49) C s (cid:60) ξ s − C t (cid:60) ξ t ≤ d k d k + η ε R + ε w . This conclude the analysis of Case 1. ase 2: (cid:60) ξ i (cid:48) for i (cid:48) ∈ L ( v k ) all have the same sign. Suppose first that (cid:60) ξ i (cid:48) ≥ for all i (cid:48) ∈ L ( v k ) . Then,we have(50) ≤ C s (cid:60) ξ s − C t (cid:60) ξ t ≤ d k + η (cid:60) ξ s ≤ d k d k + η ε R + ε I + 4∆ ε w , where the last inequality follows from item 2 of Consequence 5.6 of the induction hypothesis applied to G k at vertex v k with color s , which states that |(cid:60) ξ s | ≤ d k ( ε R + ε I ) + 4∆ ε w . Similarly, when (cid:60) ξ i (cid:48) ≤ for all i (cid:48) ∈ Γ G k ,v k , we have ≤ C s (cid:60) ξ s − C t (cid:60) ξ t = C t |(cid:60) ξ t | − C s |(cid:60) ξ s |≤ d k + η |(cid:60) ξ t | ≤ d k d k + η ε R + ε I + 4∆ ε w , (51)where the last inequality follows from item 2 of Consequence 5.6 of the induction hypothesis applied to G k at vertex v k with color t , which states that |(cid:60) ξ t | ≤ d k ( ε R + ε I ) + 4∆ ε w . This concludes the analysis ofCase 2.Now, substituting eqs. (49) to (51) into eq. (47), we get(52) (cid:12)(cid:12)(cid:12)(cid:16) (cid:60) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( i ) G k ,v k ( ˜ w ) (cid:17) (cid:17) − (cid:16) (cid:60) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17) − f ˜ γ (cid:16) a ( j ) G k ,v k ( ˜ w ) (cid:17) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε R + 3 ε I + 5∆ ε w . Substituting eq. (52) into eq. (46), we get d (cid:12)(cid:12)(cid:12) (cid:60) ln R ( i,j ) G,u ( w ) − ln R ( i,j ) G,u ( ˜ w ) (cid:12)(cid:12)(cid:12) ≤ d k d k + η ε R + 3 ε I + 7∆ ε w < ε R , where the last inequality holds since ηε R > (∆ + 1)(3 ε I + 7∆ ε w ) (recalling that ≤ d k ≤ ∆ and η = ∈ [0 . , ). This verifies item 3 of the induction hypothesis.Finally, for proving item 4, we consider the imaginary part of eq. (44). We first note that | n i − n j | |(cid:61) ln w | ≤ ∆ | ln w − ln ˜ w | ≤ ∆ ν w ≤ ε w . We then have d (cid:12)(cid:12)(cid:12) (cid:61) ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ ε w + max ≤ k ≤ d (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − (cid:61) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) . (53)Applying eq. (40) of Consequence 5.6 of the induction hypothesis to the graph G k at vertex v k with colors i, j ∈ L ( v k ) gives(54) (cid:12)(cid:12)(cid:12) (cid:61) f γ (cid:16) a ( i ) G k ,v k ( w ) (cid:17) − (cid:61) f γ (cid:16) a ( j ) G k ,v k ( w ) (cid:17)(cid:12)(cid:12)(cid:12) ≤ d k d k + η ε I + 6∆ ε w . Substituting eq. (54) into eq. (53) we then have d (cid:12)(cid:12)(cid:12) (cid:61) ln R ( i,j ) G,u ( w ) (cid:12)(cid:12)(cid:12) ≤ d k d k + η ε I + 8∆ ε w < ε I , where the last inequality holds since ηε I > ε w (recalling that ≤ d k ≤ ∆ and η ∈ [0 . , ).This proves item 4, and also completes the inductive proof of Lemma 5.3. (cid:3) We now use Lemma 5.3 to prove Theorem 5.1.
Proof of Theorem 5.1.
