Approximately counting independent sets of a given size in bounded-degree graphs
aa r X i v : . [ c s . D S ] F e b APPROXIMATELY COUNTING INDEPENDENT SETS OFA GIVEN SIZE IN BOUNDED-DEGREE GRAPHS
EWAN DAVIES AND WILL PERKINS
Abstract.
We determine the computational complexity of approxi-mately counting and sampling independent sets of a given size in bounded-degree graphs. That is, we identify a critical density α c (∆) and provide(i) for α < α c (∆) randomized polynomial-time algorithms for approx-imately sampling and counting independent sets of given size at most αn in n -vertex graphs of maximum degree ∆; and (ii) a proof thatunless NP=RP, no such algorithms exist for α > α c (∆). The criti-cal density is the occupancy fraction of hard core model on the clique K ∆+1 at the uniqueness threshold on the infinite ∆-regular tree, giving α c (∆) ∼ e e as ∆ → ∞ . Introduction
Counting and sampling independent sets in graphs are fundamental com-putational problems arising in several fields including algorithms, statisticalphysics, and combinatorics. Given a graph G , let I ( G ) denote the set ofindependent sets of G . The independence polynomial of G is Z G ( λ ) = X I ∈I ( G ) λ | I | = X k ≥ i k ( G ) λ k , where i k ( G ) is the number of independent sets of size k in G . The inde-pendence polynomial also arises as the partition function of the hard-coremodel from statistical physics.With G and λ as inputs, exact computation of Z G ( λ ) is Z G ( λ ) has been a major topic in recenttheoretical computer science research. There is a detailed understanding ofthe complexity of approximating Z G ( λ ) for the class of graphs of maximumdegree ∆, in particular showing that there is a computational threshold whichcoincides with a certain probabilistic phase transition as one varies the valueof λ .The hard-core model on G at fugacity λ is the probability distribution on I ( G ) defined by µ G,λ ( I ) = λ | I | Z G ( λ ) . Date : February 10, 2021.WP supported in part by NSF grants DMS-1847451 and CCF-1934915.
Defined on a lattice like Z d (through an appropriate limiting procedure),this is a simple model of a gas (the hard-core lattice gas) and it exhibits anorder/disorder phase transition as λ changes. The hard-core model can alsobe defined on the infinite ∆-regular tree (the Bethe lattice ). Kelly [20] deter-mined the critical threshold for uniqueness of the infinite volume measureon the tree, namely(1) λ c (∆) = (∆ − ∆ − (∆ − ∆ . This value of λ also marks a computational threshold for the complexity ofapproximating Z G ( λ ) on graphs of maximum degree ∆. One can approx-imate Z G ( λ ) up to a relative error of ε in time polynomial in n and 1 /ε with several different methods, provided G is of maximum degree ∆ and λ < λ c (∆). The first such algorithm is based on correlation decay on treesand is due to Weitz [30], but recently alternative algorithms based on polyno-mial interpolation [3, 24, 25] and Markov chains [2, 7, 6] for this problem havealso been given. Conversely, for λ > λ c (∆) a result of Sly and Sun [28] andGalanis, ˇStefankoviˇc, and Vigoda [16] (following Sly [27]) states that unlessNP=RP there is no polynomial-time algorithm for approximating Z G ( λ ) ongraphs of maximum degree ∆. Counting and sampling are closely related,and by standard reduction techniques the same computational thresholdholds for the problem of approximately sampling independent sets from thehard-core distribution.The hard-core model is an example of the grand canonical ensemble fromstatistical physics, where one studies physical systems that can freely ex-change particles and energy with a reservoir. Closely related is the canoni-cal ensemble , where one removes the reservoir and considers a system with afixed number of particles. In the context of independent sets in graphs, thiscorresponds to the uniform distribution on independent sets of some fixedsize k . Here the number i k ( G ) of independent sets of size k in G plays therole of the partition function. In this paper we answer affirmatively the nat-ural question of whether there is a similar complexity phase transition forthe problem of approximating i k ( G ), and the related problem of samplingindependent sets of size k approximately uniformly. Analogous to the criti-cal fugacity in the hard-core model, we identify a critical density α c (∆), andfor α < α c (∆) we give a fully polynomial-time randomized approximationscheme (FPRAS, defined below) for counting independent sets of size k in n -vertex graphs of maximum degree ∆, where 0 ≤ k ≤ αn . We also showthat unless NP=RP there is no such algorithm for α > α c (∆).In statistical physics the grand canonical ensemble and the canonical en-semble are known to be equivalent in some respects under certain conditions,and the present authors, Jenssen, and Roberts [12] used this idea to givea tight upper bound on i k ( G ) for large k in large ∆-regular graphs G (seealso [10] for the case of small k ). Here, the main idea in our proofs is also PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 3 to exploit the equivalence of ensembles. For algorithms at subcritical den-sities we approximately sample independent sets from the hard-core modeland show that with sufficiently high probability we get an independent setof the desired size, distributed approximately uniformly. For hardness atsupercritical densities we construct an auxiliary graph G ′ such that i k ( G ′ )is approximately proportional to Z G ( λ ) for some λ > λ c (∆), and hence ishard to approximate. Our algorithm for subcritical densities is new, and inthe sense of permitting higher densities it outperforms previous algorithmsfor this problem based on Markov chains [5, 1], and an algorithm implicitin [10] based on the cluster expansion.A pleasant feature of our methods is the incorporation of several advancesfrom recent research on related topics. From the geometry of polynomials weuse a state-of-the-art zero-free region for Z G ( λ ) due to Peters and Regts [25]and a central limit theorem of Michelen and Sahasrabudhe [23, 22] (thoughan older result of Lebowitz, Pittel, Ruelle and Speer [21] would also suffice),and we also apply the very recent development that a natural Markov chainfor sampling from the hard-core model at subcritical fugacities (the Glauberdynamics) mixes rapidly [1, 6]. Finally, our results also show a connectionbetween these algorithmic and complexity-theoretic problems and extremalcombinatorics problems for bounded-degree graphs [9, 12, 10], see also thesurvey [31].1.1. Preliminaries.
