Central limit theorems for combinatorial optimization problems on sparse Erdős-Rényi graphs
aa r X i v : . [ m a t h . P R ] J un CENTRAL LIMIT THEOREMS FOR COMBINATORIALOPTIMIZATION PROBLEMS ON SPARSE ERD ˝OS-R´ENYIGRAPHS
By Sky Cao
Stanford University
For random combinatorial optimization problems, there has beenmuch progress in establishing laws of large numbers and computinglimiting constants for the optimal value of various problems. However,there has not been as much success in proving central limit theorems.This paper introduces a method for establishing central limit theo-rems in the sparse graph setting. It works for problems that display akey property which has been variously called “endogeny”, “long-rangeindependence”, and “replica symmetry” in the literature. Examplesof such problems are maximum weight matching, λ -diluted minimummatching, and optimal edge cover. CONTENTS1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1 Background and motivation . . . . . . . . . . . . . . . . . . . 21.2 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.1 Maximum weight matching . . . . . . . . . . . . . . . . . . . 52.2 λ -diluted minimum matching . . . . . . . . . . . . . . . . . . 52.3 Optimal edge cover . . . . . . . . . . . . . . . . . . . . . . . . 62.4 The general result . . . . . . . . . . . . . . . . . . . . . . . . 73 Applications of Corollary 2.5 . . . . . . . . . . . . . . . . . . . . . 113.1 Maximum weight matching . . . . . . . . . . . . . . . . . . . 113.2 λ -diluted minimum matching . . . . . . . . . . . . . . . . . . 154 Application of Theorem 2.4 to optimal edge cover . . . . . . . . . . 184.1 Basic facts of optimal edge cover . . . . . . . . . . . . . . . . 194.2 Constructing the local approximation . . . . . . . . . . . . . 214.3 Quantitative bound for the error in the local approximation . 234.4 Completing the proof of the central limit theorem . . . . . . . 26 MSC 2010 subject classifications:
Primary 60F05; secondary 90C27, 82B44
Keywords and phrases:
Central limit theorem, combinatorial optimization, Erd˝os-R´enyigraph, Stein’s method, generalized perturbative approach, endogeny, long-range indepen-dence, replica symmetry, maximum weight matching, minimum matching, optimal edgecover. SKY CAO
1. Introduction.
Background and motivation.
Combinatorial optimization problemsare in essence functions on weighted graphs. By making the underlyingweighted graph random, we may obtain a random combinatorial optimiza-tion problem. The basic object of study then becomes the optimal value ofthe problem, which is now a random variable. There are general strategiesfor establishing laws of large numbers for this random variable in varioussettings. When the weighted graph comes from Euclidean points (say n i.i.d. points from the unit square), certain subadditive properties may beexploited. For more details and references in the Euclidean setting, see themonographs by Steele [29] or Yukich [36].Unfortunately, in the Euclidean setting, not much is known about limit-ing constants (a notable exception is the Euclidean bipartite matching prob-lem, due to recent work by Caracciolo et al. [8], and Ambrosio et al. [6]).One of the main difficulties seems to be the correlation between Euclideandistances. We can thus obtain a more tractable mathematical problem bysimply making all distances independent. I.e., the random weighted graphis now obtained by starting with a complete graph, and then giving eachedge an i.i.d. edge weight. This is called the mean field setting. A relatedsetting is where the graph is a sparse random graph (e.g. Erd˝os-R´enyi orrandom regular); this is called the sparse graph setting. In the 1980s, sta-tistical physicists obtained predictions on the limiting constants for variouscombinatorial optimization problems in the mean field setting. See W¨astlund[32, 35] for a list of references. In 2001, Aldous [2] provided the first rigorousproof of one of these predictions, for the minimum matching problem. Hisgeneral proof strategy was a rigorous version of the Cavity method fromstatistical physics. It is called the Objective method, or the Local weakconvergence method; see Aldous and Steele [4] for a survey. The Objectivemethod gives a general purpose approach to computing limiting constantsin the mean field and sparse graph settings, and has been applied to otherproblems; see [1, 16, 20, 22, 23, 28, 32, 35] for an incomplete list. Besides the PARSE OPTIMIZATION CLTS objective method, there are also other ways of computing limiting constants,see [15, 18, 25, 26, 30, 31, 33, 34] (again an incomplete list).Thus for many problems, we understand very well the first order behavior,so let us now look at the fluctuations. It is commonly believed that the op-timal value for various random combinatorial optimization problems shouldbe asymptotically Normal; see e.g. the discussion in Section 5 of Chatterjee[10]. However, in the Euclidean setting, I am only aware of two problemsfor which a central limit theorem has been proven: minimal spanning tree(Alexander [5], Kesten and Lee [21]), and Euclidean bipartite matching (delBarrio and Loubes [13]). Minimal spanning tree is particularly amenableto mathematical analysis because there is a greedy algorithm for solvingthe problem, which leads to many convenient properties. Such propertieswere also used by Chatterjee and Sen [12] to obtain rates of convergencefor minimal spanning tree in the Euclidean and lattice settings. The conve-nient property of Euclidean bipartite matching is that it may be written asan optimal transport problem, and thus techniques from optimal transporttheory may be used.There are also general central limit theorems for functions of Euclideanpoint processes; see Yukich [37] for a survey and references, and see L`achieze-Rey et al. [24] for a recent result. However, for combinatorial optimizationproblems, verifying the conditions of the general theorems seems to be anopen problem.Turning now to the mean field setting, I am only aware of a central limittheorem for minimal spanning tree (Janson [19]), whose proof also uses con-venient properties of the problem. For minimum matching, a conjecture isgiven by Hessler and W¨astlund [17].In any of the settings mentioned (Euclidean, lattice, mean field, sparsegraph), there does not seem to be a general purpose strategy for obtainingcentral limit theorems. The present paper seeks to make a dent in this di-rection, for the sparse graph setting. In particular, a general central limittheorem is proven, and is applied to give central limit theorems for variouscombinatorial optimization problems that have been previously studied inthe literature.1.2. Setting.
For λ >
0, let p n := λ/n , or more generally, np n → λ . Inwords, λ is the asymptotic average vertex degree. With p n implicit, let G n bean Erd˝os-R´enyi graph on n vertices [ n ] := { , . . . , n } , with edge probability p n . Additionally, G n will have edge weights, which are i.i.d. from some non-negative distribution F w n which may depend on n . However, we will assumethat F w n converges in total variation to some distribution F w . With p n SKY CAO and F w n implicit, we will denote this weighted graph by G n , which maybe represented by a pair ( W n , B n ), where W n , B n are independent, with W n = ( w nij , ≤ i < j ≤ n ), and B n = ( b nij , ≤ i < j ≤ n ). The entries of W n are i.i.d. from F w n , and the entries of B n are i.i.d. Bernoulli( p n ). Fornotational convenience, we will often hide the dependence on n and write W, B instead of W n , B n . Generic vertices will typically be denoted v, u , andas such for edges e = ( v, u ), we will often write w e = w ( v,u ) = w ( u,v ) , and b e = b ( v,u ) = b ( u,v ) . Note we still refer to e = ( v, u ) as an “edge”, even if itis not present in the weighted graph G n .We will study optimization problems, denoted by a function f which takesweighted graphs as input. Under certain conditions on f , we will be able toshow that f ( G n ) − E f ( G n ) p Var( f ( G n )) d −→ N (0 , . In this paper, C will denote a numerical constant which may always takento be larger, and which may change from line to line.