Let G be any graph of maximum degree ∆ satisfying Condition 1. If G has nounpinned vertices, then Z G ( w ) = 1 and there is nothing to prove. Otherwise, let u be an unpinned vertexin G . By Consequence 5.4 of the induction hypothesis (which we proved in Lemma 5.3), we then have Z w ( G ) (cid:54) = 0 for w as in the statement of the theorem. (cid:3) The proof of Theorem 1.4 is now immediate. roof of Theorem 1.4. Let the quantity ν w = ν w (∆) be as in the statements of Theorems 4.1 and 5.1. Fix themaximum degree ∆ , and suppose that w satisfies(55) − ν w / ≤ (cid:60) w ≤ ν w / and |(cid:61) w | ≤ ν w / . Let G be a graph of maximum degree ∆ satisfying Condition 1. When w satisfying eq. (55) is such that (cid:60) w ≤ ν w / , we have | w | ≤ ν w , so that Z G ( w ) (cid:54) = 0 by Theorem 4.1, while when such a w satisfies (cid:60) w ≥ ν w / , we have Z G ( w ) (cid:54) = 0 from Theorem 5.1. It therefore follows that Z G ( w ) (cid:54) = 0 for all w satisfying eq. (55), and thus the quantity τ ∆ in the statement of Theorem 1.4 can be taken to be ν w / . (cid:3) We conclude with a brief discussion of the dependence of τ ∆ on ∆ . We saw above that τ ∆ can be takento be ν w (∆) / , so it is sufficient to consider the dependence of ν w = ν w (∆) on ∆ . Let c = 10 − . As statedin the discussion following eq. (3), ν w can be chosen to be . c/ (2 ∆ ∆ ) for the case of general list colorings,or c/ (300∆ ) with the assumption of uniformly large list sizes (which, we recall from Remark 5, is satisfiedin the case of uniform q -colorings). We have not tried to optimize these bounds, and it is conceivable that amore careful accounting of constants in our proofs can improve the value of the constant c by a few ordersof magnitude. R e f e r e n c e s [1] M. Bayati, D. Gamarnik, D. Katz, C. Nair, and P. Tetali. Simple deterministic approximation algorithms for counting matchings.In Proceedings of the 39th ACM Symposium on Theory of Computing (STOC) , (2007), pp. 122–127.[2] A. Bandhyopadhyay and D. Gamarnik. Counting without sampling: Asymptotics of the log partition function for certainstatistical physics models.
Random Structures & Algorithms (4) (2008), pp. 452–479.[3] A. Barvinok. Combinatorics and Complexity of Partition Functions.
Springer International Publishing, 2016.[4] A. Barvinok and G. Regts. Weighted counting of solutions to sparse systems of equations. Preprint arXiv:1706.05423v6 (2019).[5] A. Barvinok and P. Soberón. Computing the partition function for graph homomorphisms.
Combinatorica (4) (2017),pp. 633–650.[6] F. Bencs, E. Davies, V. Patel G. Regts. On zero-free regions for the anti-ferromagnetic Potts model on bounded-degree graphs.Preprint arXiv:1812.07532v2 (2019).[7] R.L. Brooks. On coloring the nodes of a network. Mathematical Proceedings of the Cambridge Philosophical Society (1941),pp. 194–197.[8] R. Bubley and M. Dyer. Path coupling: A technique for proving rapid mixing in Markov chains. In Proceedings of the 38thIEEE Symposium on Foundations of Computer Science (FOCS) , 1997, pp. 223–231.[9] S. Chen, M. Delcourt, A. Moitra, G. Perarnau and L. Postle. Improved bounds for sampling colorings via linear programming.In
Proceedings of the 30th ACM-SIAM Symposium on Discrete Algorithms (SODA) , 2019, pp. 2216–2234.[10] P. Csikvári and P. Frenkel. Benjamini-Schramm continuity of root moments of graph polynomials.