Given an error parameter ε and real numbers z , ˆ z ,we say that ˆ z is a relative ε -approximation to z if e − ε ≤ ˆ z/z ≤ e ε . A fully polynomial-time randomized approximation scheme or FPRAS for acounting problem is a randomized algorithm that with probability at least3 / ε -approximation to the solution of the problem intime polynomial in the size of the input and 1 /ε . If the algorithm is deter-ministic (i.e. succeeds with probability 1) then it is a fully polynomial-timeapproximation scheme ( FPTAS ). An ε -approximate sampling algorithm fora probability distribution µ outputs a random sample from a distribution ˆ µ such that the total variation distance k µ − ˆ µ k T V ≤ ε , and an efficient sam-pling scheme is, for all ε > ε -approximate sampling algorithm whichruns in time polynomial in the size of the input and log(1 /ε ). Note thatapproximate sampling schemes whose running times are polynomial in 1 /ε or in log(1 /ε ) are common in the literature, but we adopt the stronger def-inition for this paper. The inputs to our algorithms are graphs, and inputsize corresponds to the number of vertices of the graph.An independent set in a graph G = ( V, E ) is a subset I ⊂ V such thatno edge of E is contained in I . The density of such an independent set I is | I | / | V | , and it will be convenient for us to parametrize independentsets by their density instead of their size. We write I ( G ) for the set of allindependent sets in G , I k ( G ) for the set of independent sets of size k in G , and i k ( G ) = |I k ( G ) | for the number of such sets. Recall the hard-coredistribution µ G,λ on I ( G ) is given by µ G,λ ( I ) = λ | I | /Z G ( λ ). We also define E. DAVIES AND W. PERKINS the occupancy fraction α G ( λ ) of the hard-core model on G at fugacity λ tobe the expected density of a random independent set drawn according to µ G,λ . Let G ∆ be the set of graphs of maximum degree ∆.The critical density that we show constitutes a computational thresholdfor the problems of counting and sampling independent sets of a given sizein graphs of maximum degree ∆ is α c (∆) = λ c (∆)1 + (∆ + 1) λ c (∆) = (∆ − ∆ − (∆ − ∆ + (∆ + 1)(∆ − ∆ − , with λ c the critical fugacity as in (1). This may seem unexpected at firstsight, but has a natural interpretation. The threshold is in fact the quantity α K ∆+1 ( λ c (∆)), the occupancy fraction of the clique on ∆ + 1 vertices at thecritical fugacity λ c (∆). This is a natural threshold because the occupancyfraction is a monotone increasing function of λ , and the clique on ∆ + 1vertices has the minimum occupancy fraction over all graphs of maximumdegree ∆. Thus, for any G ∈ G ∆ , the value of λ which makes α G ( λ ) > α c (∆)must be greater than λ c (∆). Conversely, if α < α c (∆) then for every graph G ∈ G ∆ there is some λ < λ c (∆) such that α G ( λ ) = α .1.2. Our results.
We are now ready to state our main result.
Theorem 1. (a)
For every α < α c (∆) there is an FPRAS for i ⌊ αn ⌋ ( G ) and an effi-cient sampling scheme for the uniform distribution on I ⌊ αn ⌋ ( G ) for n -vertex graphs G of maximum degree ∆ . (b) Unless
NP=RP , for every α ∈ ( α c (∆) , / there is no FPRAS for i ⌊ αn ⌋ ( G ) for n -vertex, ∆ -regular graphs G . The assumption NP =RP, which is that polynomial-time algorithms usingrandomness cannot solve all problems in NP, is standard in computationalcomplexity theory. Indeed, this assumption is used in [27, 28, 16] to showhardness of approximation for Z G ( λ ) on regular graphs at supercritical fu-gacities, which we apply directly. The upper bound of 1 / α in (b) isrequired since in a regular graph (of degree ≥
1) there are no independentsets of density greater than 1 / / α ∈ ( α c (∆) , k in n -vertex graphs of maximum degree ∆ mixes rapidly when k < n/ (2∆+2), andrecently this was slightly improved to k < n/ (2∆) via the method of high-dimensional expanders by Alev and Lau [1] (who also gave an improvedbound in terms of the smallest eigenvalue of the adjacency matrix of G ).The fast mixing of this Markov chain provides a randomized algorithm forapproximate sampling and an FPRAS for approximate counting for this PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 5 range of k . Implicit in the work of the authors and Jenssen [10] is analternative method based on the cluster expansion that yields an FPTASfor i k ( G ) when k < e − n/ (∆ + 1), and although we did not try to optimizethe constant it seems unlikely that without significant extension the clusterexpansion approach could yield a sharp result. Considering asymptotics as∆ → ∞ , these previous algorithms work for densities up to ( c + o (1)) / ∆ withthe constant c being 1 / e − ≈ .
007 respectively. Here, our algorithmswork up for densities α satisfying α < α c (∆) = (1 + o (1)) e e , where the constant e/ (1 + e ) is approximately 0 . Triangle-free graphs.
As an additional application of our techniqueswe find an approximate computational threshold for the class of triangle-freegraphs.
Theorem 2.
For every δ > there is ∆ large enough so that the followingis true. (a) For ∆ ≥ ∆ and α < − δ ∆ there is an FPRAS and efficient samplingscheme for i ⌊ αn ⌋ ( G ) for the class of triangle-free graphs of maximumdegree ∆ . (b) For ∆ ≥ ∆ and α ∈ (cid:0) δ ∆ , / (cid:1) there is no FPRAS for i ⌊ αn ⌋ ( G ) forthe class of triangle-free graphs of maximum degree ∆ . The proof of this theorem uses a result on the occupancy fraction of triangle-free graphs from [11].1.4.
Related work.