2. Main Results.
In this section, we will introduce the combinatorialoptimization problems that are considered, and collect the main results. Asan overview, Section 2.4 introduces the general central limit theorem (The-orem 2.4), and Sections 2.1-2.3 describe applications of the general theorem.The proof of the general theorem (Section 5) is by the generalized perturba-tive approach to Stein’s method (introduced by Chatterjee in [9], see also hissurvey [10]). Although the general theorem is for the sparse graph setting,Section 2.3 actually gives a central limit theorem in the mean field setting.The basic idea is that the mean field setting may be approximated by thesparse graph setting; this idea comes from W¨astlund [32, 35].The key assumption in the general theorem is introduced by Definition 2.6.Various forms of this key assumption have appeared before in the literature:Aldous and Bandyopadhyay [3] call it “endogeny”, Gamarnik et al. [16]call it “long-range independence”, and W¨astlund [32, 35] calls it “replicasymmetry”. The reason for this is because in applying the Objective methodto compute the limiting constant for a given combinatorial optimizationproblem, the key problem-specific step is in verifying endogeny/long-rangeindependence/replica symmetry. Thus this paper may be summarized asfollows: in the sparse graph setting, if we can compute the limiting constantfor a given problem by the Objective method, then assuming we are able toverify certain technical conditions, we also get a central limit theorem.The first two combinatorial optimization problems we introduce deal withmatchings on graphs. A matching of a graph is a collection of edges such
PARSE OPTIMIZATION CLTS that each vertex is incident to at most one edge of the collection. Given aweighted graph, we may naturally define the weight of a matching to be thetotal sum of edge weights over edges in the matching.2.1. Maximum weight matching.
Given a weighted graph, we define themaximum weight matching to be the matching with maximal weight. Fix λ >
0. Let G n be the weighted graph with p n = λ/n , and i.i.d. Exp(1) edgeweights (so in this case, the edge weight distribution does not depend on n ).Let M n = M ( G n ) be the weight of the maximum weight matching of G n .In [16], it was proven that maximum weight matching possesses long-rangeindependence, which allowed the authors to show M n /n p → β ( λ ), with β ( λ )explicitly characterized. The following theorem gives a central limit theoremfor M n . Theorem . We have M n − E M n p Var( M n ) d −→ N (0 , . λ -diluted minimum matching. Fix λ >
0. Let K n denote the com-plete graph with i.i.d. edge weights distributed as n Exp(1). For a matchingof K n , define its λ -diluted cost as the sum of the edge weights in the match-ing, plus λ/ M λ ( K n ) be theminimal λ -diluted cost among all matchings.A priori, this problem doesn’t seem to be an optimization problem on asparse graph. But observe that in finding the λ -diluted minimum matching,we may ignore all edges in K n of weight larger than λ . Thus M λ is actuallya function of K n ( λ ), the subgraph of K n consisting of all edges with weightat most λ . Observe that K n ( λ ) is exactly a weighted sparse Erd˝os-R´enyigraph where p n = 1 − e − λ/n ≈ λ/n , and F w n is the distribution of n Exp(1)conditioned to lie in [0 , λ ] (which converges in total variation to Unif[0 , λ ]as n → ∞ ). Thus λ -diluted minimum matching can be made to fit into ourframework, and thus we have a central limit theorem. Theorem . We have M λ ( K n ) − E M λ ( K n ) p Var( M λ ( K n )) d −→ N (0 , . To give some background, this problem was introduced by W¨astlund[32, 35], see also his paper with Parisi [27]. These papers all study theminimum matching problem, which is the λ → ∞ limit of the λ -diluted SKY CAO minimum matching problem. The λ -diluted problem was introduced as amore localized version of minimum matching, and as such proved easier toanalyze. Replica symmetry was shown for M λ , which led to the proof of M λ ( K n ) /n p → β ( λ ), with β ( λ ) explicitly characterized. Results on minimummatching were then deduced from the results on the λ -diluted problem bytaking λ → ∞ . In this way one can think of the minimum matching problemon a complete graph as “essentially” a sparse graph problem.2.3. Optimal edge cover.
As before, let K n be the complete graph withi.i.d. edge weights distributed as n Exp(1). An edge cover is a collection ofedges such that each vertex of K n is incident to at least one edge of thecollection. Naturally, the cost of an edge cover is the sum of edge weightsover all edges in the edge cover. The optimal edge cover is defined to be theedge cover of minimal weight, and its cost is denoted EC ( K n ). Theorem . We have EC ( K n ) − E EC ( K n ) p Var( EC ( K n )) d −→ N (0 , . The proof of this will be through what is essentially a truncation argu-ment. Similar to the previous section, we will define a certain relaxed versionof optimal edge cover, indexed by a parameter λ >
0, and denoted EC λ ( K n ).The relaxed problem is related to optimal edge cover bylim λ →∞ EC λ ( K n ) = EC ( K n ) . Moreover, the relaxed problem is an optimization problem on a sparse Erd˝os-R´enyi graph, so that the methods of this paper will apply to give a centrallimit theorem for EC λ ( K n ). Even more, we will have a rate of convergence,which with a little bit of work, can be shown to be robust enough to allowus to take λ to infinity with n . This will then allow us to transfer the centrallimit theorem for EC λ ( K n ) to a central limit theorem for EC ( K n ).The relaxed problem was introduced by W¨astlund [32]. First take λ > λ -diluted cost of a collection of edges (not necessarilyan edge cover) as the sum of edge weights over all edges in the collection,plus λ/ λ/ λ -diluted edge cover is defined to be the collection ofedges (again, not necessarily an edge cover) of minimal λ -diluted cost, andits cost is denoted EC λ ( K n ). Observe EC λ is actually just a function of K n ( λ ), and thus is an optimization problem on a sparse Erd˝os-R´enyi graph. PARSE OPTIMIZATION CLTS W¨astlund [32] showed replica symmetry for EC λ , which led to the proof of EC λ ( K n ) p → β ( λ ), with β ( λ ) explicitly characterized.2.4. The general result.
Given a weighted graph G , and a vertex v , let G − v denote the weighted graph obtained by deleting v and all edges inci-dent to v . Given an integer k ≥
0, let B k ( v, G ) denote the weighted graphobtained from the union of all paths in G which start at v and are of lengthat most k . We will call v the root of B k ( v, G ), even if B k ( v, G ) is not a tree.We may think of B k ( v, G ) as a (small) neighborhood of v . One key technicalfact is that we can replace B k ( v, G n ) by a limiting object, which we nowdefine. Definition . For λ >
0, let T ( ∞ , λ ) denote a Galton-Watson pro-cess with offspring distribution Poisson( λ ). For integer k >
0, let T ( k, λ )denote the depth k subtree of T ( ∞ , λ ). Additionally, given a weight dis-tribution F w , let T ( ∞ , λ, F w ) denote T ( ∞ , λ ), equipped with edge weightswhich are i.i.d. from F w . Let T ( k, λ, F w ) denote the weighted depth k subtreeof T ( ∞ , λ, F w ).The point is that B k ( v, G n ) is essentially T k , in the following sense (seeSection 6 for precise statements). Definition . Let G be a weighted graph, and let v be a vertex of G .Let T be a rooted weighted tree. Let k >
0. We say B k ( v, G ) ∼ = T if thereexists a bijection ϕ between the vertices of B k ( v, G ) and the vertices of T ,which maps v to the root of T , and preserves all edges and edge weights.I.e., if ( u, u ′ ) is an edge of B k ( v, G ) with weight w , then ( ϕ ( u ) , ϕ ( u ′ )) is anedge of T with weight w .In words, B k ( v, G ) ∼ = T if the two objects differ only by a vertex rela-beling. We extend Definition 2.2 to pairs of neighborhoods which share thesame root. Definition . Let G , G ′ be weighted graphs which share the samevertex set. Let v be a vertex of G , G ′ . Let T , T ′ be trees which share the sameroot. Let k >
0. We say that ( B k ( v, G ) , B k ( v, G ′ )) ∼ = ( T , T ′ ), if there existsa bijection ϕ between the vertices of B k ( v, G ) , B k ( v, G ′ ) and the vertices of T , T ′ , which maps v to the root of T , T ′ , and preserves all edges and edgeweights. I.e., if ( u, u ′ ) is an edge of B k ( v, G ) with weight w , then ( ϕ ( u ) , ϕ ( u ′ ))is an edge of T with weight w . Similarly, if ( u, u ′ ) is an edge of B k ( v, G ′ )with weight w , then ( ϕ ( u ) , ϕ ( u ′ )) is an edge of T ′ with weight w . SKY CAO
For technical reasons, we will also need to work with the following objects.
