Counting hypergraph matchings up to uniqueness threshold
Counting hypergraph matchings up to uniqueness threshold
arXiv:1503.05812v2 [cs.DS] 16 Jul 2015
Yitong Yin∗ Nanjing University, China [email protected]
Jinman Zhao Nanjing University, China [email protected]
Abstract We study the problem of approximately counting hypergraph matchings with an activity parameter λ in hypergraphs of bounded maximum degree and bounded maximum size of hyperedges. This problem unifies two important statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). dd We show for this model the critical activity λc = k(d−1) d+1 is the threshold for the uniqueness of Gibbs measures on the infinite (d + 1)-uniform (k + 1)-regular hypertree. And we show that when λ < λc the model exhibits strong spatial mixing at an exponential rate and there is an FPTAS for the partition function of the model on all hypergraphs of maximum degree at most k + 1 and maximum edge size at most d + 1. Assuming NP6=RP, there is no FPRAS for the partition function of the model when λ > 2λc on above family of hypergraphs . Towards closing this gap and obtaining a tight transition of approximability, we study the local weak convergence from an infinite sequence of random finite hypergraphs to the infinite uniform regular hypertree with specified symmetry, and prove a surprising result: the existence of such local convergence is fully characterized by the reversibility of the uniform random walk on the infinite hypertree projected on the symmetry classes. We also give an explicit construction for the sequence of random finite hypergraphs with proper local convergence property when the reversibility condition is satisfied.
1
Introduction
The problems of approximate counting have long been studied as statistical physics models, and an exciting accomplishment was connecting the computational complexity of approximate counting to the phase transition in the statistical physics model. The hardcore model and the monomerdimer model are perhaps the two simplest and most important statistical physics models studied for approximate counting. In the hardcore model, given a graph G = (V, E) and a vertex-activity λ, the model assigns each independent set I of G a weight wλIS (I) = λ|I| . A natural probability distribution, call the IS IS IS Gibbs measure, can be defined over P allIS independent sets of G as µλ (I) = wλ (I)/Zλ (G) where IS the normalizing factor Zλ (G) = I wλ (I) that takes the sum over all independent sets of G is the so-called partition function. In the monomer-dimer mode, given a graph G = (V, E) and an edge-activity λ, the model assigns each matching M of G a weight wλM (M ) = λ|M | . The Gibbs measure over all matchings of G can be accordingly defined, and the partition function now becomes ∗
State Key Laboratory for Novel Software Technology, Nanjing University, China. Supported by NSFC grants 61272081 and 61321491.
1
P ZλM (G) = M wλM (M ) where the sum is taken over all matchings of G. The approximate counting of (weighted) independent sets and (weighted) matchings of G can then be formulated respectively as approximately computing the partition functions ZλIS (G) and ZλM (G). It was well known that the hardcore model exhibits the following phase transition on the infinite (d+1)-regular tree: there is a critical activity λc (d) = dd /(d−1)d+1 , usually referred in the literature as the uniqueness threshold, such that the correlation between the marginal distribution at the root of the tree and any condition on all the vertices at distance t from the root decays exponentially in t when λ < λc (d), but the boundary-to-root correlation remains substantial even as t → ∞ when λ > λc (d). This property of correlation decay is also called spatial mixing, and was known to be equivalent to the uniqueness of the infinite-volume Gibbs measure on the infinite (d + 1)-regular tree [30]. In a seminal work [31], Weitz showed that for all λ < λc (d) the decay of correlation holds for the hardcore model on all graphs of maximum degree bounded by d + 1 and there is a deterministic FPTAS for the partition function of the hardcore model on all such graphs. Here the specific notion of decay of correlation established is the strong spatial mixing, which requires the boundary-to-vertex correlation have exponential decay even conditioning on the states of vertices from a subset being arbitrarily fixed. The implication of the decay of correlation on regular tree to FPTAS on graphs with bounded maximum degree for the hardcore model was established by a construction called self-avoiding walk (SAW) tree due to Weitz [31]. This connection between approximability of partition function and the phase transition of the model is further strengthened in a series of works [6,7,27,28] which show that unless NP=RP there is no FPRAS for the partition function of the hardcore model when λ > λc (d) on graphs with maximum degree bounded by d + 1. For the monomer-dimer model, it was well known that the model has no such phase transition as above [11, 12]. And analogously the well known result of Jerrum and Sinclair [13] gives an FPRAS for the partition function of the monomer-dimer model on all graphs. In [1] strong spatial mixing with an exponential rate was established for the model on all graphs with maximum degree bounded by an arbitrary constant and a deterministic FPTAS was also given for the partition function of the monomer-dimer model on all such graphs. In this paper, we study spatial mixing and approximate counting of hypergraph matchings, a model that unifies both the hardcore model and the monomer-dimer model. A hypergraph H = (V, E) consists of a vertex set V and a collection E of vertex subsets, called the hyperedges. A matching of H is a set M ⊆ E of disjoint hyperedges in H. Given a hypergraph H and an activity parameter λ > 0, a configuration is a matching M of H, and is assigned a weight wλ (M ) = λ|M | . The Gibbs measure over all matchings of H is defined as µ(M ) = wλ (M )/Zλ (H), where the normalizing factor Zλ (H) is the partition function for the model, defined as: X Zλ (H) = λ|M | . M : matching of H
This model represents an important subclass of Boolean constraint satisfaction problem (CSP): that is, the CSPs defined by the matching (packing) constraints. It also unifies the hardcore model and the monomer-dimer model in the following sense. Consider the family of hypergraphs of maximum edge size at most d + 1 and maximum degree at most k + 1: • When d = 1, the model becomes the monomer-dimer model on graphs of maximum degree at most k + 1. • When k = 1, the matchings of hypergraph H correspond to independent sets of its dual graph, and the model becomes the hardcore model on graphs of maximum degree at most d + 1. 2
For the general model on hypergraphs, the problem has only been studied when λ = 1, i.e. counting the number of matchings in a hypergraph. In a recent work, Liu and Lu [19] proves the decay of correlation and the existence of FPTAS for the matchings in 3-uniform hypergraphs of maximum degree at most 4. A similar result is proved for 3-uniform hypergraphs of maximum degree at most 3 in an independent work [4]. Both results were based on case-studies of correlation decay for specific settings of edge sizes and degrees which are insufficient to establish a transition with continuous λ. Our contributions.
We show that for hypergraph matchings the λc (d, k) =
dd k(d−1)d+1
is the
uniqueness threshold for Gibbs measure on the infinite uniform regular hypertree. Indeed, let Tkd denote the infinite (d+1)-uniform (k +1)-regular hypertree. There is a way to classify all vertices in Tkd into 2 equivalent classes and define two simple Gibbs measures µ+ , µ− on Tkd which are invariant under all class-preserving automorphisms (see Section 3) such that the followings hold. d
d k Proposition 1.1. The Gibbs measures µ+ = µ− if and only if λ ≤ λc = k(d−1) d+1 . Further, on Td there is a unique Gibbs measure on hypergraph matchings with activity λ if and only if λ ≤ λc . d
d This covers as special cases the well-known uniqueness threshold λc (d, 1) = (d−1) d+1 for the hardcore model on the infinite (d + 1)-regular tree and also the lack of phase-transition for the monomer-dimer model. The Gibbs measures µ± generalize the extremal semi-translation invariant Gibbs measures (which are invariant under all parity-preserving automorphisms on the infinite regular tree) for the hardcore model. The two equivalent classes for vertices in the definition of µ± play similar roles as the parities in the semi-translation invariant Gibbs measure. We then establish the decay of correlation for hypergraph matchings on all hypergraphs with bounded maximum edge size and bounded maximum degree when the activity λ is in the interior of the uniqueness region for uniform regular hypertrees. The specific notion of decay of correlations that we establish is the strong spatial mixing [31] (see Section 2 for a precise definition). Consequently, we give an FPTAS for the partition function for the hypergraphs with bounded maximum edge size and bounded maximum degree as long as the uniqueness condition is satisfied.