Counting independent sets of a specified size has arisenin various places as a natural fixed-parameter version of counting indepen-dent sets, and is equivalent to counting cliques of a specified size in thecomplement graph. Exact computation of i k ( G ) in an n -vertex graph H istrivially possible in time O ( k n k ), though improvements can be made viafast matrix multiplication algorithms (see e.g. [15]). Another branch of re-search concerns the complexity (in both time and number of queries to thegraph data structure) of counting and approximately counting cliques. Forexample, in [14] the authors gave a randomized approximation algorithm forapproximating the number of cliques of size k . Results of this kind performpoorly in our setting, which is equivalent to counting cliques in the com-plement of bounded-degree graphs, because such graphs are very dense. Inparticular, the main result of [14] has expected running time Ω(( nk/e ) k ) inour setting.With a focus on bounded-degree graphs and connections to statisticalphysics, our work is closer in spirit to that of Curticapean, Dell, Fomin,Goldberg, and Lapinskas [8]. There, the authors consider the problem ofcounting independent sets of size k in bipartite graphs from the perspective E. DAVIES AND W. PERKINS of parametrized complexity. They give algorithms for exact computationand approximation of i k ( G ) in bipartite graphs (of bounded degree andotherwise), including a fixed parameter tractable randomized approximationscheme, though their running times are exponential in k . We note that thecomplexity of approximately counting the total number of independent setsin bipartite graphs (a problem known as Questions and future directions.
For the hard-core model, the algo-rithm of Weitz [30] gives a deterministic approximation algorithm (FPTAS)for Z G ( λ ) for λ < λ c (∆). The approach of Barvinok along with results ofPatel and Regts and Peters and Regts give another FPTAS for the samerange of parameters [3, 24, 25]. Our algorithm for approximating the num-ber of independent sets of a given size uses randomness, but we conjecturethat there is a deterministic algorithm that works for the same range ofparameters. (The cluster expansion approach of [10] gives an FPTAS butonly for smaller values of α ). Conjecture 1.
There is an FPTAS for i ⌊ αn ⌋ ( G ) for G ∈ G ∆ and all α <α c (∆) . The Markov chain analyzed in [5, 1] is the ‘down/up’ Markov chain: start-ing from an independent set I t ∈ I k ( G ) at step t , pick a uniformly randomvertex v ∈ I t and a uniformly random vertex w ∈ V . Let I ′ = ( I t \ v ) ∪ w .If I ′ ∈ I k ( G ), let I t +1 = I ′ ; if not, let I t +1 = I t . Conjecture 2.
The down/up Markov chain for sampling from I ⌊ αn ⌋ ( G ) mixes rapidly for α < α c (∆) and all G ∈ G ∆ . One of the steps of our proof leads to a natural probabilistic conjectureconcerning the hard-core model in bounded degree graphs.
Conjecture 3.
Suppose G is a graph on n vertices of maximum degree ∆ .Then if λ < λ c (∆) and k = ⌊ E G,λ | I |⌋ , we have P G,λ [ | I | = k ] = Ω( n − / ) , where the implied constant only depends on ∆ and λ and the expectation andprobability are with respect to the hard-core model on G at fugacity λ . Lemma 5 below gives the weaker bound Ω( n − log − n ). A stronger con-jecture would be that a local central limit theorem for | I | holds whenever λ < λ c (∆).Finally, our proofs of Theorems 1 and 2 show a close connection betweenthe computational threshold for sampling independent sets of a given size inbounded-degree graphs and the extremal combinatorics problem of minimiz-ing the occupancy fraction in the hard-core model over a class of bounded-degree graphs. We expect that a rigorous connection between the two prob-lems can be proved. PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 7 Algorithms
In this section, we fix ∆ ≥ α < α c (∆). We will give an algorithmthat, for G ∈ G ∆ on n vertices and k ≤ αn , returns an ε -approximateuniform sample from I k ( G ) and runs in time polynomial in n and log(1 /ε );this proves the sampling part of Theorem 1 (a). We then use this algorithmto approximate i k ( G ) using a standard simulated annealing process to provethe approximate counting part of Theorem 1 (a).Given λ ≥
0, let I be a random independent set from the hard-core modelon G at fugacity λ . We will write P G,λ for probabilities over the hard-coremeasure µ G,λ , so e.g. P G,λ ( | I | = k ) is the probability that I is of size exactly k . Often we will suppress the dependence on G .A key tool that we use for probabilistic analysis and to approximatelysample from µ G,λ is the
Glauber dynamics . This is a Markov chain withstate space I ( G ) and stationary distribution µ G,λ . Though the algorithm ofWeitz [30] was the first to give an efficient approximate sampling algorithmfor µ G,λ for λ < λ c (∆) and all G ∈ G ∆ , a randomized algorithm withbetter running time now follows from recent results showing that the Glauberdynamics mix rapidly for this range of parameters [2, 7, 6]. The mixing time T mix ( M , ε ) of a Markov chain M is the number of steps from the worst-caseinitial state I for the resulting state to have a distribution within totalvariation distance ε of the stationary distribution. We will use the followingresult of Chen, Liu, and Vigoda [6], and the sampling algorithm that itimplies. Theorem 3 ([6]) . Given ∆ ≥ and ξ ∈ (0 , λ c (∆)) , there exists C > suchthat the following holds. For all ≤ λ < λ c (∆) − ξ and graphs G ∈ G ∆ on n vertices, the mixing time T mix ( M , ε ) of the Glauber dynamics M for thehard-core model on G with fugacity λ is at most Cn log( n/ε ) . This impliesan ε -approximate sampling algorithm for µ G,λ for G ∈ G ∆ that runs in time O ( n log n log( n/ε )) . The sampling algorithm follows from the mixing time bound; the extrafactor log n is the cost of implementing one step of Glauber dynamics (whichrequires reading O (log n ) random bits to sample a vertex uniformly). Notethat the implicit constant in the running time depends on how close λ is to λ c (∆), but in applications of this theorem we will have λ ≤ λ c (∆) − ξ forsome fixed ξ >
0, so that the implicit constant depends only on ξ , which inturn depends on α .2.1. Approximate sampling.