Definition . Given λ >
0, and a weight distribution F w , define˜ T ( ∞ , λ, F w ) as follows. Take T , T ′ i.i.d. ∼ T ( ∞ , λ, F w ). Let ∅ , ∅ ′ denote theroots of T , T ′ respectively. Construct ˜ T ( ∞ , λ, F w ) as the tree with root ∅ , obtained by starting with T , and then adding an edge between ∅ , ∅ ′ with edge weight distributed as F w , independent of everything else. Let˜ T ( k, λ, F w ) be the depth k subtree of ˜ T ( ∞ , λ, F w ).Note that ˜ T k d = ˜ T ( k, λ, F w ) may be constructed in the following manner.Take T k , T ′ k − independent, with T k d = T ( k, λ, F w ), T ′ k − d = T ( k − , λ, F w ),with roots ∅ , ∅ ′ respectively. Let ℓ ∼ F w independent of everything else.Then define ˜ T k to be the tree with root ∅ constructed by connecting ∅ , ∅ ′ with an edge of weight ℓ . When ˜ T k is defined this way, we say that ˜ T k isconstructed from ( T k , T ′ k − , ∅ , ∅ ′ , ℓ ). Remark.
Observe that the underlying graph of ˜ T ( ∞ , λ, F w ) is a Galton-Watson process, where the root has offspring distribution 1 + Poisson( λ ),and every subsequent vertex has offspring distribution Poisson( λ ). Definition . Let ( W ′ , B ′ ) be an i.i.d. copy of ( W, B ). Given an edge e , define G en to be the weighted graph obtained by using w ′ e , b ′ e in place of w e , b e . Let ∆ e f := f ( G n ) − f ( G en ).We now present the key assumption that an optimization problem mustsatisfy for us to be able to prove a central limit theorem. Roughly speaking,it says that small perturbations of the problem must be able to be locallyapproximated. See Sections 3 or 5 of [10] for the motivation for making suchan assumption. Definition . Let f be a function on weighted graphs. We say that( f, ( G n , n ≥ Property GLA (“good local approximation”) for λ, F w , if np n → λ , d T V ( F w n , F w ) →
0, and for each k > LA Lk , LA Uk , which take as input pairs of rooted weighted trees,such that the following conditions hold.(A1) For any edge e = ( v, u ), if B k := B k ( v, G n ) and B ′ k := B k ( v, G en ) aretrees, then LA Lk ( B k , B ′ k ) ≤ ∆ e f ≤ LA Uk ( B k , B ′ k ) . (A2) For any edge e , and any k >
0, if ( T , T ′ ) is such that we have( B k , B ′ k ) := ( B k ( v, G n ) , B k ( v, G en )) ∼ = ( T , T ′ ), then LA Lk ( T , T ′ ) = LA Lk ( B k , B ′ k ) , PARSE OPTIMIZATION CLTS LA Uk ( T , T ′ ) = LA Uk ( B k , B ′ k ) . (A3) Let ˜ T k d = ˜ T ( k, F w , λ ) be constructed from ( T k , T ′ k − , ∅ , ∅ ′ , ℓ ). Define δ k := max (cid:18) E ( LA Uk ( ˜ T k , T k ) − LA Lk ( ˜ T k , T k )) , E ( LA Uk ( T k , ˜ T k ) − LA Lk ( T k , ˜ T k )) (cid:19) . Then lim k →∞ δ k = 0 . Remark.
In words, Property GLA ensures that when we perturb theweighted graph G n at a single edge e , the resulting change in f may beapproximated by LA Lk . Then by (A1), the approximation error is at most LA Uk − LA Lk . In analyzing this term, (A2) allows us to replace neighbor-hoods of Erd˝os-R´enyi graphs by Galton-Watson trees. Then by (A3), wemay conclude that the approximation error goes to 0 as k goes to infinity.The construction of LA Lk , LA Uk to fulfill (A1) usually proceeds by exploit-ing certain recursive properties of the given function f . Typically, (A2) willbe trivial to check, because in constructing the local approximations, wewill never use the vertex labels. The recursive distributional properties ofGalton-Watson trees are usually used to verify (A3).As alluded to earlier, variants of assumptions (A1)-(A3) have appearedbefore in the literature. The present paper states them in a way that isconvenient for proving central limit theorems.We now state the general result, which says that any optimization prob-lem which has Property GLA, with some additional regularity conditions,satisfies a central limit theorem. Theorem . Suppose ( f, ( G n , n ≥ satisfies Property GLA for λ, F w . Suppose additionally that the following regularity condition is sat-isfied. There is a function H such that (2.1) J := max (cid:18) , sup n E H ( w e , w ′ e ) (cid:19) < ∞ , (here the the dependence on n comes from w e , w ′ e , which are distributed like F w n ), and for any e = ( v, u ) , we have (2.2) | ∆ e f | ≤ b e , b ′ e ) = 1) H ( w e , w ′ e ) . SKY CAO
Let σ n := Var( f ( G n )) , λ n := np n , Z n := f ( G n ) − E f ( G n ) σ n , and let Φ denote the standard normal cdf. There is a numerical constant C ,such that with ε k ( n ) := (2 λ + 3) k n / + C ( λ n + 1) k min( λ, (cid:18) | λ n − λ | + d T V ( F w n , F w ) + λ n (cid:19) , and ρ k ( n ) := min (cid:18) ( λ n + C ) k + C n , (cid:19) , we have for any k, n > , sup t ∈ R | P ( Z n ≤ t ) − Φ( t ) | ≤ C J / "(cid:18) nσ n (cid:19) / (cid:18) δ / k + ε k ( n ) / + ρ k ( n ) / (cid:19) + (cid:18) nσ n (cid:19) / λ / n n / . Remark.
To help parse the rate of convergence, note that by assump-tion, δ k → λ n → λ , and for all k , lim n →∞ ε k ( n ) = 0, lim n →∞ ρ k ( n ) = 0.Thus to obtain convergence, one naturally will first take n → ∞ , and then k → ∞ . This will be successful as long as one is able to show that the vari-ance σ n is at least of order n . This variance lower bound may in general benontrivial to obtain. However, for various problems, one may use the gen-eral method introduced by Chatterjee [11]. The other regularity conditionsshould be easier to verify.The following corollary gives sufficient conditions under which PropertyGLA holds. It also simplifies the conclusion of the previous theorem, at thecost of no longer giving a rate of convergence. Corollary . Suppose we have ( f, ( G n , n ≥ , such that np n → λ , d T V ( F w n , F w ) → . Suppose for each k there exists functions g Lk , g Uk suchthat when B k ( v, G n ) is a tree, we have (2.3) g Lk ( B k ( v, G n )) ≤ f ( G n ) − f ( G n − v ) ≤ g Uk ( B k ( v, G n )) . Moreover, if B k ( v, G n ) ∼ = T , then (2.4) g Lk ( T ) = g Lk ( B k ( v, G n )) , g Uk ( T ) = g Uk ( B k ( v, G n )) . PARSE OPTIMIZATION CLTS Additionally, let T k d = T ( k, λ, F w ) , ˜ T k d = ˜ T ( k, λ, F w ) , and suppose (2.5) lim k →∞ E ( g Uk ( T k ) − g Lk ( T k )) = 0 , (2.6) lim k →∞ E ( g Uk ( ˜ T k ) − g Lk ( ˜ T k )) = 0 . Suppose also that the following regularity conditions are satisfied. • The variance is at least of order n : lim inf n →∞ n − Var( f ( G n )) > . • There exists a function H such that sup n E H ( w e , w ′ e ) < ∞ , and for any e = ( v, u ) , we have | ∆ e f | ≤ b e , b ′ e ) = 1) H ( w e , w ′ e ) . Then f ( G n ) − E f ( G n ) p Var( f ( G n )) d −→ N (0 , . Remark.
Theorem 2.4 and Corollary 2.5 seek to hide away as manytechnical details involving Erd˝os-R´enyi graphs as possible. So to prove acentral limit theorem, one may work almost exclusively with Galton-Watsontrees, which due to their recursive nature, are much nicer objects.