Theorem 1.2. For every finite integers d, k ≥ 1 and any λ < λc , for the model of hypergraph matchings with activity λ on all hypergraphs of maximum edge size at most d + 1 and maximum degree at most k + 1, we have: • the model exhibits strong spatial mixing with an exponential rate; and • there exists an FPTAS for computing the partition function. Remark. The theorem unifies the strong spatial mixing and correlation-decay based FPTAS for the hardcore model [31] and the monomer-dimer model [1], and also covers as special cases the results for approximate counting hypergraph matchings in [4,19]. Further, the runningtime for the O dkˆx ln kd 1−(d−1)kˆ x FPTAS for outputting an -approximation of the partition function is n where x ˆ is −d the fixed point solution to the tree recursion x ˆ = f (ˆ x) = λ (1 + kˆ x) , achieving the best running time (if ever explicitly specified) for the deterministic algorithms in [1, 4, 19, 31]. As far as we know, Theorem 1.2 is the first time for a non-spin system that the approximability of the model is established as its parameter approaching the phase-transition threshold. We achieve 3
this by constructing a hypergraph version of Weitz’s self-avoiding walk tree [31]. As a by-product, it proves that for hypergraph matchings, the worst case for strong spatial mixing among all hypergraphs of bounded maximum edge size and bounded maximum degree, is achieved by uniform regular hypertrees. Such phenomenon was once believed to be true for most statistical physics models, but had been found false for even some simple spin systems [17, 26]. Here we show as an example for general CSPs, that the worst cases for strong spatial mixing are indeed regular trees. Proposition 1.3. If the model of hypergraph matchings with activity λ exhibits strong spatial mixing with rate δ on Tkd , then the model exhibits strong spatial mixing with the same rate on all hypergraphs of maximum edge size at most d + 1 and maximum degree at most k + 1. For the hardness of approximation, due to a simple reduction from the hardcore model, it is k easy to show that if λ > 2k+1+(−1) λc ≈ 2λc , then unless NP=RP there is no FPRAS for the k+1 partition function for the family of hypergraphs stated in Theorem 1.2. We then explore the possibility of closing this gap between λc and 2λc and finding the tight condition for the transition of approximability. The coincidence between the physical phase transition of Gibbs measures on the infinite regular tree and the transition of computational complexity on finite graphs is due to the local weak convergence: Gibbs measures on an infinite sequence of (random) finite graphs weakly converges locally to the Gibbs measure on the infinite regular tree. Consequently, the nonuniqueness of Gibbs measure on the infinite tree may imply the multimodality of local distributions in the finite graphs, which makes the local Markov chain torpid mixing on the sequence of finite graphs [5, 22] and also makes the sequence of finite graphs become suitable gadgets for the inapproximability reduction [6–8, 27, 28]. The key to the local weak convergence of measures is the local convergence of graph structures: for spin systems, this is represented by the property of being locally tree-like. More precisely, not only the graph topology of the sequence of finite graphs needs to converge locally to that of the infinite tree, but also the finite graphs should preserve the symmetry of the Gibbs measures on the infinite tree that exhibit the critical phase transition. For example, for anti-ferromagnetic 2-spin systems, the transition of uniqueness/nonuniqueness on the infinite tree is achieved by semi-translation invariant Gibbs measures — the Gibbs measures which are invariant under parity-preserving automorphisms on the infinite tree, and hence the sequence of finite graphs which converges locally to the infinite tree needs to be locally like the tree structure along with the parity assignment, which corresponds to the random regular bipartite graphs as a hardness gadget for anti-ferromagnetic spin system [5–8, 22, 27, 28]. Such graphs locally resembles the infinite regular tree with the proper symmetry (preserving the parity of vertices) under which the invariant Gibbs measure achieves the phase-transition threshold. For general CSPs the underlying structures are hypergraphs instead of graphs. And Proposition 1.3 suggests conceptually that for our model the infinite uniform regular hypertrees Tkd should play a role as the infinite regular trees to the spin system. Our next result characterizes Tkd with specific symmetry which allows such locally convergent sequences. Unlike trees, on hypertrees there is no parity-preserving symmetry, and the symmetries on hypertrees may be much more complicated. We resolve this by introducing a hypergraph version of the branching matrix of [23] and proving it fully characterize the symmetry on Tkd . With the help of this notion of branching matrices, we prove a surprising result: the existence of local convergence from any sequence of finite hypergraphs to a uniform regular hypertree with certain symmetry is determined precisely by the reversibility of a natural random walk over equivalent classes of vertices and hyperedges in the infinite hypertree (see Section 7 for more precise definitions). 4
Theorem 1.4 (informal). There exist a sequence Hn of random finite hypergraphs that converge locally to the infinite (d + 1)-uniform (k + 1)-regular hypertree Tkd with labeling that are invariant under an automorphism group G if and only if the projection of the uniform random walk on Tkd onto the orbits of G is time-reversible. The proof of this theorem is model-independent, thus the theorem holds generally for all CSPs where local convergence to the infinite hypertree is concerned. We also explicitly construct a hypergraph sequence Hn with the proper local convergence when the time-reversibility is satisfied. The construction has potential to be used as gadgets for general CSPs and is interesting by itself. Applying Theorem 1.4 gives us the following “barrier” result: there is no local convergence from any sequence of finite hypergraphs to Tkd with the symmetry assumed by the family of Gibbs measures µ+ , µ− that achieve the uniqueness/nonuniqueness threshold. Proposition 1.5. There does not exit any sequence of finite hypergraphs that converge locally to Tkd with the symmetry assumed by the Gibbs measures µ+ , µ− in Proposition1.1. Proposition 1.1, 1.3 and 1.5 combined suggest: Even for this very simple class of counting CSP, in order to precisely classify its approximability, we may need either algorithms that do not rely on the mixing of Gibbs measure, or hardness that holds beyond the current framework of local weak convergence of Gibbs measures, or both. Remark on exposition. For convenience of visualizing the results, all our results in the rest of the paper are presented for independent sets in the dual hypergraphs. Note that matchings are equivalent to independent sets under hypergraph duality. And the only effect of duality on a family of hypergraphs with bounded maximum edge size and bounded maximum degree is to switch the bounds on the edge size and the degree. We stress that we adopt a stronger version of the definition of hypergraph independent set: a vertex subset I ⊆ V is an independent set of a hypergraph H = (V, E) if no two vertices in I are contained in the same hyperedge. This should be distinguished from a weaker definition of hypergraph independent set (e.g. the one used in [2]), which only needs any independent set I to contain no hyperedge as subset. Related works. Approximate counting of hypergraph matchings was studied in [14] for hypergraphs with restrictive structures, and in [4, 19] for hypergraphs with bounded edge size and maximum degree. In [2], approximate counting of a variant of hypergraph independent sets was studied, where an independent set only need to not contain any hyperedge. The spatial mixing (decay of correlation) is already a widely studied topic in Computer Science, because it may support FPTAS for #P-hard counting problems. The decay of correlation was established via the self-avoiding walk tree for the hardcore model [25,31], monomer-dimer model [1, 24], and two-spin systems [16, 17, 24]. Similar tree-structured recursions were employed to prove the decay of correlation for multi-spin systems [9, 10, 21] and more general CSPs [18–20].
2
Preliminary
For a hypergraph H = (V, E), the size of a hyperedge e ∈ E is its cardinality |e|, and the degree of a vertex v ∈ V , denoted by deg (v) = degH (v), is the number of hyperedges e ∈ E incident to v, i.e. satisfying v ∈ e. A hypergraph H is k-uniform if all hyperedges are of the same size k, and is 5
d-regular if all vertices have the same degree d. The incidence graph of a hypergraph H = (V, E) is a bipartite graph with V and E as vertex sets on the two sides, such that each (v, e) ∈ V × E is a bipartite edge if and only if v is incident to e. A matching of hypergraph H = (V, E) is a set M ⊆ E of disjoint hyperedges in H. Given an activity parameter λ > 0, the Gibbs measure is a probability distribution over matchings of M M H proportional to the weightPwλM (M ) = λ|M | , defined as µM λ (M ) = wλ (M )/Zλ (H), where the normalizing factor ZλM (H) = M wλM (M ) is the partition function. Similarly, an independent set of hypergraph H = (V, E) is a set I ⊆ V of vertices satisfying |I ∩ e| ≤ 1 for all hyperedges e in H. The Gibbs measure over independent sets of H with activity λ > 0 is given by µIS λ (I) =
wλIS (I) λ|I| = IS , IS Zλ (H) Zλ (H)
(1)
P where the normalizing factor ZλIS (H) = I wλIS (I) is the partition function for independent sets of H with activity λ. Independent sets and matchings are equivalent under hypergraph duality. The dual of a hypergraph H = (V, E), denoted by H∗ = (E ∗ , V ∗ ), is the hypergraph whose vertex set is denoted by E ∗ and edge set is denoted by V ∗ , such that every vertex v ∈ V (and every hyperedge e ∈ E) in H is one-to-one corresponding to a hyperedge v ∗ ∈ V ∗ (and a vertex e∗ ∈ E ∗ ), such that e∗ ∈ v ∗ if and only if v ∈ e. Note that under duality, matchings and hypergraphs are the same CSP and hence result in the same Gibbs measure, which remains to be true even with activity λ. Also a family of hypergraphs of bounded maximum edge size and bounded maximum degree is transformed under duality to a family of hypergraphs with the bounds on the edge size and degree exchanged. Remark 2.1. With the above equivalence under duality, from now on we state all our results in terms of the independent sets in the dual hypergraph and omit the superscript ·IS in notations. Given the Gibbs measure over independent sets of hypergraph H and a vertex v, we define the marginal probability pv as pv = pH,v = Pr[v ∈ I] which is the probability that v is in an independent set I sampled from the Gibbs measure (such a vertex is also said to be occupied ). Given a vertex set Λ ⊂ V , a configuration is a σΛ ∈ {0, 1}Λ which corresponds to an independent set IΛ partially specified over Λ such that σΛ (v) indicates whether a v ∈ Λ is occupied by the independent set. We further define the marginal probability Λ pσH,v as Λ pσv Λ = pσH,v = Pr[v ∈ I | IΛ = σΛ ] which is the probability that v is occupied under the Gibbs measure conditioning on the configuration of vertices in Λ ⊂ V being fixed as σΛ . Definition 2.1. The independent sets of a finite hypergraph H = (V, E) with activity λ > 0 exhibit weak spatial mixing (WSM) with rate δ : N → R+ if for any v ∈ V , Λ ⊆ V , and any two configurations σΛ , τΛ ∈ {0, 1}Λ which correspond to two independent sets partially specified on Λ, |pσv Λ − pτvΛ | ≤ δ(distH (v, Λ)), where distH (v, Λ) is the shortest distance between v and any vertex in Λ in hypergraph H. 6
Definition 2.2. The independent sets of a finite hypergraph H = (V, E) with activity λ > 0 exhibit strong spatial mixing (SSM) with rate δ : N → R+ if for any v ∈ V , Λ ⊆ V , and any two configurations σΛ , τΛ ∈ {0, 1}Λ which correspond to two independent sets partially specified on Λ, |pσv Λ − pτvΛ | ≤ δ(distH (v, ∆)), where ∆ ⊆ Λ stands for the subset on which σΛ and τΛ differ and distH (v, ∆) is the shortest distance between v and any vertex in ∆ in hypergraph H. The definitions of WSM and SSM extend to infinite hypergraphs with the same conditions to be satisfied for every finite region Ψ ⊂ V conditioning on the vertices in ∂Ψ being unoccupied.