The following algorithm uses Theorem 3and a binary search on values of λ to generate samples from I k ( G ). Themain results in this section are a proof that the samples are distributedapproximately uniformly and a bound on the running time. E. DAVIES AND W. PERKINS
Algorithm: Sample- k • INPUT: α < α c ; ε > G ∈ G ∆ of size n ; integer k ≤ αn . • OUTPUT: I ∈ I k ( G ) with distribution within ε total variation dis-tance of the uniform distribution of I k ( G ).(1) Let λ ∗ = α − α (∆+1) .(2) For t = 0 , . . . , ⌊ λ ∗ n ⌋ , let λ t = t/ (2 n ).(3) Let Λ = { λ t : t = 0 , . . . , ⌊ λ ∗ n ⌋} .(4) FOR i = 1 , . . . , C log n ,(a) Let λ be a median of the set Λ i − .(b) With N = C ′ n log (cid:0) log nε (cid:1) , take N independent samples I , . . . , I N from a distribution ˆ µ λ on I ( G ).(c) Let κ = N P Nj =1 | I j | .(d) If | κ − k | ≤ / j ∈ { , . . . , N } so that | I j | = k ,then output I j for the smallest such j and HALT.(e) If κ ≤ k , let Λ i = { λ ′ ∈ Λ i − : λ ′ > λ } . If instead κ > k , letΛ i = { λ ′ ∈ Λ i − : λ ′ < λ } .(5) If no independent set of size k has been obtained by the end of theFOR loop (or if Λ j = ∅ at any step), use a greedy algorithm andoutput an arbitrary I ∈ I k ( G ). Theorem 4.
Let C be the constant in line (4) of the algorithm. If thedistributions ˆ µ λ are each within total variation distance ε/ (2 CN log n ) of µ G,λ , the output distribution of Sample- k is within total variation distance ε of the uniform distribution of I k ( G ) . The running time of Sample- k is O ( N log n · T ( n, ε )) where T ( n, ε ) is the running time required to produce asample from ˆ µ λ satisfying the above guarantee. The sampling part of Theorem 1 follows immediately from Theorem 4since by Theorem 3 we can obtain ε/ (2 CN log n )-approximate samples from µ G,λ in time O ( n log n log( n log n · N/ε )). Thus, the total running time ofSample- k with this guarantee on ˆ µ λ is O ( N · n log n · log( nN/ε )) ≤ n log n · polylog (cid:0) log nε (cid:1) . Before we prove Theorem 4, we collect a number of preliminary resultsthat we will use. The first is a bound on the probability of getting anindependent set of size close to the mean from the hard-core model when λ <λ c (∆). We use the notation nα G ( λ ) for the expected size of an independentset from the hard-core model on G at fugacity λ to avoid ambiguities. Lemma 5.
For ∆ ≥ and α < α c (∆) , there is a unique λ ∗ < λ c (∆) sothat α K ∆+1 ( λ ∗ ) = α , and the following holds. For any G ∈ G ∆ on n verticesand any ≤ k ≤ αn , there exists an integer t ∈ { , . . . , ⌊ λ ∗ n ⌋} so that (2) (cid:12)(cid:12) nα G ( t/ (2 n )) − k (cid:12)(cid:12) ≤ / . PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 9
Moreover, if t satisfies (2) then µ G,t/ (2 n ) ( I k ( G )) = Ω (cid:18) n log n (cid:19) . To prove this lemma we need several more results. The first is an ex-tremal bound on α G ( λ ) for G ∈ G ∆ . The statement of the theorem followsfrom a stronger property proved by Cutler and Radcliffe in [9]; see [12] fordiscussion. Theorem 6 ([9]) . For all G ∈ G ∆ and all λ ≥ , α G ( λ ) ≥ α K ∆+1 ( λ ) = λ λ (∆ + 1) . We next rely on a zero-free region for Z G ( λ ) due to Peters and Regts [25],so that we can apply the subsequent central limit theorem. Theorem 7 ([25]) . Let ∆ ≥ and ξ ∈ (0 , λ c (∆)) . Then there exists δ > such that for every G ∈ G ∆ the polynomial Z G has no roots in the complexplane that lie within distance δ of the real interval [0 , λ c (∆) − ξ ) . The probability generating function of a discrete random variable X dis-tributed on the non-negative integers is the polynomial in z given by f ( z ) = P j ≥ P ( X = j ) z j , and the above result shows that at subcritical fugacitythe probability generating function of | I | has no zeros close to 1 in C . Thislets us use the following result of Michelen and Sahasrabudhe [22]. Theorem 8 ([22]) . For n ≥ let X n be a random variable taking valuesin { , . . . , n } with mean µ n , standard deviation σ n , and probability gener-ating function f n . If f n has no roots within distance δ n of in C , and σ n δ n / log n → ∞ , then ( X n − µ n ) /σ n tends to a standard normal in distri-bution. The final tools we need are simple bounds on the variance of the size ofan independent set from the hard-core model.
Lemma 9.
Let G be a graph on n vertices and let I be a random independentset drawn from the hard-core model on G at fugacity λ . Then, if M is thesize of a largest independent set in G we have λ (1 + λ ) M ≤ var( | I | ) ≤ n λ λ . If G has maximum degree ∆ then this applies with M = n/ (∆ + 1) .Proof. For the upper bound note that | I | is the sum of the indicator randomvariables X v that the vertex v ∈ V ( G ) is in I . Then because P ( X v =1) ≤ λ/ (1 + λ ) for all v , from the Cauchy–Schwarz inequality in the formcov( X u , X v ) ≤ var( X u ) var( X v ) we obtainvar( | I | ) = X u ∈ V ( G ) X v ∈ V ( G ) cov( X u , X v ) ≤ n λ λ . For the lower bound, let J be some fixed independent set in G of maximumsize M . Now write X = | I | , and let K = I \ J . By the law of total variance,var( X ) = E [var( X | K )] + var( E [ X | K ]) ≥ E [var( X | K )] . But we have X = | K | + | I ∩ J | , and conditioned on K the set | I ∩ J | isdistributed according to the hard-core model on J \ N G ( K ), the subset of J uncovered by K . Since J is independent, this is a sum of at most | J | inde-pendent, identically distributed Bernoulli random variables with probability λ/ (1 + λ ).Now, writing U = | J \ N G ( K ) | for the number variables in the sum wehave var( X ) ≥ E [var( X | K )] = λ (1 + λ ) E U . A vertex u ∈ J is uncovered by K precisely when N ( u ) ∩ K = ∅ . Then bysuccessive conditioning and the maximum degree condition, the probabilitythat u is uncovered by K is at least (1+ λ ) − ∆ . This means E U ≥ | J | (1+ λ ) − ∆ and hence var( X ) ≥ λ (1 + λ ) M .