3. Applications of Corollary 2.5.
In this section, we will apply thesimpler Corollary 2.5 to the first two combinatorial optimization problemslisted in Section 2. In both cases, assumption (2.4) will be clear from con-struction of the g Lk , g Uk .For rooted weighted trees T , we will denote the root by ∅ . For vertices u ∈ T , we will denote the set of children of u by C ( u ). Additionally, for edges( v, u ) in T with u the child, we will denote the edge weight by ℓ u .3.1. Maximum weight matching.
In this problem, we have that p n = λ/n , and the weight distribution is Exp(1) for all n . The ideas of [16] willallow us to verify Property GLA. I detail them here with no claims of orig-inality. SKY CAO
Construction of g Lk , g Uk . For v ∈ G , observe that we have the recur-sion M ( G ) = max (cid:18) M ( G − v ) , max u :( v,u ) ∈ G w ( v,u ) + M ( G − { v, u } ) (cid:19) . Defining h ( G , v ) := M ( G ) − M ( G − v ), we thus have(3.1) h ( G , v ) = max (cid:18) , max u :( v,u ) ∈ G w ( v,u ) − h ( G − v, u ) (cid:19) . We will use this recursion to define the local approximations. Given aninteger k >
0, and a rooted weighted tree T of depth at most k , define h k ( · ; T ) : T → R in the following manner. For all leaf vertices u ∈ T ,set h k ( u ; T ) := 0. Then use (3.1) to define h k at all other vertices. I.e., fornon-leaf vertices u ∈ T , set h k ( u ; T ) := max (cid:18) , max u ′ ∈C ( u ) ℓ u ′ − h k ( u ′ ; T ) (cid:19) . One may verify by using induction that for any even k such that B k ( v, G n )is a tree, we have h k ( v ; B k ( v, G n )) ≤ h ( G n , v ) , and for any odd k such that B k ( v, G n ) is a tree, we have h ( G n , v ) ≤ h k ( v ; B k ( v, G n )) . Thus for any odd k such that B k ( v, G n ) is a tree, we have h k − ( v ; B k − ( v, G n )) ≤ h ( G n , v ) ≤ h k ( v ; B k ( v, G n )) . Now to define g Lk , g Uk , let i L := 2 ⌊ ( k − / ⌋ , and i U := 2 ⌊ ( k − / ⌋ + 1. I.e., i U is the largest odd number less than or equal to k , and i L = i U −
1. Thisdefinition ensures that i L , i U ≤ k , and thus we may set (when B k ( v, G n ) isa tree) g Lk ( B k ( v, G n )) := h i L ( v ; B i L ( v, G n )) , g Uk ( B k ( v, G n )) := h i U ( v ; B i U ( v, G n )) . With this definition, (2.3) is satisfied.
Verification of (2.5) , (2.6) . Let T ∞ d = T ( ∞ , λ, Exp(1)), and let T k be the depth k subtree of T ∞ . For brevity, we write h k ( ∅ ) instead of h k ( ∅ ; T k ). To verify (2.5), it suffices to show(3.2) lim r →∞ E ( h r +1 ( ∅ ) − h r ( ∅ )) = 0 . PARSE OPTIMIZATION CLTS With this coupling of the trees ( T k , k ≥ h r +1 ( ∅ ) is non-increasing in r , h r ( ∅ ) is non-decreasing in r .Observe also that for all r , h r ( ∅ ) ≤ h r +1 ( ∅ ) . Defining h U := lim r →∞ h r +1 ( ∅ ), h L := lim r →∞ h r ( ∅ ), we thus have that h r +1 ( ∅ ) − h r ( ∅ ) ↓ h U − h L . Note 0 ≤ h r +1 ( ∅ ) − h r ( ∅ ) ≤ h r +1 ( ∅ ) ≤ h ( ∅ ) ≤ max u ∈C ( ∅ ) ℓ u , and the quantity on the right hand side has finite second moment. Thus bydominated convergence, to verify (3.2), it suffices to show that h U − h L = 0a.s. As h L ≤ h U , the following lemma suffices. Lemma . E h L = E h U . Proof.
It follows by Theorem 3 and Proposition 1 of [16] that h k ( ∅ ) d −→ X ∗ , for some X ∗ . This implies h L d = h U , and thus E h L = E h U . Remark.
In a sense, everything before this lemma is routine, while theassertion that h L d = h U is nontrivial. This is one of the major results of[16], and it is essentially this assertion that is refered to as “long-rangeindependence” by Gamarnik et al.Once we’ve verified (2.5), (2.6) follows easily. Let ˜ T k be constructed from( T k , T ′ k − , ∅ , ∅ ′ , ℓ ). Moreover, we may assume that ˜ T k , ˜ T k +1 are coupled sothat ˜ T k is the depth k subtree of ˜ T k +1 . It then suffices to showlim r →∞ E ( h r +1 ( ∅ ; ˜ T r +1 ) − h r ( ∅ ; ˜ T r )) = 0 . Observe h r +1 ( ∅ ; ˜ T r +1 ) = max (cid:18) h r +1 ( ∅ ; T r +1 ) , ℓ − h r ( ∅ ′ ; T ′ r ) (cid:19) , and h r ( ∅ ; ˜ T r ) = max (cid:18) h r ( ∅ ; T r ) , ℓ − h r − ( ∅ ′ ; T ′ r − ) (cid:19) . SKY CAO
Letting X r := h r +1 ( ∅ ; T r +1 ) − h r ( ∅ ; T r ), X ′ r := h r − ( ∅ ′ ; T ′ r − ) − h r ( ∅ ′ ; T ′ r ), we have0 ≤ h r +1 ( ∅ ; ˜ T r +1 ) − h r ( ∅ ; ˜ T r ) ≤ X r + X ′ r , and thus (cid:18) h r +1 ( ∅ ; ˜ T r +1 ) − h r ( ∅ ; ˜ T r ) (cid:19) ≤ X r + 2( X ′ r ) . We’ve already shown E X r →
0, and a small modification of the proof alsoshows E ( X ′ r ) →
0, and thus (2.6) is verified.With Property GLA established, we proceed to verify the regularity con-ditions of Corollary 2.5. Fix e = ( v, u ). To determine the function H , bysplitting into the cases b e = 0 , b ′ e = 0 ,
1, we may obtain | M ( G n ) − M ( G en ) | ≤ b e , b ′ e ) = 1) max( w e , w ′ e ) . Thus we may take H ( w e , w ′ e ) := max( w e , w ′ e ). As w e , w ′ e i.i.d. ∼ Exp(1), clearly E H ( w e , w ′ e ) < ∞ .The application of Corollary (2.5) to prove Theorem (2.1) will now becomplete as soon as we show the following variance lower bound. Lemma . We have lim inf n →∞ n − Var( M ( G n )) > . Proof.
We use the general framework of [11]. For brevity, let M n := M ( G n ). As observed in [11], it suffices to find constants c , c > n , for b − a ≤ c √ n , we have P ( a ≤ M n ≤ b ) ≤ − c . To find c , c , first observe that conditional on the underlying graph G n ,the law of G n is some structured collection of i.i.d. Exp(1) random variables,call them w , . . . , w E n , where E n is the number of edges in G n . For α > ε := ε n := αn − / , and w ′ i := w i / (1 − ε ), 1 ≤ i ≤ E n .Let M ′ n be the maximum weight matching of G n with the edge weights w ′ , . . . , w ′ E n . Lemma 1.2 of [11] implies that for −∞ < a ≤ b < ∞ , we have(3.4) P ( a ≤ M n ≤ b ) ≤
12 (1 + P ( (cid:12)(cid:12) M n − M ′ n (cid:12)(cid:12) ≤ b − a ) + d T V ( L M n , L M ′ n )) , PARSE OPTIMIZATION CLTS where d T V ( · , · ) is total variation distance, and L M n , L M ′ n are the laws of M n , M ′ n , respectively. Let d T V ( · , · | G n ) denote total variation distance con-ditional on G n . Then it follows by Corollary 1.8 of [11] that d T V ( L M n , L M ′ n | G n ) ≤ C ( E n α /n ) / = C ( E n /n ) / α. Thus d T V ( L M n , L M ′ n ) ≤ E d T V ( L M n , L M ′ n | G n ) ≤ Cα E ( E n /n ) / ≤ C √ λα, where the final inequality follows by noting E n ∼ Binomial( n ( n − / , λ/n ).Observe now that M ′ n = M n / (1 − ε ), and thus | M n − M ′ n | = M n ε/ (1 − ε ).By Theorem 3 of [16], we have M n /n p → β ( λ ) >
0. In particular, for some c > P ( M n ε/ (1 − ε ) ≤ c √ n ) ≤ P ( M n /n ≤ c /α ) → . We now choose α small so that d T V ( L M n , L M ′ n ) ≤ / c small depending on α so that the above holds. Now by (3.4), wehave that for large enough n , for any b − a ≤ c √ n , P ( a ≤ M n ≤ b ) ≤
12 (1 + 1 / /
2) = 7 / . As detailed at the beginning of the proof, this implies the desired variancelower bound.3.2. λ -diluted minimum matching. With λ implicit, let G n := K n ( λ ).Recall that p n = 1 − e − λ/n , and F w n is distributed as n Exp(1), conditionedto lie in [0 , λ ]. We have np n → λ and d T V ( F w n , Unif[0 , λ ]) →
0. To ver-ify Property GLA, we follow the ideas of [27, 32, 35], with no claims oforiginality.
Verification of Property GLA.
For v ∈ G n , observe M λ ( G n ) = min (cid:18) λ M λ ( G n − v ) , min u :( v,u ) ∈ G n w ( v,u ) + M λ ( G n − { v, u } ) (cid:19) . Defining h λ ( G , v ) := M λ ( G ) − M λ ( G − v ), we have(3.5) h λ ( G n , v ) = min u :( v,u ) ∈ G n (cid:18) λ , w ( v,u ) − h ( G n − { v, u } ) (cid:19) . Note as w e ∈ [0 , λ ] for all edges e , we have that h λ ∈ [ − λ/ , λ/ g Lk , g Uk . SKY CAO
And again, the key step in verifying (2.5), (2.6) is showing that with T k d = T ( k, λ, Unif[0 , λ ]), we have that g Lk ( T k ) , g Uk ( T k ) converge in distribution tothe same limit. It is essentially this condition that W¨astlund calls “replicasymmetry”, and it is given by Theorem 3.3 of [27] (and in a more generalsetting in [32, 35]).To verify the regularity conditions of Corollary 2.5, first note M λ ( G n ) − M λ ( G en ) = h λ ( G n , v ) − h λ ( G en , v ), and recall h λ ( G n , v ) ∈ [ − λ/ , λ/ H ( w e , w ′ e ) = λ . So really the only thing that needs proving isthe variance lower bound. Lemma . For fixed λ > , we have lim inf n →∞ n − Var( M λ ( G n )) > . Proof.
The proof is a small adaptation of the proof of Theorem 2.9 of[11]. To follow that proof more closely, we first do some rescaling. Let ˜ G n be G n with all edge weights divided by n , so that the edge weights of ˜ G n are Exp(1). We then consider M λ/n ( ˜ G n ), which is equal to n − M λ ( G n ). Itsuffices to show lim inf n →∞ n Var( M λ/n ( ˜ G n )) > . For brevity, denote M n := M λ/n ( ˜ G n ). As mentioned in the proof of Lemma3.2, it suffices to find constants c , c > n , for b − a ≤ c / √ n , we have P ( a ≤ M n ≤ b ) ≤ − c . Towards this end, define the function φ : [0 , ∞ ) → [0 , ∞ ), φ ( x ) = ( √ nx if 0 ≤ x ≤ /nx + 1 / √ n − /n if x > /n. Let α > A = ( a ij , ≤ i < j ≤ n ) be the edge weightsof ˜ G n , and define A ′ = ( a ′ ij , ≤ i < j ≤ n ), where a ′ ij is such that a ′ ij + αn − φ ( a ′ ij ) = a ij . Note as the map x x + αn − φ ( x ) is continuous and strictly increasing, a ′ ij exists and is unique. The proof of Theorem 2.9 of [11] shows that d T V ( L A , L A ′ ) ≤ Cα.
PARSE OPTIMIZATION CLTS Defining M ′ n to be the cost of the λ/n -diluted minimum matching with theweights A ′ , we thus have d T V ( L M n , L M ′ n ) ≤ Cα.
Now by Lemma 1.2 of [11], for all −∞ < a ≤ b < ∞ , we have(3.6) P ( a ≤ M n ≤ b ) ≤
12 (1 + P ( (cid:12)(cid:12) M n − M ′ n (cid:12)(cid:12) ≤ b − a ) + Cα ) , so our goal now is to bound P ( | M n − M ′ n | ≤ b − a ).Observe that a ′ ij ≤ a ij for all i < j , so that M ′ n ≤ M n , so that we have | M n − M ′ n | = M n − M ′ n . Fix 1 ≥ β > b i := min j = i a ij (where a ij = a ji if i > j ). Let D n := { i : b i ≥ β/n } . For i ∈ D n , we have a ij ≥ β/n for all j = i . As x x + αn − φ ( x ) is increasing, we have that a ′ ij ≥ x n , where x n is the unique solution of x n + αn − φ ( x n ) = βn − . From the definition of φ , and as β ≤
1, we have x n = βn + α √ n . Thus for i ∈ D n , and j = i , a ij − a ′ ij = αn − φ ( a ′ ij ) ≥ αn − φ βn + α √ n ! = αβn / + αn . Now let B n := { vertices that are matched in the λ/n -diluted minimum matching of ˜ G n } . We have M n − M ′ n ≥ X i = j i is matched to j in M n )( a ij − a ′ ij ) ≥ X i ∈ B n X j = i i is matched to j in M n )( a ij − a ′ ij ) ≥ X i ∈ D n ∩ B n αβn / + αn = 12 αβ | D n ∩ B n | n / + αn . SKY CAO
Now suppose for the moment that | D n ∩ B n | /n p → κ >
0. Then there exists c depending on α, β, κ such that P ( M n − M ′ n ≤ c / √ n ) → . Thus recalling (3.6), by taking α small so that d T V ( L M n , L M ′ n ) ≤ Cα ≤ / n , and any b − a ≤ c / √ n , P ( a ≤ M n ≤ b ) ≤
12 (1 + 1 / /
4) = 34 . Thus the proof is complete once we show that we may take β > | D n ∩ B n | /n p → κ >
0. As | D n ∩ B n | ≥ | D n | − | B cn | , this be immediate oncewe establish the following two lemmas. Lemma . We have | D n | /n p → exp( − β ) . Proof.
This follows by computing the first and second moments.
Lemma . We have that | B cn | /n converges in probability to a constantstrictly less than 1. Proof.
Note | B cn | is the number of unmatched vertices in the λ -dilutedminimum matching. By Proposition 3.1 of [35], we have that | B cn | /n p → F λ ( λ/ F λ : [ − λ/ , λ/ → [0 ,
1] is some function. Moreover, fromthe proof of Proposition 2.11 of [35], we have λ ≤ − log F λ ( λ/ F λ ( λ/ , which implies that F λ ( λ/ < λ >
4. Application of Theorem 2.4 to optimal edge cover.
We devotea separate section for Optimal edge cover because unlike in Section 3, wewill need to spend some time establishing some basic facts before we canapply Theorem 2.4.The approach to proving Theorem 2.3 will be as follows. Because Theorem2.4 gives a rate of convergence, we will be able to first prove a central limittheorem for the quantity EC λ n ( K n ) − E EC λ n ( K n ) p Var( EC λ n ( K n )) , PARSE OPTIMIZATION CLTS where now λ n is taken to infinity with n . Moreover, we will show that λ n islarge enough so that(4.1) Var( EC ( K n ))Var( EC λ n ( K n )) → , and(4.2) E EC ( K n ) − E EC λ n ( K n ) p Var( EC ( K n )) → . This will then allow us to conclude Theorem 2.3.4.1.