3
Gibbs measures on the infinite tree
We follow Remark 2.1 and state our discoveries in terms of independent sets in the dual hypergraphs. Let Tdk be the infinite (k +1)-uniform (d+1)-regular hypertree, whose incidence graph is the infinite tree in which all vertices with parity 0 are of degree (k + 1) and all vertices with parity 1 are of degree (d + 1). A probability measure µ on hypergraph independent sets of Tdk is Gibbs if for any finite sub-hypertree T , conditioning µ upon the event that all vertices on the outer boundary of T are unoccupied gives the same distribution on independent sets of T as defined by (1) with H = T . We further consider the simple Gibbs measures satisfying conditional independence: Conditioning µ on a configuration of a subset Λ of vertices results in a measure in which the configurations on the components separated by Λ are independent of each other. The Gibbs distribution on a finite hypergraph is always simple. A Gibbs measure on Tdk is translation-invariant if it is invariant under dd all automorphisms of Tdk . We will show that λc (d, k) = k(d−1) d+1 is the uniqueness threshold for the Gibbs measures on hypergraph independent sets of Tdk . Theorem 3.1. There is always a unique simple translation-invariant Gibbs measure on independent dd sets of Tdk . Let λc = λc (d, k) = k(d−1) d+1 . There is a unique Gibbs measure on independent sets of Tdk with activity λ if and only if λ ≤ λc . This proves the uniqueness threshold stated in Proposition 1.1. The construction of extremal Gibbs measures µ+ , µ− can be described with the notion of branching matrices.
3.1
Branching matrices
Fix an automorphism group G of Tdk . A G-translation-invariant Gibbs measure on Tdk is a measure that is invariant under all automorphisms from G. For example, the semi-translation-invariant Gibbs measures on regular tree are invariant under all parity-preserving automorphisms on Td1 . The natural group actions of G respectively on vertices and hyperedges partition the sets of vertices and hyperedges into orbits. Let τv and τe be the respective numbers of orbits for vertices and hyperedges. For each i ∈ [τv ], we say a vertex is of type-i if it is in the i-th orbit for vertices; and the same also applies to hyperedges. Assuming the symmetry on Tdk given by automorphism group G, the hypergraph branching matrices, or just branching matrices, are the following two nonnegative integral matrices: D = D τv ×τe = [dij ] and K = K τe ×τv = [kji ], which satisfy that for any i ∈ [τv ] and j ∈ [τe ]: 7
• every vertex in Tdk of type-i is incident to precisely dij hyperedges of type-j; • every hyperedge in Tdk of type-j contains precisely kji vertices of type-i. The D and K are transition matrices from vertex-types to hyperedge-types and vice versa in The definition can be seen as a hypergraph generalization of the branching matrix for multitype Galton-Watson tree [23]. Since types (orbits) are invariant under all automorphisms from G, it is clear that the above D and K are well-defined for every automorphism group G on Tdk with finitely many orbits. Tdk .
Proposition 3.2. Every automorphism group G on Tdk with finitely many orbits has a pair of P branching matrices D and K which are as described above and satisfy: (1) j dij = d + 1 and P k = k + 1; (2) dij = 0 if and only if kji = 0; and (3) DK and KD are irreducible. i ji Conversely, any pair of nonnegative integral matrices D and K satisfying these conditions are branching matrices for some automorphism group G on Tdk . Proof. Let G be an automorphism group on Tdk with finitely many P to see that P orbits. It is trivial the branching matrices D and K are well-defined and satisfy j dij = d + 1 and i kji = k + 1. A vertex v of type-i is incident to a hyperedge e of type-j if and only if e of type-j contains a vertex v of type i, thus kji 6= 0 if and only if dij 6= 0. 0 D , which is a consequence The irreducibility of DK and KD follows that of the matrix K 0 to the that every type of vertex and hyperedge is accessible from all other types of vertices and hyperedges, which follows the simple fact that the incidence graph Tdk is strongly connected. Conversely, let D and K be a pair of nonnegative integral matrices satisfying the conditions above. We can start from any vertex (or hyperedges) o of type-i and construct an infinite hypertree rooted at o with each vertex and hyperedge labeled with the respective type according to the rules specified by the branching Pmatrices D and K. P Since dij = 0 if and only if kji = 0, the construction is always possible. Since j dij = d + 1 and i kji = k + 1, the resulting infinite hypertree must be k-uniform and d-regular. Since DK and KD are irreducible, no matter how we choose the type for the root o, the resulting hypertree contains all types of vertices and hyperedges. We can then construct an automorphism group G on Tdk according with orbits being the types just specified. For every pair of vertices (or hyperedges) u, v with the same type, by generating the hypertree according to D, K starting from u and v respectively, we obtain an automorphism φu→v on Tdk which maps u to v and preserves the types of all vertices and hyperedges. Let G = h {φu→v | ∀u, v with the same type} i be the group generated from all such automorphisms. Then D and K are branching matrices for automorphism group G on Tdk .
3.2
Extremal Gibbs measures
Assume that there are two vertex-types and two hyperedge-types, both denoted as {+, −}, and the 1 d k 1 b = c= branching matrices are defined as D and K , i.e.: d 1 1 k 1. every ‘±’-vertex is incident to a ‘±’-hyperedge and d ‘∓’-hyperedges; 2. every ‘±’-hyperedge contains k ‘±’-vertices and a ‘∓’-vertex. 8
Figure 1: Classifying vertices and hyperedges of T23 into two types ‘+’(black) and ‘−’(white). The hypergraph is represented as its incidence graph where circles stand for vertices and squares stand for hyperedges.
See Figure 1 for an illustration. Fix a ‘+’-vertex v in Tdk as the root. Let µ+ (resp. µ− ) be d the Gibbs measure on Tk defined by conditioning on all vertices to be unoccupied for the t-th ‘−’-vertices (resp. ‘+’-vertices) along all path from the root and taking the weak limit as t → ∞. b and K, c on any path any ‘±’-vertex has a ‘∓’-vertex within Note that for the 2-coloring given by D 2 steps, so the limiting sequence is well-defined. And by symmetry, starting from a root of type-‘−’ gives the same pair of measures. The µ± generalize the extremal semi-translation-invariant Gibbs measures on infinite regular trees. For hypertree Tdk with k ≥ 2, there are no parity-preserving automorphisms. Nevertheless, b and K c generalizes the parity-preserving automorphisms to hypertrees the symmetry given by D and has the similar phase-transition as semi-translation-invariant Gibbs measures on trees. The b b with orbits given by µ± are simple and are G-translation-invariant for the automorphism group G b b and K. c In fact, they are extremal G-translation-invariant D Gibbs measures on Tdk . We will + − see that the model has uniqueness if and only if µ = µ , which together with Theorem 3.1 gives Proposition 1.1.
3.3
Uniqueness of Gibbs measures
Lemma 3.3. Let µ be a simple Gibbs measure on independent sets of Tdk . Let v be a vertex in Tdk and vij the j-th vertex (besides v) in the i-th hyperedge incident to v, for i = 1, 2, . . . , d + 1 and j = 1, 2, . . . , k. Let pv = µ[ v is occupied ] and pvij = µ[ vij is occupied ]. It holds that d+1 k Y X 1 − pv − pv = λ(1 − pv )−d pvij . (2) j=1
i=1
Proof. Since µ is a Gibbs measure, for any vertex v in Tdk , it holds that pv = µ[ v is occupied ] =
λ · µ[ all the neighbors of v are unoccupied ] 1+λ 9
On the other hand, since µ is simple, conditioning on the root being unoccupied the sub-hypertrees are independent of each other, thus µ[ all the neighbors of v are unoccupied ] =µ[ v is occupied ] · µ[ all the neighbors of v are unoccupied | v is occupied ] + µ[ v is unoccupied ]
d+1 Y
µ[ ∀1 ≤ j ≤ k, vij is unoccupied | v is unoccupied ]
i=1
=pv + (1 − pv )
d+1 Y i=1
1 −
k X
µ[ vij is occupied | v is unoccupied ] .