The final assertion follows from the fact that any n -vertex graph of maximumdegree ∆ contains an independent set of size at least n/ (∆ + 1), which iseasy to prove by analyzing a greedy algorithm. (cid:3) Proof of Lemma 5.
A standard calculation gives ∂∂λ α G ( λ ) = 1 n ∂∂λ λZ ′ G ( λ ) Z G ( λ ) = 1 nλ var( | I | ) , and so Lemma 9 gives that 0 < α ′ G ( λ ) ≤ n for all λ > λ ∗ < λ c (∆) be the solution to the equation α K ∆+1 ( λ ∗ ) = α .This means λ ∗ = α − α (∆ + 1) , as defined in Sample- k . The fact that λ ∗ < λ c (∆) follows from the factthat α < α c (∆) = α K ∆+1 ( λ c (∆)), and that occupancy fractions are strictlyincreasing. Then using Theorem 6 we have that(3) α G ( λ ∗ ) ≥ α K ∆+1 ( λ ∗ ) = α , and so there exists λ ∈ (0 , λ ∗ ] such that nα G ( λ ) = k . Using the upper boundon α ′ G ( λ ), we see that as λ increases over an interval of length 1 / (2 n ), thefunction nα G ( λ ) can increase by at most 1 /
2. Hence, there is at least oneinteger t ∈ { , . . . , ⌊ λ ∗ n ⌋} such that | nα G ( t/ (2 n )) − k | ≤ / | I | and rapid mixing of the Glauber dynamics. There is a close connection PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 11 between zeros of the probability generating function of | I | and the zeros ofthe partition function itself. The probability generating function of | I | is f ( z ) = X j ≥ P λ ( | I | = j ) z j = X j ≥ i j ( G ) λ j z j Z G ( λ ) = Z G ( λz ) Z G ( λ ) . Then for λ such that Z G ( λ ) = 0, z is a root of f if and only if zλ is aroot of Z G ( λ ). By our assumptions on t , when λ = t/ (2 n ) Theorem 7gives the existence of δ > G ∈ G ∆ there are no complexzeros of f within distance δ/λ of 1. This is because Theorem 7 means that Z G ( zλ ) = 0 implies | zλ − λ | ≥ δ . The condition of Theorem 8 which statesthat σ n δ n / log n → ∞ is met because λ < λ c (∆) ≤ σ n δ n ≥ s λ (1 + λ ) n ∆ + 1 · δλ ≥ Ω (cid:16)p n/λ (cid:17) > ω (log n ) . Now, given λ = t/ (2 n ) such that (2) holds, we have n λ λ ≥ nα G ( λ ) ≥ k − / ≥ / , and so λ ≥ Ω(1 /n ). Together with the lower bound on the standard devia-tion of the size of the independent set drawn according to µ G,λ of Ω( √ λn )from Lemma 9, condition (2) thus implies that k is within some constantnumber r > nα G ( λ ). The centrallimit theorem and standard properties of the normal distribution mean thatthere are constants ρ > r ) and n suchthat for all n ≥ n , with probability at least ρ , | I | is at least r standarddeviations below the mean, and similarly with probability at least ρ it is atleast r standard deviations above the mean. So we have P G,λ ( | I | ≥ k ) ≥ ρ and P G,λ ( | I | ≤ k ) ≥ ρ .The transition probabilities when we are at state I in the Glauber dynam-ics are given by the following random experiment. Choose a vertex v ∈ V ( G )uniformly at random and let I ′ = ( I ∪ { v } with probability λ/ (1 + λ ) , I \ { v } with probability 1 / (1 + λ ) .Now if I ′ is independent in G move to state I ′ , otherwise stay in state I .This means that the sequence of sizes of set visited must take consecutiveinteger values. By Theorem 3, there is a constant C ′′ such that from anarbitrary starting state, in C ′′ n log n steps the distribution π of the currentstate is within total variation distance ρ/ k , with probabilityat least ρ/ C ′′ n log n steps is an independent set ofsize at least k . (ii) Starting from an independent set of size at least k , with probabilityat least ρ/ C ′′ n log n steps is an independent set ofsize at most k .Consider starting from an initial state distributed according to µ G,λ . Thenevery subsequent state is also distributed according to µ G,λ , and the abovefacts mean that for any sequence of C ′′ n log n consecutive steps, with proba-bility at least ρ/ k . Recalling that λ = t/ (2 n ),this immediately implies that µ G,t/ (2 n ) ( I k ( G )) ≥ ρ C ′′ n log n , as required. (cid:3) Now we prove Theorem 4.
Proof.