European Journal ofCombinatorics (2016), pp. 302–320.[11] M. Dyer and A. Frieze. Randomly coloring graphs with lower bounds on girth and maximum degree. Random Structures &Algorithms (2003), pp. 167–179.[12] M. Dyer, A. Frieze, T. Hayes and E. Vigoda. Randomly coloring constant degree graphs. In Proceedings of the 45th IEEESymposium on Foundations of Computer Science (FOCS) , 2004, pp. 582–589.[13] L. Eldar and S. Mehraban. Approximating the permanent of a random matrix with vanishing mean. In
Proceedings of the 59thIEEE Symposium on Foundations of Computer Science (FOCS) , 2018, pp. 23–34.[14] R. Fernández and A. Procacci. Regions without complex zeros for chromatic polynomials on graphs with bounded degree.
Combinatorics, Probability & Computing (2008), pp. 225–238.[15] A. Frieze and J. Vera. On randomly coloring locally sparse graphs. Discrete Mathematics and Theoretical Computer Science ,2006.[16] A. Frieze and E. Vigoda. A survey on the use of Markov chains to randomly sample colorings. In Combinatorics, Complexityand Chance: A Tribute to Dominic Welsh , Oxford University Press, G. Grimmett and C. McDiarmid (eds.), 2007, pp. 53–71.[17] A. Galanis, D. Štefankovič and E. Vigoda. Inapproximability for antiferromagnetic spin systems in the tree non-uniquenessregion.
Journal of the ACM (2015), article Journal ofDiscrete Algorithms (2012), pp. 29–47.[19] D. Gamarnik, D. Katz and S. Misra. Strong spatial mixing for list coloring of graphs. Random Structures & Algorithms (2013), pp. 599–613.[20] Q. Ge and D. Štefankovič. Strong spatial mixing of q -colorings on Bethe lattices. Prepint arXiv:1102.2886v3 (2011).
21] L.A. Goldberg, R. Martin and M. Paterson. Strong spatial mixing for graphs with fewer colors. In
Proceedings of the 45th IEEESymposium on Foundations of Computer Science (FOCS) , 2004, pp. 562–571.[22] T. Hayes. Randomly coloring graphs of girth at least five. In
Proceedings of the 35th ACM Symposium on Theory of Computing(STOC) , 2003, pp. 269–278.[23] T. Hayes and E. Vigoda. A non-Markovian coupling for randomly sampling colorings. In
Proceedings of the 44th IEEESymposium on Foundations of Computer Science (FOCS) , 2003, pp. 618–627.[24] T. Hayes and E. Vigoda. Coupling with the stationary distribution and improved sampling for colorings and independent sets.In
Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms (SODA) , 2005, pp. 971–979.[25] T. Helmuth, W. Perkins, and G. Regts. Algorithmic Pirogov-Sinai theory. To appear in
Proceedings of the 51st ACM Symposiumon Theory of Computing (STOC) , 2019. Preprint arXiv:1806.11548.[26] B. Jackson, A. Procacci and A. Sokal. Complex zero-free regions at large | q | for multivariate Tutte polynomials (alias Pottsmodel partition functions) with general complex edge weights. Journal of Combinatorial Theory B (2013), pp. 21–45.[27] M. Jenssen, P. Keevash, and W. Perkins. Algorithms for
Proceedings of the 30thACM-SIAM Symposium on Discrete Algorithms (SODA) , 2019, pp. 2235–2247.[28] M. Jerrum. A very simple algorithm for estimating the number of k -colourings of a low-degree graph. Random Structures &Algorithms (1995), pp. 157–165.[29] T.D. Lee and C.N. Yang. Statistical theory of equations of state and phase transitions II. Lattice gas and Ising model. PhysicsReview (1952), pp. 410–419.[30] J. Liu, A. Sinclair and P. Srivastava. The Ising partition function: Zeros and deterministic approximation. In Proceedings of the58th IEEE Symposium on Foundations of Computer Science (FOCS) , 2017, pp. 986–997.[31] P. Lu and Y. Yin. Improved FPTAS for multi-spin systems. In
Proceedings of APPROX , 2013, pp. 639–654.[32] P. Lu, K. Yang, C. Zhang and M. Zhu. FPTAS for counting proper four colorings on cubic graphs. In
Proceedings of the 28thACM-SIAM Symposium on Discrete Algorithms (SODA) , 2017, pp. 1798–1817.[33] M. Molloy. The Glauber dynamics on colorings of a graph with high girth and maximum degree.