j=1
Note that for any two adjacent vertices v, vij , we have µ[ vij is occupied ] = µ[ vij is occupied | v is unoccupied ] · µ[ v is unoccupied ], thus µ[ vij is occupied | v is unoccupied ] =
pvij µ[ vij is occupied ] = . 1 − µ[ v is occupied ] 1 − pv
The lemma follows by combining everything together. Equation (2) gives an infinite system involving all vertices in Tdk . If the simple Gibbs measure µ is G-translation-invariant for some automorphism group G on Tdk , the marginal probability pv = µ[ v is occupied ] depends only on the type (orbit) of v. Corollary 3.4. Let µ be a simple G-translation-invariant Gibbs measure on Tdk with branching matrices D τv ×τe = [dij ] and K τe ×τv = [kji ]. For every i ∈ [τv ], let pi = µ[ v is occupied ] for vertex v in Tdk of type-i. It holds for every s ∈ [τv ] that dij
ps = λ(1 − ps )−d
Y
1 −
j∈[τe ]
X
kji · pi
.
i∈[τv ]
b and K c defined in Section 3.2, the system in CorolApplying with the branching matrices D lary 3.4 becomes ( p+ = λ(1 − p+ )−d (1 − k p+ − p− )(1 − p+ − k p− )d , p− = λ(1 − p− )−d (1 − k p− − p+ )(1 − p− − k p+ )d . ( y = f (x) kp+ kp− kλ Let x = 1−p− −k p+ and y = 1−p+ −k p− . The system becomes , where f (x) = (1+x) d x = f (y) is the hardcore tree-recursion. Since f (x) is positive and decreasing in x, it follows that there is a unique positive x ˆ such that x ˆ = f (ˆ x), which means there is always a unique simple translationd invariant Gibbs measure on Tk . It is well-known (see [7] and [15, 29]) the system has three distinct solutions (ˆ x, x ˆ), (x+ , x− ) and (x− , x+ ) where 0 < x− < x ˆ < x+ , when kλ > dd /(d − 1)d+1 , i.e. λ > λc (d, k); and the three solutions collide into a unique solution (ˆ x, x ˆ) when λ ≤ λc (d, k), which means b there is a unique simple G-translation-invariant Gibbs measure on Tdk if and only if λ ≤ λc . The only b if part of Theorem 3.1 is proved. Recall that µ± are simple and are extremal G-translation-invariant + − Gibbs measures, and hence it also holds that µ = µ if and only if λ ≤ λc (d, k). 10
For the if part of Theorem 3.1, we first prove the weak spatial mixing of the model on Tdk b d . According to the when λ ≤ λc . Consider instead the (k + 1)-uniform d-ary hypertree T = T k Qd tree recursion (3) in Section 4, we have RT = λ i Pk1 where RT represents the ratio 1+
j=1
RTij
between probabilities of being occupied and unoccupied at the root of T and Tij represents the sub-hypertree rooted by the j-th child in the i-th incident hyperedge of the root of T , which is isomorphic to T . Due to the monotonicity of the recursion, the extremal boundary conditions (the boundary conditions that achieve the maximum and minimum of RT ) at level t are that exactly one of each hyperedge at level t is occupied in each hyperedge at level t, and that all vertices at level t are unoccupied respectively. By symmetry, the system is then simplified to that Rt = (1+kRλ )d . t−1
Substituting xt = kRt we have xt =
kλ . (1+xt−1 )d
It is well known [15, 29] that the system converges
to its unique fixed point regardless of the initial value as long as kλ ≤ dd /(d − 1)(d+1) , hence the b d when λ ≤ λc (d, k). The weak spatial mixing on Td is an weak spatial mixing of the model on T k k easy consequence. With this weak spatial mixing of the model on Tdk when λ ≤ λc , the uniqueness of the Gibbs measure is implied by the following generic equivalence between weak spatial mixing and uniqueness of Gibbs measure. The if part of Theorem 3.1 is proved. Proposition 3.5 (Weitz [30]). If the independent sets of Tdk with activity λ exhibit weak spatial mixing with a rate δ(·) that goes to zero, then there is a unique Gibbs measure µ on independent sets of Tdk with activity λ.
4
The hypergraph self-avoiding walk tree
We call a hypergraph a hypertree if its incidence graph has no cycles. Let T = (V, E) be a rooted hypertree with vertex v as its root. We assume that root v is incident to d distinct hyperedges e1 , e2 , . . . , ed , such that for i = 1, 2, . . . , d, • |ei | = ki + 1; and • ei = {v, vi1 , vi2 , . . . , viki }. For 1 ≤ i ≤ d and 1 ≤ j ≤ ki , let Tij be the sub-hypertree rooted at vij . Recall that all hypertrees considered by us satisfy the property that any two hyperedges share at most one common vertex, thus all vij are distinct and the sub-hypertrees Tij are disjoint. Let σΛ ∈ {0, 1}Λ be a configuration indicating an independent set partially specified on vertex set Λ, and for each 1 ≤ i ≤ d and 1 ≤ j ≤ ki , let σΛij be the restriction of σΛ on the sub-hypertree Tij . Consider the ratios of marginal probabilities: σΛ σΛ σΛ RTσΛ = pσTΛ,v / 1 − pσTΛ,v and RTijij = pTijij,vij / 1 − pTijij,vij . The following recursion can be easily verified due to the disjointness between sub-hypertrees: RTσΛ
=λ
d Y i=1
1+
11
1 Pki
σΛij j=1 RTij
.
(3)
This is the “tree recursion” for hypergraph independent sets. The tree recursions for the hardcore model [31] and the monomer-dimer model [1] can both be interpreted as special cases. For general hypergraphs which are not trees, we construct a hypergraph version of self-avoidingwalk tree, which allows computing marginal probabilities in arbitrary hypergraphs with the tree recursion. Moreover, we show that the uniform regular hypertree is the worst case for SSM among all hypergraphs of bounded maximum edge-size and bounded maximum degree. Theorem 4.1 (Proposition 1.3 restated). For any positive integers k, d and any positive λ, if the independent sets of Tdk with activity λ exhibit strong spatial mixing with rate δ(·), then the independent sets of any hypergraph of maximum edge size at most (k + 1) and maximum degree at most (d + 1), with activity λ, exhibit strong spatial mixing with the same rate δ(·). Under duality, the same holds for the hypergraph matchings. We then define the hypergraph self-avoiding walk tree. A walk in a hypergraph H = (V, E) is a sequence (v0 , e1 , v1 , . . . , e` , v` ) of alternating vertices and hyperedges such that every two consecutive vertices vi−1 , vi are incident to the hyperedge ei between them. A walk w = (v0 , e1 , v1 , . . . , e` , v` ) is called self-avoiding if: • w = (v0 , e1 , v1 , . . . , e` , v` ) forms a simple path in the incidence graph of H; and • for every i = 1, 2, . . . , `, vertex vi is incident to none of {e1 , e2 , . . . , ei−1 }. Note that the second requirement is new to the hypergraphs. A self-avoiding walk w = (v0 , e1 , v1 , . . . , e` , v` ) can be extended to a cycle-closing walk w0 = (v0 , e1 , v1 , . . . , e` , v` , e0 , v 0 ) so that for some 0 ≤ i ≤ ` − 1, the suffix (vi , ei+1 , vi+1 , . . . , e` , v` , e0 , v 0 ) of the walk forms a simple cycle in the incidence graph of H. We call v 0 the cycle-closing vertex. Given a hypergraph H = (V, E), an ordering of incident hyperedges at every vertex can be arbitrarily fixed, so that for any two hyperedges e1 , e2 incident to a vertex u we use e1 i. We can then write this in a form of telescopic product: σΛ RH,v
=
d Y
σΛ τi RH v ,v , i
i=1
where σΛ τi means the combination of the two configurations σΛ and τi . σ Λ τi We can obtain the value of RH v ,v by further fix vertices in ei , the hyperedge containing vi . i Since now vi is contained only in ei , we can see that σΛ τi RH v ,v = i
λ1/d 1+
σΛ τi ρij j=1 RHv /vi ,uij
Pki
,
where ki is the number of the vertices other than vi which is incident to ei and ρij is the configuration at vertices of ei in which all the vertices uij 0 other than uij are fixed to unoccupied. σΛ Combining above two equations, we get a recursive procedure for calculating RH,v in the same manner that equation (4) has: σΛ RH,v =λ
d Y
1
i=1 1 +
Pki
σ τρ
Λ i ij j=1 RHv /vi ,uij
.