We first prove the theorem under the assumption that each ˆ µ λ isexactly the hard-core measure µ G,λ , taking note of how many times wesample from any ˆ µ λ .We say a failure occurs at step i in the FOR loop if either of the followingoccur:(1) | nα G ( λ ) − κ | > / | nα G ( λ ) − k | ≤ / k in step i .We show that the probability that a failure occurs at any time during thealgorithm is at most ε/
2. By a union bound, it is enough to show that theprobability of either type of failure at a given step i is at most ε C log n .Consider an arbitrary step i with its value of λ . To bound the quantity P ( | nα G ( λ ) − κ | > / κ is the mean of N independent samplesfrom ˆ µ λ , which we currently assume to be µ G,λ . Then we have E κ = nα G ( λ )and Hoeffding’s inequality gives P ( | nα G ( λ ) − κ | > / ≤ e − N/ (8 n ) , so for this to be at most ε/ (4 C log n ) we need only N ≥ Ω (cid:16) n log (cid:0) log nε (cid:1)(cid:17) . To bound the probability that the current step involves λ such that | nα G ( λ ) − k | ≤ /
2, but we fail to get a set of size k in the N samples,observe that we have N independent trials for getting a set of size k , andeach trial succeeds with probability p ≥ c/ ( n log n ) by Lemma 5. Then theprobability we see no successful trials is (cid:18) − cn log n (cid:19) N , which is at most ε/ (4 C log n ) for N ≥ Ω (cid:16) n log n · log (cid:0) log nε (cid:1)(cid:17) . PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 13
Thus, we can take N = Θ (cid:0) n log (cid:0) log nε (cid:1)(cid:1) , as in line (5) of Sample- k .Next we show that in the event that no failure occurs during the running ofthe algorithm, the algorithm outputs an independent set I with distributionwithin ε/ I k ( G ).We first observe that if no failure occurs, the algorithm at some pointreaches a value of λ so that | nα G ( λ ) − k | ≤ /
2. This is a simple consequenceof Lemma 5, which means there exists some t with this property, and thebinary search structure of the algorithm. In particular, in each iteration ofthe FOR loop, at line (e) the size of the set Λ i being searched goes down by(at least) half. Conditioned on no failures, the search also proceeds in thecorrect half of λ i because we search the upper half only when κ < k − / nα G ( λ ) ≤ κ + 1 / < k and henceusing a larger value of λ must bring nα G ( λ ) closer to k . The case κ > k +1 / λ such that | nα G ( λ ) − k | ≤ / µ G,λ conditioned on getting a set of size exactly k is preciselythe uniform distribution on I k ( G ), hence if the algorithm outputs an inde-pendent set of size k during the FOR loop, its distribution is exactly uniformdistribution on I k ( G ). Thus, under the assumption that each ˆ µ λ is precisely µ G,λ we have shown that with probability at least 1 − ε/ I k ( G ) is output during the FORloop.We do not have access to an efficient exact sampler for µ G,λ , so we make dowith the approximate sampler from Theorem 3. One interpretation of totalvariation distance is that when each ˆ µ λ has total variation distance at most ξ from µ G,λ , there is a coupling between ˆ µ λ and µ G,λ such that the probabilitythey disagree is at most ξ . Then to prove Theorem 4 we consider a thirdfailure condition: that during any of the calls to a sampling algorithm for anyˆ µ λ the output differs from what would have been given by µ G,λ under thiscoupling. Since we make at most CN log n calls to such sampling algorithms,provided ξ ≤ ε/ (2 CN log n ) the probability of any failure of this kind is atmost ε/
2. Together with the above proof for samples distributed exactlyaccording to µ G,λ which successfully returns uniform samples from I k ( G )with probability 1 − ε/
2, we have now shown the existence of a sampler thatwith probability 1 − ε returns uniform samples from I k ( G ), and makes atmost CN log n calls to a ε/ (2 CN log n )-approximate sampler for µ G,λ (atvarious values of λ ). Interpreting this in terms of total variation distance,this means we have an ε -approximate sampler for the uniform distributionon I k ( G ) with running time O ( N log n · T ( n, ε )). (cid:3) Approximate counting via sampling.
Given a graph G = ( V, E )on n vertices and j ≥
0, let f j ( G ) = ( j + 1) i j +1 ( G ) /i j ( G ). This f j ( G ) hasan interpretation as the expected free volume over a uniform random inde-pendent set J ∈ I j ( G ), that is, f j = E | V \ ( J ∪ N ( J )) | . This holds becauseeach vertex in V \ ( J ∪ N ( J )) can be added to J to make an independent set of size j + 1, and each such set is counted j + 1 times in this way. Thenby a simple telescoping product we have(4) i k ( G ) = k − Y j =0 f j ( G ) j + 1 , and hence if for 0 ≤ j ≤ k − ε/k approxima-tion to f j in time polynomial in n and 1 /ε then we can obtain a relative ε -approximation to i k ( G ) in time polynomial in n and 1 /ε . By the defini-tion of f j as an expectation over a uniform random independent set of size j , we can use an efficient sampling scheme for this distribution to approxi-mate f j , which is provided by Theorem 4. That is, by repeatedly samplingindependent sets of size j approximately uniformly and recording the freevolume we can approximate the expected free volume f j ( G ), and hence thecorresponding term of the product in (4). Doing this for all 0 ≤ j ≤ k − i k ( G ). This scheme is an example of simulated annealing , which can be used as a general technique for obtain-ing approximation algorithms from approximate sampling algorithms. Formore details, see e.g. [19, 26]. Here the integer j is playing the role of inversetemperature, and we approximate i k ( G ) by estimating f j ( G ) (by samplingfrom I j ( G )) with the cooling schedule j = 0 , , . . . , k − ≤ j ≤ k − − δ ′ returns a relative ε/k -approximationˆ t j to f j ( G ) / ( j + 1) in time T ′ . Then (4) implies that with probability atleast 1 − kδ , the product ˆ ı k = Q k − j =0 ˆ t j is a relative ε -approximation to i k ( G ),and this takes time kT ′ to compute. For the FPRAS in Theorem 1, ittherefore suffices to design the hypothetical algorithm with δ ′ = 1 / (4 k ) and T ′ polynomial in n and 1 /ε .First, suppose that we have access to an exactly uniform sampler for I j ( G ) for 0 ≤ j ≤ k −
1, but impose the smaller failure probability boundof δ ′ /
2. Then, for each j , let ˆ t j be the sample mean of m computationsof | V \ ( J ∪ N ( J )) | / ( j + 1) where J is a uniform random independent setof size j . We note that as a random variable | V \ ( J ∪ N ( J )) | / ( j + 1) hasa range of at most j ∆ / ( j + 1) in a graph of maximum degree ∆ because0 ≤ | N ( J ) | ≤ j ∆, and for j ≤ k − k ≤ αn < α c (∆) n < e e n ∆ , we have | V \ ( J ∪ N ( J )) | j + 1 ≥ n − j (∆ + 1) j + 1 ≥ ∆ e − . Let S j be the mean of m samples of | V \ ( J ∪ N ( J )) | / ( j + 1). Then, usingthat for ε ′ ≤ | S j − µ | ≤ ε ′ µ/ S j to be a relative PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 15 ε ′ -approximation to µ , by Hoeffding’s inequality, m ≥ Ω( ε − k log(1 /δ ′ )) = Ω( ε − k log k )samples are sufficient to obtain the required approximation accuracy ε ′ withthe required success probability 1 − δ ′ /
2. Since we do not have an exactsampler, we use the approximate sampler obtained in this section with to-tal variation distance δ ′ /
2. Using the coupling between the exact and theapproximate sampler that we used in the proof of Theorem 4, this sufficesto obtain the required sampling accuracy with failure probability at most δ ′ . Recalling that k ≤ n , it is now simple to check that the running time ofthe entire annealing scheme is polynomial in n and 1 /ε . This completes theproof of Theorem 1 (a).2.3. Triangle-free graphs.