SIAM Journal on Computing (2004), pp. 712–737.[34] V. Patel and G. Regts. Deterministic polynomial-time approximation algorithms for partition functions and graph polynomials. SIAM Journal on Computing (2017), pp. 1893–1919.[35] J. Salas and A. Sokal. Absence of phase transition for antiferromagnetic Potts models via the Dobrushin uniqueness theorem. Journal of Statistical Physics (1997), pp. 551–579.[36] A. Sokal. Bounds on the complex zeros of (di)chromatic polynomials and Potts-model partition functions. Combinatorics,Probability & Computing (2001), pp. 41–77.[37] A. Sokal. The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. Surveys in Combinatorics (2005),pp. 173–226.[38] E. Vigoda. Improved bounds for sampling colorings.
Journal of Mathematical Physics (2000), pp. 1555–1569.[39] D. Weitz. Combinatorial criteria for uniqueness of Gibbs measures. Random Structures & Algorithms (4) , pp. 445–475.[40] D. Weitz. Counting independent sets up to the tree threshold. In Proceedings of the 38th ACM Symposium on Theory ofComputing (STOC) , 2006, pp. 140–149.[41] C.N. Yang and T.D. Lee. Statistical theory of equations of state and phase transitions I. Theory of condensation.
PhysicsReview (1952), pp. 404–409. A p p e n d i x A . S k e tc h o f t h e a l g o r i t h mIn this section we outline how to apply Barvinok’s algorithmic paradigm to translate our zero-freenessresult (Theorem 1.4) into the FPTAS claimed in Theorem 1.3. Let G be a graph with n vertices and m edgesand maximum degree ∆ . Recall that our goal is to obtain a ± ε approximation of the Potts model partitionfunction Z G ( w ) at any point w ∈ [0 , . Note that Z G is a polynomial of degree m , and that computing Z G at w = 1 is trivial since Z G (1) = q n . Recall also that Theorem 1.4 ensures that Z G has no zeros in theregion D ∆ of width τ ∆ around the real interval [0 , . For technical convenience we will actually workwith a slightly smaller zero-free region consisting of the rectangle D (cid:48) ∆ = { w ∈ C : − τ (cid:48) ∆ ≤ (cid:60) w ≤ τ (cid:48) ∆ ; |(cid:61) w | ≤ τ (cid:48) ∆ } , where τ (cid:48) ∆ = τ ∆ / √ . Note that D (cid:48) ∆ ⊂ D ∆ so D (cid:48) ∆ is also zero-free. In the rest of this section, we drop thesubscript ∆ from these quantities.Now let f ( z ) be a complex polynomial of degree d for which f (0) is easy to evaluate, and suppose wewish to approximate f (1) . Barvinok’s basic paradigm [3, Section 2.2] achieves this under the assumptionthat f has no zeros in the open disk B (0 , δ ) of radius δ centered at : the approximation simply onsists of the first k = O ( δ log( dεδ )) terms of the Taylor expansion of log f around 0. (Note that thisexpansion is absolutely convergent within B (0 , δ ) by the zero-freeness of f .) These terms can in turnbe expressed as linear combinations of the first k coefficients of f itself. We now sketch how to reduce ourcomputation of Z G ( w ) to this situation.First, for any fixed w ∈ [0 , , define the polynomial g ( z ) := Z G ( z ( w −
1) + 1) . Note that g (0) = Z G (1) is trivial, while g (1) = Z G ( w ) is the value we are trying to compute. Moreover, plainly g ( z ) (cid:54) = 0 for all z ∈ D (cid:48) . Next, define a polynomial φ : C → C that maps the disk B (0 , δ ) into the rectangle D (cid:48) , so that φ (0) = 0 and φ (1) = 1 ; Barvinok [3, Lemma 2.2.3] gives an explicit construction of such a polynomial,with degree N = exp(Θ( τ − )) and with δ = exp( − Θ( τ − )) . Now we have reduced the computation of Z G ( w ) to that of f (1) , where f ( z ) := g ( φ ( z )) is a polynomial of degree deg( g ) · deg( φ ) = mN that isnon-zero on the disk B (0 , δ ) , so the framework of the previous paragraph applies. Note that the numberof terms required in the Taylor expansion of log f is k = O ( δ log( mNεδ )) = exp( O ( τ − )) log( n ∆ ε ) .Naive computation of these k terms requires time n Θ( k ) , which yields only a quasi-polynomial algorithmsince k contains a factor of log n . This complexity comes from the need to enumerate all colorings ofsubgraphs induced by up to k edges. However, a technique of Patel and Regts [34], based on Newton’sidentities and an observation of Csikvari and Frenkel [10], can be used to reduce this computation to anenumeration over subgraphs induced by connected sets of edges (see [34, Section 6] for details). Since G has bounded degree, this reduces the complexity to ∆ O ( k ) = ( n ∆ ε ) log(∆) exp( O ( τ − )) . For any fixed ∆ this ispolynomial in ( n/ε ) , thus satisfying the requirement of a FPTAS.Note that the degree of the polynomial is exponential in τ − ; since τ − in turn is exponential in ∆ (seethe discussion following the proof of Theorem 1.4), the degree of the polynomial is doubly exponential in ∆ .The same discussion explains how this can be improved to singly exponential for the case of uniformlylarge list sizes. A p p e n d i x B . P r o o f o f L e m m a 2 . 5We restate the lemma for convenience. The domain D as is in the discussion preceding the statement ofLemma 2.5; in particular a domain D that is rectangular and symmetric about the real line suffices. Lemma 2.5 ( Mean value theorem for complex functions ) . Let f be a holomorphic function on D suchthat for z ∈ D , (cid:61) f ( z ) has the same sign as (cid:61) z . Suppose further that there exist positive constants ρ I and ρ R such that • for all z ∈ D , |(cid:61) f (cid:48) ( z ) | ≤ ρ I ; • for all z ∈ D , (cid:60) f (cid:48) ( z ) ∈ [0 , ρ R ] .Then for any z , z ∈ D , there exists C z ,z ∈ [0 , ρ R ] such that |(cid:60) ( f ( z ) − f ( z )) − C z ,z · (cid:60) ( z − z ) | ≤ ρ I · |(cid:61) ( z − z ) | , and |(cid:61) ( f ( z ) − f ( z )) | ≤ ρ R · (cid:40) |(cid:61) ( z − z ) | , when ( (cid:61) z ) · ( (cid:61) z ) ≤ . max {|(cid:61) z | , |(cid:61) z | } otherwise.Proof. We write f = u + ιv , where u, v : D → R are seen as differentiable functions from R to R satisfyingthe Cauchy-Riemann equations u (1 , = v (0 , and u (0 , = − v (1 , . This implies in particular that (cid:60) f (cid:48) ( z ) = u (1 , ( z ) = v (0 , ( z ) and (cid:61) f (cid:48) ( z ) = v (1 , ( z ) = − u (0 , ( z ) .Let z be a point in D such that (cid:60) ( z − z ) = 0 and (cid:61) ( z − z ) = 0 (by the conditions imposed on D ,such a z exists, possibly after interchanging z and z ). Now we have (cid:60) ( f ( z ) − f ( z )) = u ( z ) − u ( z ) + u ( z ) − u ( z )= u (1 , ( z (cid:48) ) · (cid:60) ( z − z ) + u ( z ) − u ( z ) , here z (cid:48) is a point lying on the segment [ z , z ] , obtained by applying the standard mean value theorem tothe function u along this segment (note that the segment is parallel to the real axis). On the other hand,since the segment [ z , z ] is parallel to the imaginary axis, we apply the standard mean value theorem tothe real valued function u to get (after recalling that (cid:12)(cid:12) u (0 , ( z ) (cid:12)(cid:12) = |(cid:61) f (cid:48) ( z ) | ≤ ρ I for all z ∈ D ) | u ( z ) − u ( z ) | ≤ ρ I |(cid:61) ( z − z ) | = ρ I |(cid:61) ( z − z ) | . This proves the first part, once we set C z ,z = u (1 , ( z (cid:48) ) = (cid:60) f (cid:48) ( z (cid:48) ) , which must lie in [0 , ρ R ] since z (cid:48) ∈ D .For the second part, we note