(5)
Notice that the recursion does terminate, since the number of unfixed vertices reduces at least by σ τi ρij one in each step because in calculating RHΛv /v all copies vi0 of v is either fixed (when i0 6= i) or i ,uij erased (when i0 = i) from the hypergraph Hv /vi . σΛ We now show that the procedure described above for calculating RH,v results in the same value as using the hypertree procedure for TSAW (H, v) with corresponding condition of σΛ imposed on it. First notice that the calculation carried out by the two procedure is the same, since they share the same function (Equation (4) and (5)) when we view them as recursive calls. Furthermore, we have the same stopping values for the both recursive procedures. During constructing TSAW (H, v), if node u corresponding to walk is not included in the hypertree, which is equivalent to fix u to unoccupied in the sense of causing the same effect on the ratio of occupation to its parent node. 14
And when node u in the hypertree corresponding to a self-avoiding walk w = (v, e1 , v1 . . . , e` , v` ), with that w can be extended as w0 = (w, e`+1 , v`+1 ) to a cycle-closing vertex v`+1 = vi for some 0 ≤ i < ` via a new hyperedge e`+1 6∈ {e0 , e1 , . . . , e` }, and e`+1
arXiv:1503.05812v2 [cs.DS] 16 Jul 2015
Yitong Yin∗ Nanjing University, China [email protected]
Jinman Zhao Nanjing University, China [email protected]
Abstract We study the problem of approximately counting hypergraph matchings with an activity parameter λ in hypergraphs of bounded maximum degree and bounded maximum size of hyperedges. This problem unifies two important statistical physics models in approximate counting: the hardcore model (graph independent sets) and the monomer-dimer model (graph matchings). dd We show for this model the critical activity λc = k(d−1) d+1 is the threshold for the uniqueness of Gibbs measures on the infinite (d + 1)-uniform (k + 1)-regular hypertree. And we show that when λ < λc the model exhibits strong spatial mixing at an exponential rate and there is an FPTAS for the partition function of the model on all hypergraphs of maximum degree at most k + 1 and maximum edge size at most d + 1. Assuming NP6=RP, there is no FPRAS for the partition function of the model when λ > 2λc on above family of hypergraphs . Towards closing this gap and obtaining a tight transition of approximability, we study the local weak convergence from an infinite sequence of random finite hypergraphs to the infinite uniform regular hypertree with specified symmetry, and prove a surprising result: the existence of such local convergence is fully characterized by the reversibility of the uniform random walk on the infinite hypertree projected on the symmetry classes. We also give an explicit construction for the sequence of random finite hypergraphs with proper local convergence property when the reversibility condition is satisfied.
1
Introduction
The problems of approximate counting have long been studied as statistical physics models, and an exciting accomplishment was connecting the computational complexity of approximate counting to the phase transition in the statistical physics model. The hardcore model and the monomerdimer model are perhaps the two simplest and most important statistical physics models studied for approximate counting. In the hardcore model, given a graph G = (V, E) and a vertex-activity λ, the model assigns each independent set I of G a weight wλIS (I) = λ|I| . A natural probability distribution, call the IS IS IS Gibbs measure, can be defined over P allIS independent sets of G as µλ (I) = wλ (I)/Zλ (G) where IS the normalizing factor Zλ (G) = I wλ (I) that takes the sum over all independent sets of G is the so-called partition function. In the monomer-dimer mode, given a graph G = (V, E) and an edge-activity λ, the model assigns each matching M of G a weight wλM (M ) = λ|M | . The Gibbs measure over all matchings of G can be accordingly defined, and the partition function now becomes ∗
State Key Laboratory for Novel Software Technology, Nanjing University, China. Supported by NSFC grants 61272081 and 61321491.
1
P ZλM (G) = M wλM (M ) where the sum is taken over all matchings of G. The approximate counting of (weighted) independent sets and (weighted) matchings of G can then be formulated respectively as approximately computing the partition functions ZλIS (G) and ZλM (G). It was well known that the hardcore model exhibits the following phase transition on the infinite (d+1)-regular tree: there is a critical activity λc (d) = dd /(d−1)d+1 , usually referred in the literature as the uniqueness threshold, such that the correlation between the marginal distribution at the root of the tree and any condition on all the vertices at distance t from the root decays exponentially in t when λ < λc (d), but the boundary-to-root correlation remains substantial even as t → ∞ when λ > λc (d). This property of correlation decay is also called spatial mixing, and was known to be equivalent to the uniqueness of the infinite-volume Gibbs measure on the infinite (d + 1)-regular tree [30]. In a seminal work [31], Weitz showed that for all λ < λc (d) the decay of correlation holds for the hardcore model on all graphs of maximum degree bounded by d + 1 and there is a deterministic FPTAS for the partition function of the hardcore model on all such graphs. Here the specific notion of decay of correlation established is the strong spatial mixing, which requires the boundary-to-vertex correlation have exponential decay even conditioning on the states of vertices from a subset being arbitrarily fixed. The implication of the decay of correlation on regular tree to FPTAS on graphs with bounded maximum degree for the hardcore model was established by a construction called self-avoiding walk (SAW) tree due to Weitz [31]. This connection between approximability of partition function and the phase transition of the model is further strengthened in a series of works [6,7,27,28] which show that unless NP=RP there is no FPRAS for the partition function of the hardcore model when λ > λc (d) on graphs with maximum degree bounded by d + 1. For the monomer-dimer model, it was well known that the model has no such phase transition as above [11, 12]. And analogously the well known result of Jerrum and Sinclair [13] gives an FPRAS for the partition function of the monomer-dimer model on all graphs. In [1] strong spatial mixing with an exponential rate was established for the model on all graphs with maximum degree bounded by an arbitrary constant and a deterministic FPTAS was also given for the partition function of the monomer-dimer model on all such graphs. In this paper, we study spatial mixing and approximate counting of hypergraph matchings, a model that unifies both the hardcore model and the monomer-dimer model. A hypergraph H = (V, E) consists of a vertex set V and a collection E of vertex subsets, called the hyperedges. A matching of H is a set M ⊆ E of disjoint hyperedges in H. Given a hypergraph H and an activity parameter λ > 0, a configuration is a matching M of H, and is assigned a weight wλ (M ) = λ|M | . The Gibbs measure over all matchings of H is defined as µ(M ) = wλ (M )/Zλ (H), where the normalizing factor Zλ (H) is the partition function for the model, defined as: X Zλ (H) = λ|M | . M : matching of H
This model represents an important subclass of Boolean constraint satisfaction problem (CSP): that is, the CSPs defined by the matching (packing) constraints. It also unifies the hardcore model and the monomer-dimer model in the following sense. Consider the family of hypergraphs of maximum edge size at most d + 1 and maximum degree at most k + 1: • When d = 1, the model becomes the monomer-dimer model on graphs of maximum degree at most k + 1. • When k = 1, the matchings of hypergraph H correspond to independent sets of its dual graph, and the model becomes the hardcore model on graphs of maximum degree at most d + 1. 2
For the general model on hypergraphs, the problem has only been studied when λ = 1, i.e. counting the number of matchings in a hypergraph. In a recent work, Liu and Lu [19] proves the decay of correlation and the existence of FPTAS for the matchings in 3-uniform hypergraphs of maximum degree at most 4. A similar result is proved for 3-uniform hypergraphs of maximum degree at most 3 in an independent work [4]. Both results were based on case-studies of correlation decay for specific settings of edge sizes and degrees which are insufficient to establish a transition with continuous λ. Our contributions.
We show that for hypergraph matchings the λc (d, k) =
dd k(d−1)d+1
is the
uniqueness threshold for Gibbs measure on the infinite uniform regular hypertree. Indeed, let Tkd denote the infinite (d+1)-uniform (k +1)-regular hypertree. There is a way to classify all vertices in Tkd into 2 equivalent classes and define two simple Gibbs measures µ+ , µ− on Tkd which are invariant under all class-preserving automorphisms (see Section 3) such that the followings hold. d
d k Proposition 1.1. The Gibbs measures µ+ = µ− if and only if λ ≤ λc = k(d−1) d+1 . Further, on Td there is a unique Gibbs measure on hypergraph matchings with activity λ if and only if λ ≤ λc . d
d This covers as special cases the well-known uniqueness threshold λc (d, 1) = (d−1) d+1 for the hardcore model on the infinite (d + 1)-regular tree and also the lack of phase-transition for the monomer-dimer model. The Gibbs measures µ± generalize the extremal semi-translation invariant Gibbs measures (which are invariant under all parity-preserving automorphisms on the infinite regular tree) for the hardcore model. The two equivalent classes for vertices in the definition of µ± play similar roles as the parities in the semi-translation invariant Gibbs measure. We then establish the decay of correlation for hypergraph matchings on all hypergraphs with bounded maximum edge size and bounded maximum degree when the activity λ is in the interior of the uniqueness region for uniform regular hypertrees. The specific notion of decay of correlations that we establish is the strong spatial mixing [31] (see Section 2 for a precise definition). Consequently, we give an FPTAS for the partition function for the hypergraphs with bounded maximum edge size and bounded maximum degree as long as the uniqueness condition is satisfied.