Here we prove Theorem 2. We use the follow-ing lower bound on the occupancy fraction of triangle-free graphs.
Theorem 10 ([11]) . For every δ > , there is ∆ large enough so that forevery ∆ ≥ ∆ , and every triangle-free G ∈ G ∆ , α G ( λ c (∆) − / ∆ ) ≥ − δ ∆ . This statement follows from [11, Theorem 3] and some asymptotic analysisof the bound for λ = λ c (∆) − / ∆ as ∆ → ∞ . Now the algorithm forTheorem 2 is essentially the same as for Theorem 1, but since we assume thegraph G is triangle free we can use a stronger lower bound on the occupancyfraction than Theorem 6. Let δ > α < (1 − δ ) / ∆ as in Theorem 2.Then Theorem 10 means that for ∆ ≥ ∆ and any triangle-free graph G ∈G ∆ we have α G ( λ c (∆) − / ∆ ) ≥ − δ ∆ > α . But occupancy fractions are continuous and strictly increasing, so with λ ∗ = λ c (∆) − / ∆ there exists λ ∈ (0 , λ ∗ ] such that k = nα G ( λ ), as in the proofof Lemma 5 but permitting larger α . The analysis of the algorithm can thenproceed exactly as in the proofs of Lemma 5 and Theorem 1.3. Hardness
To prove hardness we will use the notion of an ‘approximation-preservingreduction’ from [13]. We reduce the problem of approximating the hard-corepartition function Z G ( λ ) on a ∆-regular graph G , which we recall is hard for λ > λ c (see [16, 28]), to the problem of approximating i k ( G ′ ) for ∆-regulargraph G ′ that can be constructed in time polynomial in the size of G . Inparticular, we show that it suffices to find an ε/ i k ( G ′ )in order to obtain an ε -approximation to Z G ( λ ).Let IS( α, ∆) be the problem of computing i ⌊ αn ⌋ ( G ) for a ∆-regular graph G on n vertices. Let HC( λ, ∆) be the problem of computing Z G ( λ ) for a∆-regular graph G . Theorem 11.
For every ∆ ≥ and α ∈ ( α c (∆) , / , there exists λ > λ c (∆) so that there is an approximation-preserving reduction from HC( λ, ∆) to IS( α, ∆) . Theorem 11 immediately implies the hardness part of Theorem 1 as theresults of [16, 28] show that there is no FPRAS for HC( λ, ∆) for any λ >λ c (∆) unless NP=RP. Proof of Theorem 11.
Fix ∆ ≥
3, and let α ∈ ( α c (∆) , /
2) be given. Wewill construct a ∆-regular graph H on n H vertices such that for some value λ ∈ ( λ c (∆) , ∞ ) we have(5) α H ( λ ) = α . Our reduction is then as follows: given a ∆-regular graph G on n verticesand ε >
0, let G ′ be the disjoint union of G with H ( r ) , the graph of r disjoint copies of H , with r = ⌈ C ∆ n /ε ⌉ for some absolute constant C . Let N = | V ( G ′ ) | = n + rn H . We will prove that(6) e − ε/ i k ( G ′ ) i k ( H ( r ) ) ≤ Z G ( λ ) ≤ e ε/ i k ( G ′ ) i k ( H ( r ) ) , where k = ⌊ αN ⌋ .Since G ′ can be constructed and i k ( H ( r ) ) computed in time polynomialin n , this provides the desired approximation-preserving reduction. Whatremains is to construct the graph H satisfying (5) and then to prove (6). Constructing H . The graph H = H a,b will consist of the union of a copiesof the complete bipartite graph K ∆ , ∆ and b copies of the clique K ∆+1 .Clearly H is ∆-regular. We can compute α H a,b ( λ ) = a λ (1+ λ ) ∆ − λ ) ∆ − + b (∆+1) λ λ a ∆ + b (∆ + 1) . Since the occupancy fraction of any graph is a strictly increasing function of λ , α K ∆+1 ( λ c (∆)) = α c (∆), and lim λ →∞ α K ∆ , ∆ ( λ ) = 1 /
2, we see that thereexist integers a, b ≥ λ > λ c (∆) so that α H a,b ( λ ) = α . A given pair ( a, b ) provides a suitable H a,b when α H a,b ( λ c (∆)) < α < lim λ →∞ α H a,b ( λ ) = a ∆ + b a ∆ + b (∆ + 1) , and hence it can be shown that for all ∆ ≥ , , , , , , ,
0) suffices for ( a, b ), and a suitable pair is easyto find efficiently. This provides us with the desired graph H . From hereon, fix these values a, b, λ and let n H = 2 a ∆ + b (∆ + 1). PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 17