Basic facts of optimal edge cover.
As detailed in Section 2.3, EC λ is actually a function of K n ( λ ), and so is an optimization problem on asparse Erd˝os-R´enyi graph. However, unless the situation demands, we willcontinue writing K n for brevity. We first investigate how large λ needs tobe for EC λ ( K n ) to be a good approximation of EC ( K n ). We will see thatthe answer is λ = C log n . Lemma . Suppose there is a number K such that every vertex v ∈ K n has at least one incident edge e with w e ≤ K . Then every edge in the optimaledge cover has weight at most K . Proof.
Let e = ( v, u ) be in the optimal edge cover. By hypothesis, thereare edges e v , e u incident to v, u respectively, such that w e v , w e u ≤ K . Nowby optimality, we must have w e ≤ w e v + w e u ≤ K . Lemma . If every edge in the optimal edge cover has weight at most K , then we have EC K ( K n ) = EC ( K n ) . Proof. As EC λ ( K n ) ≤ EC ( K n ) for all λ >
0, only one direction needsto be proven. Let C be the optimal 4 K -diluted edge cover. If the collection C covers every vertex, then it is in fact an edge cover and thus equality isautomatic. So suppose C leaves a vertex v un-covered. Let e v be the edgein the optimal edge cover incident to v . Then by adding the edge e v to C ,the λ -diluted cost of C increases by at most w e v − K ≤
0. Repeating for allun-covered vertices, we obtain EC ( K n ) ≤ EC K ( K n ), as desired.These two lemmas show that if K n ( λ ) has no isolated vertices, then wehave EC λ ( K n ) = EC ( K n ). Let p n ( λ ) := 1 − e − λ/n , and let deg λ ( v ) be thedegree of vertex v in K n ( λ ). SKY CAO
Lemma . We have P (cid:18) deg λ ( v ) > np n ( λ ) , ∀ v ∈ V (cid:19) ≥ − ne − np n ( λ ) / . Proof.
By Theorem 8.1 of [7], we have for a given v ∈ K n P (cid:18) deg λ ( v ) ≤ np n ( λ ) (cid:19) ≤ e − np n ( λ ) / . We conclude by applying the Union bound.
Proposition . For any constant C , there is a constant C possiblydepending on C such that for λ n = C log n , and large enough n , we have P ( EC λ n ( K n ) = EC ( K n )) ≥ − n − C . Proof.
By the previous few lemmas, we have P ( EC λ n ( K n ) = EC ( K n )) ≤ P (cid:18) ∃ v, deg λ ( v ) ≤ np n ( λ n ) (cid:19) ≤ ne − np n ( λ n ) / . With λ n = C log n , we have p n ( λ n ) = 1 − e − λ n /n ≥ λ n n for large enough n . Thus we see that it suffices to take C = 64( C + 1).With λ n = C log n , we now proceed to show (4.1), (4.2). First, we needa variance lower bound. Lemma . We have lim inf n →∞ n Var( EC ( K n )) > . Proof.
The proof of Theorem 2.9 of [11] carries over with a slight mod-ification (the argument is for complete bipartite graphs, but it also worksfor complete graphs) to showlim inf n →∞ n Var( n − EC ( K n )) > . Proposition . There is a numerical constant C such that with λ n = C log n , we have Var( EC ( K n ))Var( EC λ n ( K n )) → , and E EC ( K n ) − E EC λ n ( K n ) p Var( EC ( K n )) → . PARSE OPTIMIZATION CLTS Proof.
Observe that EC ( K n ) is bounded by the sum of n i.i.d. n Exp(1)random variables (this is a very loose bound, since EC ( K n ) is order n ,as shown in [32]). Then apply Proposition 4.4 and Lemma 4.5, along withmultiple applications of Cauchy-Schwarz.4.2. Constructing the local approximation.
In this section, we begin toconstruct the local approximations LA Lk , LA Uk that are needed for Theorem2.4. As before, this is done by finding a recursion for EC λ ( K n ). With λ implicit, let G n := K n ( λ ), so that EC λ ( K n ) = EC λ ( G n ). The main dif-ference between optimal edge cover and the problems considered in Section3 is that for optimal edge cover, the recursion we derive will not be for EC λ ( G n ) − EC λ ( G n − v ), and instead will be for a slightly different quan-tity. Indeed, this is the main reason why we can not use Corollary 2.5, andinstead have to resort to Theorem 2.4.For now, fix λ >
0. What we will eventually do is apply Theorem 2.4 toobtain a rate of convergence for fixed λ . This rate of convergence will bequantitative enough that we may actually take λ n = C log n (from Propo-sition 4.6) and still have the rate converge to 0.For the rest of Section 4.2, we follow [32], with no claims of originality.Let V n denote the vertex set of G n . For a subset of vertices S ⊆ V n , define EC λ ( G n , S ) to be the optimal λ -diluted edge cover of S , which uses edges of G n . In particular, one may use edges which connect S to V n − S . For example,if S consists of a single vertex, then EC ( G n , S ) will be the distance fromthat vertex to its nearest neighbor in G n , if that distance is less than λ/ λ/ EC λ ( G n , V n ) = EC λ ( G n ). Define the function h λ ( v, G n , S ) := EC λ ( G n , S ) − EC λ ( G n , S − { v } ) . Observe that(4.3) 0 ≤ h λ ≤ λ . Now the motivation for introducing h λ is because for e = ( v, u ), we maywrite EC λ ( G n ) − EC λ ( G en ) = h λ ( v, G n , V n ) − h λ ( v, G en , V n ) + h λ ( u, G n , V n − { v } ) − h λ ( u, G en , V n − { v } ) . (4.4)The proof follows by noting EC λ ( G n , V n − { v, u } ) = EC λ ( G en , V n − { v, u } ) , SKY CAO because if the vertices v, u do not need to be covered, then there is no needto use the edge ( v, u ).We now proceed to derive a recursion for h λ , from which we will be able toconstruct local approximations to h λ , and thus also to EC λ ( G n ) − EC λ ( G en ). Lemma . Let v have neighbors v , . . . , v d in G n . Assume v ∈ S . Wehave h λ ( v, G n , S ) = min ≤ m ≤ d λ , w ( v,v m ) − h λ ( v m , G n , S − { v } ) ! . Proof.
The edge collection which gives EC λ ( G n , S ) either uses at leastone of the edges ( v, v m ), 1 ≤ m ≤ d , or does not cover v , which incurs a costof λ/
2. Thus EC λ ( G n , S ) = min ≤ m ≤ d λ EC λ ( G n , S − { v } ) ,w ( v,v m ) + EC λ ( G n , S − { v, v m } ) ! . Now subtract EC λ ( G n , S − { v } ) on both sides.Of course, for v / ∈ S , h λ ( v, G n , S ) = 0. Now by combining the aboveLemma with (4.3), we obtain the following:(4.5) h λ ( v, G n , S ) = max , min ≤ m ≤ d λ , w ( v,v m ) − h λ ( v m , G n , S − { v } ) !! , (4.6) h λ ( v, G n , S ) ≤ min λ , min ≤ m ≤ d w ( v,v m ) ! . We now use the recursion (4.5) to construct the local approximations. With λ implicit, we define functions h Lk , h Uk . Let T be a rooted weighted tree ofdepth at most k . For vertices v ∈ T at depth k , define h Lk ( v ; T ) := 0 , h Uk ( v ; T ) := λ k is even, and h Lk ( v ; T ) := λ , h Uk ( v ; T ) := 0 PARSE OPTIMIZATION CLTS if k is odd. For leaf vertices v ∈ T at depth less than k , define h Lk ( v ; T ) = h Uk ( v ; T ) := λ . For non-leaf vertices v ∈ T , define h Lk ( v ; T ) := max , min u ∈C ( v ) λ , ℓ u − h Lk ( u ; T ) !! , and h Uk ( v ; T ) := max , min u ∈C ( v ) λ , ℓ u − h Uk ( u ; T ) !! . Observe in particular we have(4.7) 0 ≤ h Lk , h Uk ≤ λ/ Lemma . Suppose B k := B k ( v, G n ) is a tree. We have h Lk ( v ; B k ) ≤ h λ ( v, G n , V n ) ≤ h Uk ( v ; B k ) . Moreover, for any u connected to v in G n , we have min( h Uk ( u ; B k ) , w ( v,u ) ) ≤ h λ ( u, G n , V n − { v } ) ≤ min( h Lk ( u ; B k ) , w ( v,u ) ) . Proof.