Theorem 1.2. For every finite integers d, k ≥ 1 and any λ < λc , for the model of hypergraph matchings with activity λ on all hypergraphs of maximum edge size at most d + 1 and maximum degree at most k + 1, we have: • the model exhibits strong spatial mixing with an exponential rate; and • there exists an FPTAS for computing the partition function. Remark. The theorem unifies the strong spatial mixing and correlation-decay based FPTAS for the hardcore model [31] and the monomer-dimer model [1], and also covers as special cases the results for approximate counting hypergraph matchings in [4,19]. Further, the runningtime for the O dkˆx ln kd 1−(d−1)kˆ x FPTAS for outputting an -approximation of the partition function is n where x ˆ is −d the fixed point solution to the tree recursion x ˆ = f (ˆ x) = λ (1 + kˆ x) , achieving the best running time (if ever explicitly specified) for the deterministic algorithms in [1, 4, 19, 31]. As far as we know, Theorem 1.2 is the first time for a non-spin system that the approximability of the model is established as its parameter approaching the phase-transition threshold. We achieve 3
this by constructing a hypergraph version of Weitz’s self-avoiding walk tree [31]. As a by-product, it proves that for hypergraph matchings, the worst case for strong spatial mixing among all hypergraphs of bounded maximum edge size and bounded maximum degree, is achieved by uniform regular hypertrees. Such phenomenon was once believed to be true for most statistical physics models, but had been found false for even some simple spin systems [17, 26]. Here we show as an example for general CSPs, that the worst cases for strong spatial mixing are indeed regular trees. Proposition 1.3. If the model of hypergraph matchings with activity λ exhibits strong spatial mixing with rate δ on Tkd , then the model exhibits strong spatial mixing with the same rate on all hypergraphs of maximum edge size at most d + 1 and maximum degree at most k + 1. For the hardness of approximation, due to a simple reduction from the hardcore model, it is k easy to show that if λ > 2k+1+(−1) λc ≈ 2λc , then unless NP=RP there is no FPRAS for the k+1 partition function for the family of hypergraphs stated in Theorem 1.2. We then explore the possibility of closing this gap between λc and 2λc and finding the tight condition for the transition of approximability. The coincidence between the physical phase transition of Gibbs measures on the infinite regular tree and the transition of computational complexity on finite graphs is due to the local weak convergence: Gibbs measures on an infinite sequence of (random) finite graphs weakly converges locally to the Gibbs measure on the infinite regular tree. Consequently, the nonuniqueness of Gibbs measure on the infinite tree may imply the multimodality of local distributions in the finite graphs, which makes the local Markov chain torpid mixing on the sequence of finite graphs [5, 22] and also makes the sequence of finite graphs become suitable gadgets for the inapproximability reduction [6–8, 27, 28]. The key to the local weak convergence of measures is the local convergence of graph structures: for spin systems, this is represented by the property of being locally tree-like. More precisely, not only the graph topology of the sequence of finite graphs needs to converge locally to that of the infinite tree, but also the finite graphs should preserve the symmetry of the Gibbs measures on the infinite tree that exhibit the critical phase transition. For example, for anti-ferromagnetic 2-spin systems, the transition of uniqueness/nonuniqueness on the infinite tree is achieved by semi-translation invariant Gibbs measures — the Gibbs measures which are invariant under parity-preserving automorphisms on the infinite tree, and hence the sequence of finite graphs which converges locally to the infinite tree needs to be locally like the tree structure along with the parity assignment, which corresponds to the random regular bipartite graphs as a hardness gadget for anti-ferromagnetic spin system [5–8, 22, 27, 28]. Such graphs locally resembles the infinite regular tree with the proper symmetry (preserving the parity of vertices) under which the invariant Gibbs measure achieves the phase-transition threshold. For general CSPs the underlying structures are hypergraphs instead of graphs. And Proposition 1.3 suggests conceptually that for our model the infinite uniform regular hypertrees Tkd should play a role as the infinite regular trees to the spin system. Our next result characterizes Tkd with specific symmetry which allows such locally convergent sequences. Unlike trees, on hypertrees there is no parity-preserving symmetry, and the symmetries on hypertrees may be much more complicated. We resolve this by introducing a hypergraph version of the branching matrix of [23] and proving it fully characterize the symmetry on Tkd . With the help of this notion of branching matrices, we prove a surprising result: the existence of local convergence from any sequence of finite hypergraphs to a uniform regular hypertree with certain symmetry is determined precisely by the reversibility of a natural random walk over equivalent classes of vertices and hyperedges in the infinite hypertree (see Section 7 for more precise definitions). 4
Theorem 1.4 (informal). There exist a sequence Hn of random finite hypergraphs that converge locally to the infinite (d + 1)-uniform (k + 1)-regular hypertree Tkd with labeling that are invariant under an automorphism group G if and only if the projection of the uniform random walk on Tkd onto the orbits of G is time-reversible. The proof of this theorem is model-independent, thus the theorem holds generally for all CSPs where local convergence to the infinite hypertree is concerned. We also explicitly construct a hypergraph sequence Hn with the proper local convergence when the time-reversibility is satisfied. The construction has potential to be used as gadgets for general CSPs and is interesting by itself. Applying Theorem 1.4 gives us the following “barrier” result: there is no local convergence from any sequence of finite hypergraphs to Tkd with the symmetry assumed by the family of Gibbs measures µ+ , µ− that achieve the uniqueness/nonuniqueness threshold. Proposition 1.5. There does not exit any sequence of finite hypergraphs that converge locally to Tkd with the symmetry assumed by the Gibbs measures µ+ , µ− in Proposition1.1. Proposition 1.1, 1.3 and 1.5 combined suggest: Even for this very simple class of counting CSP, in order to precisely classify its approximability, we may need either algorithms that do not rely on the mixing of Gibbs measure, or hardness that holds beyond the current framework of local weak convergence of Gibbs measures, or both. Remark on exposition. For convenience of visualizing the results, all our results in the rest of the paper are presented for independent sets in the dual hypergraphs. Note that matchings are equivalent to independent sets under hypergraph duality. And the only effect of duality on a family of hypergraphs with bounded maximum edge size and bounded maximum degree is to switch the bounds on the edge size and the degree. We stress that we adopt a stronger version of the definition of hypergraph independent set: a vertex subset I ⊆ V is an independent set of a hypergraph H = (V, E) if no two vertices in I are contained in the same hyperedge. This should be distinguished from a weaker definition of hypergraph independent set (e.g. the one used in [2]), which only needs any independent set I to contain no hyperedge as subset. Related works. Approximate counting of hypergraph matchings was studied in [14] for hypergraphs with restrictive structures, and in [4, 19] for hypergraphs with bounded edge size and maximum degree. In [2], approximate counting of a variant of hypergraph independent sets was studied, where an independent set only need to not contain any hyperedge. The spatial mixing (decay of correlation) is already a widely studied topic in Computer Science, because it may support FPTAS for #P-hard counting problems. The decay of correlation was established via the self-avoiding walk tree for the hardcore model [25,31], monomer-dimer model [1, 24], and two-spin systems [16, 17, 24]. Similar tree-structured recursions were employed to prove the decay of correlation for multi-spin systems [9, 10, 21] and more general CSPs [18–20].
2
Preliminary
For a hypergraph H = (V, E), the size of a hyperedge e ∈ E is its cardinality |e|, and the degree of a vertex v ∈ V , denoted by deg (v) = degH (v), is the number of hyperedges e ∈ E incident to v, i.e. satisfying v ∈ e. A hypergraph H is k-uniform if all hyperedges are of the same size k, and is 5
d-regular if all vertices have the same degree d. The incidence graph of a hypergraph H = (V, E) is a bipartite graph with V and E as vertex sets on the two sides, such that each (v, e) ∈ V × E is a bipartite edge if and only if v is incident to e. A matching of hypergraph H = (V, E) is a set M ⊆ E of disjoint hyperedges in H. Given an activity parameter λ > 0, the Gibbs measure is a probability distribution over matchings of M M H proportional to the weightPwλM (M ) = λ|M | , defined as µM λ (M ) = wλ (M )/Zλ (H), where the normalizing factor ZλM (H) = M wλM (M ) is the partition function. Similarly, an independent set of hypergraph H = (V, E) is a set I ⊆ V of vertices satisfying |I ∩ e| ≤ 1 for all hyperedges e in H. The Gibbs measure over independent sets of H with activity λ > 0 is given by µIS λ (I) =
wλIS (I) λ|I| = IS , IS Zλ (H) Zλ (H)
(1)
P where the normalizing factor ZλIS (H) = I wλIS (I) is the partition function for independent sets of H with activity λ. Independent sets and matchings are equivalent under hypergraph duality. The dual of a hypergraph H = (V, E), denoted by H∗ = (E ∗ , V ∗ ), is the hypergraph whose vertex set is denoted by E ∗ and edge set is denoted by V ∗ , such that every vertex v ∈ V (and every hyperedge e ∈ E) in H is one-to-one corresponding to a hyperedge v ∗ ∈ V ∗ (and a vertex e∗ ∈ E ∗ ), such that e∗ ∈ v ∗ if and only if v ∈ e. Note that under duality, matchings and hypergraphs are the same CSP and hence result in the same Gibbs measure, which remains to be true even with activity λ. Also a family of hypergraphs of bounded maximum edge size and bounded maximum degree is transformed under duality to a family of hypergraphs with the bounds on the edge size and degree exchanged. Remark 2.1. With the above equivalence under duality, from now on we state all our results in terms of the independent sets in the dual hypergraph and omit the superscript ·IS in notations. Given the Gibbs measure over independent sets of hypergraph H and a vertex v, we define the marginal probability pv as pv = pH,v = Pr[v ∈ I] which is the probability that v is in an independent set I sampled from the Gibbs measure (such a vertex is also said to be occupied ). Given a vertex set Λ ⊂ V , a configuration is a σΛ ∈ {0, 1}Λ which corresponds to an independent set IΛ partially specified over Λ such that σΛ (v) indicates whether a v ∈ Λ is occupied by the independent set. We further define the marginal probability Λ pσH,v as Λ pσv Λ = pσH,v = Pr[v ∈ I | IΛ = σΛ ] which is the probability that v is occupied under the Gibbs measure conditioning on the configuration of vertices in Λ ⊂ V being fixed as σΛ . Definition 2.1. The independent sets of a finite hypergraph H = (V, E) with activity λ > 0 exhibit weak spatial mixing (WSM) with rate δ : N → R+ if for any v ∈ V , Λ ⊆ V , and any two configurations σΛ , τΛ ∈ {0, 1}Λ which correspond to two independent sets partially specified on Λ, |pσv Λ − pτvΛ | ≤ δ(distH (v, Λ)), where distH (v, Λ) is the shortest distance between v and any vertex in Λ in hypergraph H. 6
Definition 2.2. The independent sets of a finite hypergraph H = (V, E) with activity λ > 0 exhibit strong spatial mixing (SSM) with rate δ : N → R+ if for any v ∈ V , Λ ⊆ V , and any two configurations σΛ , τΛ ∈ {0, 1}Λ which correspond to two independent sets partially specified on Λ, |pσv Λ − pτvΛ | ≤ δ(distH (v, ∆)), where ∆ ⊆ Λ stands for the subset on which σΛ and τΛ differ and distH (v, ∆) is the shortest distance between v and any vertex in ∆ in hypergraph H. The definitions of WSM and SSM extend to infinite hypergraphs with the same conditions to be satisfied for every finite region Ψ ⊂ V conditioning on the vertices in ∂Ψ being unoccupied.