that since (cid:61) f ( z ) = 0 when (cid:61) z = 0 , we have for z ∈ D , (cid:61) f ( z ) = (cid:61) ( f ( z ) − f ( (cid:60) z )) = v ( z ) − v ( (cid:60) z )= v (0 , ( z (cid:48) ) · (cid:61) z, where z (cid:48) is a point lying on the segment [ z, (cid:60) z ] , obtained by applying the standard mean value theorem tothe function v along this segment (note that the segment is parallel to the imaginary axis).Since v (0 , ( z (cid:48) ) = u (1 , ( z (cid:48) ) ∈ [0 , ρ R ] for all z (cid:48) ∈ D , there exist a, b ∈ [0 , ρ R ] such that |(cid:61) ( f ( z ) − f ( z )) | = | a (cid:61) z − b (cid:61) z | , so that we get |(cid:61) ( f ( z ) − f ( z )) | = | a (cid:61) z − b (cid:61) z | ≤ ρ R · (cid:40) |(cid:61) ( z − z ) | , when ( (cid:61) z ) · ( (cid:61) z ) ≤ . max {|(cid:61) z | , |(cid:61) z | } otherwise. (cid:3) A p p e n d i x C . P r o o f o f L e m m a 2 . 6We restate the lemma for convenience.
Lemma 2.6.
Consider the domain D given by D := { z | (cid:60) z ∈ ( −∞ , − ζ ) and |(cid:61) z | < τ } , where τ < / and ζ are positive real numbers such that τ + e − ζ < . Suppose κ ∈ [0 , and consider thefunction f κ as defined in eq. (2) . Then,(1) The function f κ and the domain D satisfy the hypotheses of Lemma 2.5, if ρ R and ρ I in the statementof the theorem are taken to be e − ζ − e − ζ and τ · e − ζ ( − e − ζ ) , respectively.(2) If ε > and κ (cid:48) are such that | κ (cid:48) − κ | < ε and (1 + ε ) < e ζ , then for any z ∈ D , | f κ (cid:48) ( z ) − f κ ( z ) | ≤ εe ζ − − ε . Proof.
The domain D is rectangular and symmetric about the real axis, so it clearly satisfies the conditions.We also note that since κ ≤ , f κ ( z ) is well defined when (cid:60) z < , and maps real numbers in D to realnumbers. Further, a direct calculation shows that (cid:61) f κ ( z ) = − arg(1 − κe z ) has the same sign as sin( (cid:61) z ) when (cid:60) z < (since κ ∈ [0 , ). Since |(cid:61) z | ≤ τ < π , we see therefore that (cid:61) f κ ( z ) has the same sign as (cid:61) z ,and hence f κ satisfies the hypothesis of Lemma 2.5.Note that f (cid:48) κ ( z ) = κe z − κe z . A direct calculation then shows that (cid:60) f (cid:48) κ ( z ) = κ (cid:60) e z − κ | e z | | − κe z | and (cid:61) f (cid:48) κ ( z ) = κ (cid:61) e z | − κe z | . Now, for z ∈ D , | arg e z | ≤ τ , so that (cid:60) e z ≥ | e z | cos arg e z ≥ | e z | (1 − τ ) . Thus, we see that κ (cid:60) e z − κ | e z | ≥ κ | e z | (cid:0) − τ − κ | e z | (cid:1) ≥ κ | e z | (cid:0) − τ − κe − ζ (cid:1) . Since κ ∈ [0 , and τ + e − ζ < by assumption, we therefore have (cid:60) f (cid:48) κ ( z ) ≥ . Further (cid:60) f (cid:48) κ ( z ) ≤ | f (cid:48) κ ( z ) | = κ | e z || − κe z | ≤ κ | e z | − κ | e z | ≤ κe − ζ − e − ζ ,since κ ∈ [0 , . Together, these show that (cid:60) f (cid:48) κ ( z ) ∈ (cid:104) , e − ζ − e − ζ (cid:105) for z ∈ D , so that the claimed choice ofthe parameter ρ R in Lemma 2.5 is justified. imilarly, for the imaginary part, we have |(cid:61) f (cid:48) κ ( z ) | = κ |(cid:61) e z || − κe z | , which in turn is at most κ · τ · e − ζ (1 − κe − ζ ) for z ∈ D . Since κ ∈ [0 , , this justifies the choice of the parameter ρ I .We now turn to the second item of the observation. The derivative of f x ( z ) with respect to x is e z − xe z ,which for x within distance ε (satisfying (1 + ε ) < e ζ ) of κ and z ∈ D has length at most e ζ − − ε . Thus,the standard mean valued theorem applied along the segment [ κ, κ (cid:48) ] (which is of length at most ε ) yieldsthe claim. (cid:3) A p p e n d i x D . P r o o f o f L e m m a 2 . 7We restate the lemma here for convenience.