By (4.5), the first inequality follows from the second inequality.The second inequality follows by induction on k .With h Lk , h Uk defined, we could proceed (using (4.4)) to define the localapproximations LA Lk , LA Uk . However, we decide to delay this to Section 4.4.4.3. Quantitative bound for the error in the local approximation.
To ap-ply Theorem 2.4, we need to bound the error in local approximation (i.e. δ k in (A3) of Property GLA). With λ implicit, let T ∞ d = T ( ∞ , λ, Exp(1)),and let T k be the depth k subtree of T ∞ . We write h Lk ( ∅ ) := h Lk ( ∅ ; T k ), h Uk ( ∅ ) := h Uk ( ∅ ; T k ) for brevity. One of the main terms in the error turnsout to be E ( h Uk ( ∅ ) − h Lk ( ∅ )) . Now the results of [32] immediately implythat lim k →∞ E ( h Uk ( ∅ ) − h Lk ( ∅ )) = 0 for fixed λ >
0, but we will need amore quantitative bound, due to the fact that we are trying to take λ → ∞ with n .An inductive argument shows h Lk ( ∅ ) ≤ h Uk ( ∅ ) for all k , which implies(4.8) E ( h Uk ( ∅ ) − h Lk ( ∅ )) ≤ E ( h Uk ( ∅ )) − E ( h Lk ( ∅ )) , which we will use later. SKY CAO
Proposition . For λ > , we have E ( h Uk ( ∅ ) − h Lk ( ∅ )) ≤ Cλα ( λ ) k . Here α ( λ ) < , and even more, sup λ ≥ δ α ( λ ) < for all δ > . Remark.
The immediate consequence of this proposition is that if wetake λ n = C log n (as in Proposition 4.6), then upon taking k n = C ′ log λ n for some large enough C ′ , we have that E ( h Uk n ( ∅ ) − h Lk n ( ∅ )) → h Lk ( ∅ ) , h Uk ( ∅ ) are defined by setting some initial conditions atthe leaf vertices of T k , and then recursively defining the values of h Lk , h Uk for all non-leaf vertices. This proposition is essentially saying that the effectof the initial conditions is swept away exponentially quickly in the depth ofthe tree T k .To prove Proposition 4.9, we first need to establish the following relationbetween the distributions of h Lk , h Uk . For λ >
0, define the operator V λ onfunctions F : [0 , λ/ → [0 ,
1] as follows:( V λ F )( x ) := exp − Z λ/ F ( ℓ ) dℓ ! e − x . For notational purposes, define E λ ( F ) := R λ/ F ( ℓ ) dℓ , so that ( V λ F )( x ) = e − E λ ( F ) e − x . For k ≥ x ∈ [0 , λ/ F k ( x ) := P ( h Lk ( ∅ ) ≥ x ) , G k ( x ) := P ( h Uk ( ∅ ) ≥ x ) . Lemma . We have F k +1 = V λ G k , and G k +1 = V λ F k . This result is implicit in Section 4 of [32]. The proof takes advantage of therecursive properties of T k , as well as the fact that the offpsring distributionis Poisson( λ ), to reduce to a Poisson process calculation. A detailed proofin the case of λ -diluted minimum matching (with more general edge costdistribution) is given in Section 2.7 of [35].We now collect several simple facts about the operator V λ . PARSE OPTIMIZATION CLTS Lemma . A fixed point of V λ is the function F λ ( x ) := e − A λ e − x , where A λ = E λ ( F λ ) satisfies A λ = e − A λ (1 − e − λ/ ) . Lemma . For functions F ≤ G , we have V λ F ≥ V λ G . As F ≤ F λ ≤ G , we then have F k ≤ F λ ≤ G k for all k . Moreover, F k ≤ F k +1 , G k +1 ≤ G k for all k . We now analyze the operator V λ . In light of (4.8), Proposition 4.9 is aconsequence of the following slightly more general proposition. Proposition . We have for x ∈ [0 , λ/ , | F λ ( x ) − F k ( x ) | ≤ Cλα ( λ ) k e − x , and | G k ( x ) − F λ ( x ) | ≤ Cα ( λ ) k e − x , where α ( λ ) < for all λ > , and even more, sup λ ≥ δ α ( λ ) < for all δ > . Proof.
To start, observe G k +2 ( x ) − F λ ( x ) = e − E λ ( F k +1 ( x )) e − x − e − A λ e − x = e − ( x + E λ ( F k +1 )) (cid:18) − e − ( A λ − E λ ( F k +1 )) (cid:19) . As F k +1 ≤ F λ by Lemma 4.12, we have E λ ( F k +1 ) ≤ E λ ( F λ ) = A λ , and thus ≤ e − ( x + E λ ( F k +1 )) ( A λ − E λ ( F k +1 )) . Integrating over 0 ≤ x ≤ λ/
2, we obtain E λ ( G k +2 ) − A λ ≤ e − E λ ( F k +1 ) ( A λ − E λ ( F k +1 )) . The same argument also shows A λ − E λ ( F k +1 ) ≤ e − A λ ( E λ ( G k ) − A λ ) . Now as E λ ( F k +1 ) ≥ E λ ( F ) = 0, combining the above two displays, we have E λ ( G k +2 ) − A λ ≤ e − A λ ( E λ ( G k ) − A λ ) . SKY CAO
The same argument implies A λ − E λ ( F k +2 ) ≤ e − A λ ( A λ − E λ ( F k )) . Iterating these inequalities, we obtain E λ ( G k ) − A λ ≤ e − kA λ ( E λ ( G ) − A λ ) ≤ λe − kA λ , and A λ − E λ F k ≤ e − kA λ ( A λ − E λ ( F )) = A λ e − kA λ . As F k ≤ F k +1 , and G k +1 ≤ G k , we obtain E λ ( G k ) − A k ≤ λe −⌊ k/ ⌋ A λ ,A λ − E λ ( F k ) ≤ A λ e −⌊ k/ ⌋ A λ . Substituting back into our previously derived inequalities, and using Lemma4.12, we obtain 0 ≤ G k +1 ( x ) − F λ ( x ) ≤ A λ e −⌊ k/ ⌋ A λ e − x , ≤ F λ ( x ) − F k +1 ( x ) ≤ λA λ e −⌊ k/ ⌋ A λ e − x . To finish, observe that by using the definition of A λ and the intermediatevalue theorem, we have that A λ is increasing in λ . Thus for any δ >
0, wehave inf λ ≥ δ A λ = A δ >
0. Observe also that A λ ≤ A ∞ , where A ∞ satisfies A ∞ = e − A ∞ .4.4. Completing the proof of the central limit theorem.
We now haveall the pieces in place to deduce Theorem 2.3 from Theorem 2.4. We firstdefine the local approximations LA Lk , LA Uk . The main idea is to use (4.4),and approximate h λ by h Lk , h Uk . Fix e = ( v, u ), and let B k := B k ( v, G n ), B ′ k := B k ( v, G en ) for brevity. When B k , B ′ k are both trees, we define LA Lk ( B k , B ′ k ) := h Lk ( v ; B k ) − h Uk ( v ; B ′ k ) +1( b e = 1 , b ′ e = 0) (cid:18) min( h Uk ( u ; B k ) , w e ) − h Lk ( u ; B k ) (cid:19) +1( b e = 0 , b ′ e = 1) (cid:18) h Uk ( u ; B ′ k ) − min( h Lk ( u ; B ′ k ) , w ′ e ) (cid:19) +1( b e = 1 , b ′ e = 1) (cid:18) min( h Uk ( u ; B k ) , w e ) − min( h Lk ( u ; B ′ k ) , w ′ e ) (cid:19) . PARSE OPTIMIZATION CLTS The function LA Uk is defined similarly, by swapping the roles of h Lk , h Uk .One may verify (A1) by using Lemma 4.8. By (4.3), (4.4), we may take H ( w e , w ′ e ) := 2 λ , so that J = 64 λ .The following lemma gives quantitative bounds on the numbers ( δ k , k ≥ Lemma . Let δ k ( λ ) be defined as in (A3) of Property GLA, for theoptimization problem EC λ . Then δ k ( λ ) ≤ Cλα ( λ ) k − , with α ( λ ) < , and even more, sup λ ≥ c α ( λ ) < for all c > . Before we prove this lemma, we first show how Theorem 2.3 follows.