3
Gibbs measures on the infinite tree
We follow Remark 2.1 and state our discoveries in terms of independent sets in the dual hypergraphs. Let Tdk be the infinite (k +1)-uniform (d+1)-regular hypertree, whose incidence graph is the infinite tree in which all vertices with parity 0 are of degree (k + 1) and all vertices with parity 1 are of degree (d + 1). A probability measure µ on hypergraph independent sets of Tdk is Gibbs if for any finite sub-hypertree T , conditioning µ upon the event that all vertices on the outer boundary of T are unoccupied gives the same distribution on independent sets of T as defined by (1) with H = T . We further consider the simple Gibbs measures satisfying conditional independence: Conditioning µ on a configuration of a subset Λ of vertices results in a measure in which the configurations on the components separated by Λ are independent of each other. The Gibbs distribution on a finite hypergraph is always simple. A Gibbs measure on Tdk is translation-invariant if it is invariant under dd all automorphisms of Tdk . We will show that λc (d, k) = k(d−1) d+1 is the uniqueness threshold for the Gibbs measures on hypergraph independent sets of Tdk . Theorem 3.1. There is always a unique simple translation-invariant Gibbs measure on independent dd sets of Tdk . Let λc = λc (d, k) = k(d−1) d+1 . There is a unique Gibbs measure on independent sets of Tdk with activity λ if and only if λ ≤ λc . This proves the uniqueness threshold stated in Proposition 1.1. The construction of extremal Gibbs measures µ+ , µ− can be described with the notion of branching matrices.
3.1
Branching matrices
Fix an automorphism group G of Tdk . A G-translation-invariant Gibbs measure on Tdk is a measure that is invariant under all automorphisms from G. For example, the semi-translation-invariant Gibbs measures on regular tree are invariant under all parity-preserving automorphisms on Td1 . The natural group actions of G respectively on vertices and hyperedges partition the sets of vertices and hyperedges into orbits. Let τv and τe be the respective numbers of orbits for vertices and hyperedges. For each i ∈ [τv ], we say a vertex is of type-i if it is in the i-th orbit for vertices; and the same also applies to hyperedges. Assuming the symmetry on Tdk given by automorphism group G, the hypergraph branching matrices, or just branching matrices, are the following two nonnegative integral matrices: D = D τv ×τe = [dij ] and K = K τe ×τv = [kji ], which satisfy that for any i ∈ [τv ] and j ∈ [τe ]: 7
• every vertex in Tdk of type-i is incident to precisely dij hyperedges of type-j; • every hyperedge in Tdk of type-j contains precisely kji vertices of type-i. The D and K are transition matrices from vertex-types to hyperedge-types and vice versa in The definition can be seen as a hypergraph generalization of the branching matrix for multitype Galton-Watson tree [23]. Since types (orbits) are invariant under all automorphisms from G, it is clear that the above D and K are well-defined for every automorphism group G on Tdk with finitely many orbits. Tdk .
Proposition 3.2. Every automorphism group G on Tdk with finitely many orbits has a pair of P branching matrices D and K which are as described above and satisfy: (1) j dij = d + 1 and P k = k + 1; (2) dij = 0 if and only if kji = 0; and (3) DK and KD are irreducible. i ji Conversely, any pair of nonnegative integral matrices D and K satisfying these conditions are branching matrices for some automorphism group G on Tdk . Proof. Let G be an automorphism group on Tdk with finitely many P to see that P orbits. It is trivial the branching matrices D and K are well-defined and satisfy j dij = d + 1 and i kji = k + 1. A vertex v of type-i is incident to a hyperedge e of type-j if and only if e of type-j contains a vertex v of type i, thus kji 6= 0 if and only if dij 6= 0. 0 D , which is a consequence The irreducibility of DK and KD follows that of the matrix K 0 to the that every type of vertex and hyperedge is accessible from all other types of vertices and hyperedges, which follows the simple fact that the incidence graph Tdk is strongly connected. Conversely, let D and K be a pair of nonnegative integral matrices satisfying the conditions above. We can start from any vertex (or hyperedges) o of type-i and construct an infinite hypertree rooted at o with each vertex and hyperedge labeled with the respective type according to the rules specified by the branching Pmatrices D and K. P Since dij = 0 if and only if kji = 0, the construction is always possible. Since j dij = d + 1 and i kji = k + 1, the resulting infinite hypertree must be k-uniform and d-regular. Since DK and KD are irreducible, no matter how we choose the type for the root o, the resulting hypertree contains all types of vertices and hyperedges. We can then construct an automorphism group G on Tdk according with orbits being the types just specified. For every pair of vertices (or hyperedges) u, v with the same type, by generating the hypertree according to D, K starting from u and v respectively, we obtain an automorphism φu→v on Tdk which maps u to v and preserves the types of all vertices and hyperedges. Let G = h {φu→v | ∀u, v with the same type} i be the group generated from all such automorphisms. Then D and K are branching matrices for automorphism group G on Tdk .
3.2
Extremal Gibbs measures
Assume that there are two vertex-types and two hyperedge-types, both denoted as {+, −}, and the 1 d k 1 b = c= branching matrices are defined as D and K , i.e.: d 1 1 k 1. every ‘±’-vertex is incident to a ‘±’-hyperedge and d ‘∓’-hyperedges; 2. every ‘±’-hyperedge contains k ‘±’-vertices and a ‘∓’-vertex. 8
Figure 1: Classifying vertices and hyperedges of T23 into two types ‘+’(black) and ‘−’(white). The hypergraph is represented as its incidence graph where circles stand for vertices and squares stand for hyperedges.
See Figure 1 for an illustration. Fix a ‘+’-vertex v in Tdk as the root. Let µ+ (resp. µ− ) be d the Gibbs measure on Tk defined by conditioning on all vertices to be unoccupied for the t-th ‘−’-vertices (resp. ‘+’-vertices) along all path from the root and taking the weak limit as t → ∞. b and K, c on any path any ‘±’-vertex has a ‘∓’-vertex within Note that for the 2-coloring given by D 2 steps, so the limiting sequence is well-defined. And by symmetry, starting from a root of type-‘−’ gives the same pair of measures. The µ± generalize the extremal semi-translation-invariant Gibbs measures on infinite regular trees. For hypertree Tdk with k ≥ 2, there are no parity-preserving automorphisms. Nevertheless, b and K c generalizes the parity-preserving automorphisms to hypertrees the symmetry given by D and has the similar phase-transition as semi-translation-invariant Gibbs measures on trees. The b b with orbits given by µ± are simple and are G-translation-invariant for the automorphism group G b b and K. c In fact, they are extremal G-translation-invariant D Gibbs measures on Tdk . We will + − see that the model has uniqueness if and only if µ = µ , which together with Theorem 3.1 gives Proposition 1.1.
3.3
Uniqueness of Gibbs measures
Lemma 3.3. Let µ be a simple Gibbs measure on independent sets of Tdk . Let v be a vertex in Tdk and vij the j-th vertex (besides v) in the i-th hyperedge incident to v, for i = 1, 2, . . . , d + 1 and j = 1, 2, . . . , k. Let pv = µ[ v is occupied ] and pvij = µ[ vij is occupied ]. It holds that d+1 k Y X 1 − pv − pv = λ(1 − pv )−d pvij . (2) j=1
i=1
Proof. Since µ is a Gibbs measure, for any vertex v in Tdk , it holds that pv = µ[ v is occupied ] =
λ · µ[ all the neighbors of v are unoccupied ] 1+λ 9
On the other hand, since µ is simple, conditioning on the root being unoccupied the sub-hypertrees are independent of each other, thus µ[ all the neighbors of v are unoccupied ] =µ[ v is occupied ] · µ[ all the neighbors of v are unoccupied | v is occupied ] + µ[ v is unoccupied ]
d+1 Y
µ[ ∀1 ≤ j ≤ k, vij is unoccupied | v is unoccupied ]
i=1
=pv + (1 − pv )
d+1 Y i=1
1 −
k X
µ[ vij is occupied | v is unoccupied ] .