Lemma 2.7.
Let z , z , . . . , z n be complex numbers such that the angle between any two non-zero z i is atmost α ∈ [0 , π/ . Then | (cid:80) ni =1 z i | ≥ cos( α/ (cid:80) ni =1 | z i | .Proof. Fix a non-zero z i , and without loss of generality let z and z be the non-zero elements giving themaximum and minimum values, respectively, of the quantity arg( z j /z i ) , as z j varies over all the non-zeroelements (breaking ties arbitrarily). Consider the ray z bisecting the angle between z and z . Then, by theassumption, the angle made by z and any of the non-zero z i is at most α/ , so that the projection of z i on z is of length at least | z i | cos( α/ and is in the same direction as z . Thus, denoting by S (cid:48) the projection of S = (cid:80) ni =1 z i on z , we have | S | ≥ | S (cid:48) | ≥ n (cid:88) i =1 | z i | cos( α/ . (cid:3) A p p e n d i x E . P r o o f o f L e m m a 3 . 4We restate Lemma 3.4 here for convenience.
Lemma 3.4.
Let G be any graph that satisfies item 2 of Condition 1, and let u be any unpinned vertex in G ,then u is nice in G .Proof. We show first that Pr G,w [ c ( u ) = i ] ≤ β whenever L G ( u ) ≥ deg G ( u ) + β ; this will be required laterin the proof. To do so, we repeat the arguments in the proof of Lemma 3.3 to see that Pr G,w [ c ( u ) = i ] ≤ | L ( u ) |− deg G ( u ) . The claimed bound then follows since | L ( u ) | − deg G ( u ) ≥ β .Next we show that the upper bound of d +2 , where d is the number of unpinned neighbors of u in G ,holds conditioned on every coloring of the neighbors of the (unpinned) neighbors of u , by following asimilar path as in [19]. Consider any valid coloring σ (cid:48) of the vertices at distance two from u . Since G istriangle free, we claim that conditional on σ (cid:48) there is a tree T of depth rooted at u , with all the leavespinned according to σ (cid:48) , such that(56) Pr G,w (cid:2) c ( u ) = i | σ (cid:48) (cid:3) = Pr T,w [ c ( u ) = i ] . To see this, notice that once we condition on the coloring of the vertices at distance from u , the distributionof the color at u becomes independent of the distribution of colors of vertices at distance or more. Further,because of triangle freeness, no two neighbors of u have an edge between them, and hence any cycle inthe distance- neighborhood, if one exists, must go through at least one pinned vertex. We then observethat such a cycle can be broken by replacing any pinned vertex v (cid:48) in it with deg( v (cid:48) ) copies, one for eachof its neighbor: as discussed earlier, this operation cannot change the partition function or probabilities.This operation therefore ensures that every pinned vertex in the resulting graph is now a leaf of a tree T ofdepth rooted at u . Further, in T , the root u has d unpinned children, and all vertices at depth are pinnedaccording to σ (cid:48) . Here, we say that a coloring σ is valid if the color σ assign to any vertex