Proving (6) . We now form G ′ by taking the union of G (a ∆-regular graphon n vertices) and r copies of H . Let N = n + rn H be the number of verticesof G ′ , and write k = ⌊ αN ⌋ . Let H ( r ) be the union of r copies of H . We canwrite: i k ( G ′ ) = n X j =0 i j ( G ) i k − j ( H ( r ) )= i k ( H ( r ) ) n X j =0 i j ( G ) i k − j ( H ( r ) ) i k ( H ( r ) ) . Now to prove (6) it suffices to show that for r ≥ C ∆ n /ε and 0 ≤ j ≤ n , wehave(7) e − ε/ λ j ≤ i k − j ( H ( r ) ) i k ( H ( r ) ) ≤ e ε/ λ j . We have the exact formula (for any 0 ≤ j ≤ k ) i k − j ( H ( r ) ) = Z H ( r ) ( λ ) λ k − j P H ( r ) ,λ ( | I | = k − j )and so i k − j ( H ( r ) ) i k ( H ( r ) ) = λ j P H ( r ) ,λ ( | I | = k − j ) P H ( r ) ,λ ( | I | = k ) , where P H ( r ) ,λ denotes probabilities with respect to an independent set I drawn according to the hard-core model on H ( r ) at fugacity λ . It is thenenough to show e − ε/ ≤ P H ( r ) ,λ ( | I | = k − j ) P H ( r ) ,λ ( | I | = k ) ≤ e ε/ . This will follow from a Local Central Limit theorem (e.g. [17]) since | I | isthe sum of r i.i.d. random variables and the fact that E H ( r ) ,λ | I | is close toboth k and k − j . The following theorem gives us what we need. Theorem 12 (Gnedenko [17]) . Let X , . . . , X r be i.i.d. integer valued ran-dom variables with mean µ and variance σ , and suppose that the supportof X includes two consecutive integers. Let S r = X + · · · + X r . Then P ( S r = k ) = 1 √ πrσ exp (cid:2) − ( k − nµ ) / (2 rσ ) (cid:3) + o ( r − / ) , with the error term o ( r − / ) uniform in k . This immediately implies that with µ and σ the mean and standarddeviation of the hard-core model on H at fugacity λ , P H ( r ) ,λ ( | I | = k − j ) P H ( r ) ,λ ( | I | = k ) = e − [ j − k − rµ ) j ] / (2 rσ ) + o ( e ( k − rµ ) / (2 rσ ) /r )1 + o ( e ( k − rµ ) / (2 rσ ) /r ) . It therefore suffices to show that for large enough r , namely r ≥ C ∆ n /ε ,we can make [ j − k − rµ ) j ] / (2 rσ ) small compared to ε and show that( k − rµ ) / (2 rσ ) is bounded above by some absolute constant. Note that µ = αn H , and by Lemma 9 we have for all ∆ ≥ α, λ, a, b made according to our conditions), σ ≥ λ (1 + λ ) ( a ∆ + b ) ≥ λ c (∆)(1 + λ c (∆)) ( a ∆ + b ) ≥ . . Since k = ⌊ αN ⌋ = ⌊ αn + rαn H ⌋ , we then have ( k − rµ ) ≤ α n < n , andhence ( k − rµ ) rσ ≤ C ′ ∆ n r , where C ′ is an absolute constant. Now since 0 ≤ j ≤ n we also have (cid:12)(cid:12)(cid:12)(cid:12) j − k − rµ ) j rσ (cid:12)(cid:12)(cid:12)(cid:12) ≤ C ′ ∆ n r . This means that provided we take C to be a large enough absolute constantand r ≥ C ∆ n /ε , we have (6) as required. (cid:3) Triangle-free graphs.
The proof of hardness for triangle-free graphsis the same, but we replace K ∆+1 with a (constant-sized) random regulargraph in the construction. Bhatnagar, Sly, and Tetali [4] showed that thelocal distribution of the hard-core model on the random regular graph con-verges to that of the unique translation-invariant hard-core measure on theinfinite regular tree for a range of λ including λ = λ c (∆). This means thatif K is a random ∆-regular graph on n vertices and α T ∆ denotes the occu-pancy fraction of the unique translation-invariant hard-core measure on theinfinite ∆-regular tree (see [4, 11]) we have with probability 1 − o n (1), α G ( λ c (∆)) = α T ∆ ( λ c (∆)) + o n (1) = 1 + o n, ∆ (1)∆ , where o n (1) → n → ∞ and o n, ∆ (1) → n and ∆ tend toinfinity. Thus, for fixed δ ∈ (0 , n = n ( δ ) and ∆ = ∆ ( δ )such that with probability at least 1 − δ a random ∆-regular graph K on n vertices has α G ( λ c (∆)) ≤ (1 + δ ) / ∆. This means that in time bounded by afunction of δ an exhaustive search over ∆-regular graphs on n vertices mustyield a K with the property α K ( λ c (∆)) ≤ (1 + δ ) / ∆. Now we replace K ∆+1 with the random ∆-regular graph K in the proof above, which for ∆ ≥ ∆ allows us to work with any α ∈ ((1 + δ ) / ∆ , /
2) by the above argument. Tofinish the proof, we require that approximating Z G ( λ ) is hard for ∆-regular triangle-free graphs G when λ > λ c . This follows directly from the proof ofSly and Sun [28], as their gadget which shows hardness for ∆-regular graphscontains no triangles. Thus, we have the following analogue of Theorem 11,where we let IS ′ ( α, ∆) be the problem of computing i ⌊ αn ⌋ ( G ) for a ∆-regulartriangle-free graph G on n vertices. PPROXIMATELY COUNTING INDEPENDENT SETS OF A GIVEN SIZE 19
Theorem 13.
Given δ > there exists ∆ such that the following holds forall ∆ ≥ ∆ . For every α ∈ ((1 + δ ) / ∆ , / , there exists λ > λ c (∆) so thatthere is an approximation-preserving reduction from HC( λ, ∆) to IS ′ ( α, ∆) . This implies Theorem 2 (b).
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Department of Computer Science, University of Colorado Boulder, USA
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