j=1
Note that for any two adjacent vertices v, vij , we have µ[ vij is occupied ] = µ[ vij is occupied | v is unoccupied ] · µ[ v is unoccupied ], thus µ[ vij is occupied | v is unoccupied ] =
pvij µ[ vij is occupied ] = . 1 − µ[ v is occupied ] 1 − pv
The lemma follows by combining everything together. Equation (2) gives an infinite system involving all vertices in Tdk . If the simple Gibbs measure µ is G-translation-invariant for some automorphism group G on Tdk , the marginal probability pv = µ[ v is occupied ] depends only on the type (orbit) of v. Corollary 3.4. Let µ be a simple G-translation-invariant Gibbs measure on Tdk with branching matrices D τv ×τe = [dij ] and K τe ×τv = [kji ]. For every i ∈ [τv ], let pi = µ[ v is occupied ] for vertex v in Tdk of type-i. It holds for every s ∈ [τv ] that dij
ps = λ(1 − ps )−d
Y
1 −
j∈[τe ]
X
kji · pi
.
i∈[τv ]
b and K c defined in Section 3.2, the system in CorolApplying with the branching matrices D lary 3.4 becomes ( p+ = λ(1 − p+ )−d (1 − k p+ − p− )(1 − p+ − k p− )d , p− = λ(1 − p− )−d (1 − k p− − p+ )(1 − p− − k p+ )d . ( y = f (x) kp+ kp− kλ Let x = 1−p− −k p+ and y = 1−p+ −k p− . The system becomes , where f (x) = (1+x) d x = f (y) is the hardcore tree-recursion. Since f (x) is positive and decreasing in x, it follows that there is a unique positive x ˆ such that x ˆ = f (ˆ x), which means there is always a unique simple translationd invariant Gibbs measure on Tk . It is well-known (see [7] and [15, 29]) the system has three distinct solutions (ˆ x, x ˆ), (x+ , x− ) and (x− , x+ ) where 0 < x− < x ˆ < x+ , when kλ > dd /(d − 1)d+1 , i.e. λ > λc (d, k); and the three solutions collide into a unique solution (ˆ x, x ˆ) when λ ≤ λc (d, k), which means b there is a unique simple G-translation-invariant Gibbs measure on Tdk if and only if λ ≤ λc . The only b if part of Theorem 3.1 is proved. Recall that µ± are simple and are extremal G-translation-invariant + − Gibbs measures, and hence it also holds that µ = µ if and only if λ ≤ λc (d, k). 10
For the if part of Theorem 3.1, we first prove the weak spatial mixing of the model on Tdk b d . According to the when λ ≤ λc . Consider instead the (k + 1)-uniform d-ary hypertree T = T k Qd tree recursion (3) in Section 4, we have RT = λ i Pk1 where RT represents the ratio 1+
j=1
RTij
between probabilities of being occupied and unoccupied at the root of T and Tij represents the sub-hypertree rooted by the j-th child in the i-th incident hyperedge of the root of T , which is isomorphic to T . Due to the monotonicity of the recursion, the extremal boundary conditions (the boundary conditions that achieve the maximum and minimum of RT ) at level t are that exactly one of each hyperedge at level t is occupied in each hyperedge at level t, and that all vertices at level t are unoccupied respectively. By symmetry, the system is then simplified to that Rt = (1+kRλ )d . t−1
Substituting xt = kRt we have xt =
kλ . (1+xt−1 )d
It is well known [15, 29] that the system converges
to its unique fixed point regardless of the initial value as long as kλ ≤ dd /(d − 1)(d+1) , hence the b d when λ ≤ λc (d, k). The weak spatial mixing on Td is an weak spatial mixing of the model on T k k easy consequence. With this weak spatial mixing of the model on Tdk when λ ≤ λc , the uniqueness of the Gibbs measure is implied by the following generic equivalence between weak spatial mixing and uniqueness of Gibbs measure. The if part of Theorem 3.1 is proved. Proposition 3.5 (Weitz [30]). If the independent sets of Tdk with activity λ exhibit weak spatial mixing with a rate δ(·) that goes to zero, then there is a unique Gibbs measure µ on independent sets of Tdk with activity λ.
4
The hypergraph self-avoiding walk tree
We call a hypergraph a hypertree if its incidence graph has no cycles. Let T = (V, E) be a rooted hypertree with vertex v as its root. We assume that root v is incident to d distinct hyperedges e1 , e2 , . . . , ed , such that for i = 1, 2, . . . , d, • |ei | = ki + 1; and • ei = {v, vi1 , vi2 , . . . , viki }. For 1 ≤ i ≤ d and 1 ≤ j ≤ ki , let Tij be the sub-hypertree rooted at vij . Recall that all hypertrees considered by us satisfy the property that any two hyperedges share at most one common vertex, thus all vij are distinct and the sub-hypertrees Tij are disjoint. Let σΛ ∈ {0, 1}Λ be a configuration indicating an independent set partially specified on vertex set Λ, and for each 1 ≤ i ≤ d and 1 ≤ j ≤ ki , let σΛij be the restriction of σΛ on the sub-hypertree Tij . Consider the ratios of marginal probabilities: σΛ σΛ σΛ RTσΛ = pσTΛ,v / 1 − pσTΛ,v and RTijij = pTijij,vij / 1 − pTijij,vij . The following recursion can be easily verified due to the disjointness between sub-hypertrees: RTσΛ
=λ
d Y i=1
1+
11
1 Pki
σΛij j=1 RTij
.
(3)
This is the “tree recursion” for hypergraph independent sets. The tree recursions for the hardcore model [31] and the monomer-dimer model [1] can both be interpreted as special cases. For general hypergraphs which are not trees, we construct a hypergraph version of self-avoidingwalk tree, which allows computing marginal probabilities in arbitrary hypergraphs with the tree recursion. Moreover, we show that the uniform regular hypertree is the worst case for SSM among all hypergraphs of bounded maximum edge-size and bounded maximum degree. Theorem 4.1 (Proposition 1.3 restated). For any positive integers k, d and any positive λ, if the independent sets of Tdk with activity λ exhibit strong spatial mixing with rate δ(·), then the independent sets of any hypergraph of maximum edge size at most (k + 1) and maximum degree at most (d + 1), with activity λ, exhibit strong spatial mixing with the same rate δ(·). Under duality, the same holds for the hypergraph matchings. We then define the hypergraph self-avoiding walk tree. A walk in a hypergraph H = (V, E) is a sequence (v0 , e1 , v1 , . . . , e` , v` ) of alternating vertices and hyperedges such that every two consecutive vertices vi−1 , vi are incident to the hyperedge ei between them. A walk w = (v0 , e1 , v1 , . . . , e` , v` ) is called self-avoiding if: • w = (v0 , e1 , v1 , . . . , e` , v` ) forms a simple path in the incidence graph of H; and • for every i = 1, 2, . . . , `, vertex vi is incident to none of {e1 , e2 , . . . , ei−1 }. Note that the second requirement is new to the hypergraphs. A self-avoiding walk w = (v0 , e1 , v1 , . . . , e` , v` ) can be extended to a cycle-closing walk w0 = (v0 , e1 , v1 , . . . , e` , v` , e0 , v 0 ) so that for some 0 ≤ i ≤ ` − 1, the suffix (vi , ei+1 , vi+1 , . . . , e` , v` , e0 , v 0 ) of the walk forms a simple cycle in the incidence graph of H. We call v 0 the cycle-closing vertex. Given a hypergraph H = (V, E), an ordering of incident hyperedges at every vertex can be arbitrarily fixed, so that for any two hyperedges e1 , e2 incident to a vertex u we use e1 i. We can then write this in a form of telescopic product: σΛ RH,v
=
d Y
σΛ τi RH v ,v , i
i=1
where σΛ τi means the combination of the two configurations σΛ and τi . σ Λ τi We can obtain the value of RH v ,v by further fix vertices in ei , the hyperedge containing vi . i Since now vi is contained only in ei , we can see that σΛ τi RH v ,v = i
λ1/d 1+
σΛ τi ρij j=1 RHv /vi ,uij
Pki
,
where ki is the number of the vertices other than vi which is incident to ei and ρij is the configuration at vertices of ei in which all the vertices uij 0 other than uij are fixed to unoccupied. σΛ Combining above two equations, we get a recursive procedure for calculating RH,v in the same manner that equation (4) has: σΛ RH,v =λ
d Y
1
i=1 1 +
Pki
σ τρ
Λ i ij j=1 RHv /vi ,uij
.
(5)
Notice that the recursion does terminate, since the number of unfixed vertices reduces at least by σ τi ρij one in each step because in calculating RHΛv /v all copies vi0 of v is either fixed (when i0 6= i) or i ,uij erased (when i0 = i) from the hypergraph Hv /vi . σΛ We now show that the procedure described above for calculating RH,v results in the same value as using the hypertree procedure for TSAW (H, v) with corresponding condition of σΛ imposed on it. First notice that the calculation carried out by the two procedure is the same, since they share the same function (Equation (4) and (5)) when we view them as recursive calls. Furthermore, we have the same stopping values for the both recursive procedures. During constructing TSAW (H, v), if node u corresponding to walk is not included in the hypertree, which is equivalent to fix u to unoccupied in the sense of causing the same effect on the ratio of occupation to its parent node. 14
And when node u in the hypertree corresponding to a self-avoiding walk w = (v, e1 , v1 . . . , e` , v` ), with that w can be extended as w0 = (w, e`+1 , v`+1 ) to a cycle-closing vertex v`+1 = vi for some 0 ≤ i < ` via a new hyperedge e`+1 6∈ {e0 , e1 , . . . , e` }, and e`+1