v is from L ( v ) , and further, in case w = 0 , no twoneighbors are assigned the same color by σ . et v , · · · , v d be the d unpinned neighbors of u in T , and let T , · · · , T d be the subtrees rooted at v , · · · , v d respectively. For each k ∈ L G ( u ) , let n k be the number of neighbors of u that are pinned tocolor k . Then by Lemma 2.4, R ( j,i ) T,u ( w ) = w n j · (cid:81) dk =1 (1 − γ · P T k ,w [ c ( v k ) = j ] ) w n i · (cid:81) dk =1 (1 − γ · P T k ,w [ c ( v k ) = i ] ) . Define t kj := γ · Pr T k ,w [ c ( v k ) = j ] , and note that from the calculation at the beginning of the proof, wehave ≤ t kj ≤ γβ ≤ β ≤ / . Note also that t kj = 0 if j (cid:54)∈ L ( v k ) . Thus, we have(57) (cid:88) j ∈ Γ u t kj = γ (cid:88) j ∈ Γ u ∩ L ( v k ) Pr T k ,w [ c ( v k ) = j ] ≤ γ ≤ . Therefore,(58) Pr T,w [ c ( u ) = i ] = 1 (cid:80) j ∈ L ( v ) R ( j,i ) T,v ( w ) = w n i · (cid:81) dk =1 (1 − t ki ) (cid:80) j ∈ L ( u ) w n j · (cid:81) dk =1 (1 − t kj ) ≤ (cid:80) j ∈ Γ u (cid:81) dk =1 (1 − t kj ) , where, in the last inequality we use that n j = 0 when j is good for u in G , and also that w ∈ [0 , .Since Pr G,w [ c ( u ) = i | σ (cid:48) ] = Pr T,w [ c ( u ) = i ] , it remains to lower bound (cid:80) j ∈ Γ u (cid:81) dk =1 (1 − t kj ) . Webegin by recalling the following standard consequence of the Taylor expansion of ln(1 − x ) around : when ≤ x ≤ β < , and β is such that (1 − /β ) ≥ / ,(59) ln(1 − x ) ≥ − x − x − /β ) ≥ − x − x ≥ − (cid:18) β (cid:19) x. Note that the condition required of β is satisfied since β ≥ α ≥ / , as stipulated in item 2 of Condition 1.Since ≤ t kj ≤ /β , we therefore obtain, for every j ∈ Γ u , d (cid:89) k =1 (1 − t kj ) ≥ d (cid:89) k =1 exp (cid:18) − (cid:18) β (cid:19) t kj (cid:19) = exp (cid:32) − (cid:18) β (cid:19) d (cid:88) k =1 t kj (cid:33) . (60)For convenience of notation, we denote | Γ u | by q u . Note that since | L ( u ) | ≥ α deg( u ) + β , and u has deg( u ) − d pinned neighbors, we have q u ≥ | L ( u ) | − (deg( u ) − d ) ≥ | L ( u ) | − α (deg( u ) − d ) ≥ αd + β, where in the second inequality we use α ≥ . Now, by the AM-GM inequality, we get (cid:88) j ∈ Γ u d (cid:89) k =1 (1 − t kj ) ≥ q u (cid:89) j ∈ Γ u d (cid:89) k =1 (1 − t kj ) qu ≥ q u exp − q u (1 + 1 /β ) · d (cid:88) k =1 (cid:88) j ∈ Γ u t kj , using eq. (60) ≥ ( αd + β ) exp (cid:18) − d (1 + 1 /β ) αd + β (cid:19) , using eq. (57) and q u ≥ αd + β ≥ ( d + 2) α · exp (cid:18) − (1 + 1 /β ) α (cid:19) , using β ≥ α ≥ ( d + 2) , here the last line uses the stipulation in item 2 of Condition 1 that α and β satisfy α · exp (cid:16) − (1+1 /β ) α (cid:17) ≥ .From eqs. (56) and (58) we therefore get Pr G,w (cid:2) c ( u ) = i | σ (cid:48) (cid:3) ≤ d + 2 . Since this holds for any conditioning σ (cid:48) of the colors of the neighbors of the neighbors of u in G , we thenhave Pr G,w [ c ( u ) = i ] ≤ d + 2 , which concludes the proof. (cid:3)(cid:3)