Proof of Theorem 2.3.
Observe p n = 1 − e − λ/n ≤ λ/n . To bound d T V ( F w n , F w ), one may upper bound the L distance between the densitiesof F w n , F w . Here F w n is the distribution of n Exp(1) conditioned to lie in[0 , λ ], and F w is Unif[0 , λ ]. A calculation shows that the L distance may bebounded by Cλ/n . We thus have for fixed λ , the term ε k ( n ) from Theorem2.4 may be bounded ε k ( n ) ≤ C k ( λ + C ) k + C n / min( λ, . Let ( σ λn ) := Var( EC λ ( K n )), r λn := n/ ( σ λn ) , and Z λn := EC λ ( K n ) − E EC λ ( K n ) σ λn = EC λ ( G n ) − E EC λ ( G n ) σ λn . Using Lemma 4.14, upon applying Theorem 2.4 we obtainsup t ∈ R (cid:12)(cid:12)(cid:12) P ( Z λn ≤ t ) − Φ( t ) (cid:12)(cid:12)(cid:12) ≤ C ( r λn ) / (cid:20) λ C α ( λ ) ( k − / + C k ( λ + C ) k + C n /C min( λ, (cid:21) + C ( r λn ) / λ C n / . Now take λ n = C log n as in Proposition 4.6, and take k n = C ′ log λ n forsome C ′ large enough depending on C , such thatlim n →∞ λ Cn α ( λ n ) ( k n − / = 0 . Note as k n grows like log log n , we have thatlim n →∞ C k n ( λ n + C ) k n + C n /C = 0 . SKY CAO
Finally, by Lemma 4.5 and Proposition 4.6, we have thatlim sup n r λ n n < ∞ . Upon combining these observations, we obtainlim n →∞ sup t ∈ R (cid:12)(cid:12)(cid:12) P ( Z λ n n ≤ t ) − Φ( t ) (cid:12)(cid:12)(cid:12) = 0 . Thus Z λ n n d −→ N (0 , EC ( K n ) − E EC ( K n ) p Var( K n ) d −→ N (0 , . Proof of Lemma 4.14.
Recall the definitions of T k , ˜ T k in (A3) of Prop-erty GLA. We will show how to obtain the bound for the pair ( ˜ T k , T k ). Thecase of ( T k , ˜ T k ) will have the exact same proof.Recall ˜ T k is constructed from ( T k , T ′ k − , ∅ , ∅ ′ , ℓ ). Thus we have (notingthat h Lk ( ∅ ′ , ˜ T k ) = h Uk − ( ∅ ′ , T ′ k − )) LA Uk ( ˜ T k , T k ) = h Uk ( ∅ ; ˜ T k ) − h Lk ( ∅ ; T k ) +min( h Uk − ( ∅ ′ , T ′ k − ) , ℓ ) − h Lk − ( ∅ ′ , T ′ k − ) . Similarly, LA Lk ( T , T ′ ) = h Lk ( ∅ ; ˜ T k ) − h Uk ( ∅ ; T k ) +min( h Lk − ( ∅ ′ , T ′ k − ) , ℓ ) − h Uk − ( ∅ ′ , T ′ k − ) . Thus( LA Uk ( T , T ′ ) − LA Lk ( T , T ′ )) ≤ C ( h Uk ( ∅ ; ˜ T k ) − h Lk ( ∅ ; ˜ T k )) + C ( h Uk ( ∅ ; T k ) − h Lk ( ∅ ; T k )) + C ( h Uk − ( ∅ ′ , T ′ k − ) − h Lk − ( ∅ ′ , T ′ k − )) . Upon taking expectations, the last two terms in the right hand side abovemay be handled by Proposition 4.14. To handle the first term, observe h Uk ( ∅ , ˜ T k ) = max (cid:18) , min (cid:18) h Uk ( ∅ , T k ) , ℓ − h Lk − ( ∅ ′ , T ′ k − ) (cid:19)(cid:19) , and similarly, h Lk ( ∅ , ˜ T k ) = max (cid:18) , min (cid:18) h Lk ( ∅ , T k ) , ℓ − h Uk − ( ∅ ′ , T ′ k − ) (cid:19)(cid:19) . PARSE OPTIMIZATION CLTS Thus( h Uk ( ∅ , ˜ T k ) − h Lk ( ∅ , ˜ T k )) ≤ ( h Uk ( ∅ , T k ) − h Lk ( ∅ , T k )) +( h Uk − ( ∅ ′ , T ′ k − ) − h Lk − ( ∅ ′ , T ′ k − )) . Upon applying Proposition 4.14, we obtain E ( h Uk ( ∅ , ˜ T k ) − h Lk ( ∅ , ˜ T k )) ≤ Cλα ( λ ) k − . Collecting the previous results allows us to obtain E ( LA Uk ( ˜ T k , T k ) − LA Lk ( ˜ T k , T k )) ≤ Cλα ( λ ) k − , as desired.
5. Proofs.
In this section, we prove Corollary 2.5 and Theorem 2.4.The proof of Theorem 2.4 will rely on certain facts about neighborhoods ofErd˝os-R´enyi graphs, which are covered in Section 6.5.1.
Proof of Corollary 2.5.
Let e = ( v, u ). When B k := B k ( v, G n ), B ′ k := B k ( v, G en ) are trees, define LA Lk ( B k , B ′ k ) := g Lk ( B k ) − g Uk ( B ′ k ) , and LA Uk ( B k , B ′ k ) := g Uk ( B k ) − g Lk ( B ′ k ) . We proceed to verify Property GLA. (A1) follows by (2.3). To show (A2),note if ( B k , B ′ k ) ∼ = ( T , T ′ ), then B k ∼ = T , B ′ k ∼ = T ′ , and thus by (2.4), wehave LA Lk ( T , T ′ ) = g Lk ( T ) − g Uk ( T ′ ) = g Lk ( B k ) − g Uk ( B ′ k ) = LA Lk ( B k , B ′ k ) , and similarly for LA Uk . Finally, to show (A3), observe that for trees T , T ′ ,we have | LA Uk ( T , T ′ ) − LA Lk ( T , T ′ ) | ≤ | g Uk ( T ) − g Lk ( T ) | + | g Uk ( T ′ ) − g Lk ( T ′ ) | . (A3) now follows by (2.5), (2.6).For ε k ( n ) as defined in Theorem 2.4, since np n → λ and d T V ( F w n , F w ) →
0, we have that for all k , lim n →∞ ε k ( n ) = 0. Upon applying Theorem 2.4,because we’ve assumed the variance lower bound, we obtain for all k > n →∞ sup t ∈ R | P ( Z n ≤ t ) − Φ( t ) | ≤ C ′ δ / k , where C ′ is some finite number which may depend on ( f, ( G n , n ≥ k , and δ k is as in Property GLA. To finish, take k → ∞ . SKY CAO
Proof of Theorem 2.4.
Let E n := { ( i, j ) : 1 ≤ i < j ≤ n } . Given e = ( i, j ), let G n − e be the weighted graph obtained by deleting edge e if it is present, else doing nothing. Similarly define G n − F for a set ofedges F ⊆ E n . Let ( W ′ , B ′ ) be an independent copy of ( W, B ). Recallingthe definition of G en , for a set F ⊆ E n , define G Fn to be the weighted graphobtained by using w ′ e , b ′ e instead of w e , b e for e ∈ F . For singleton sets { e } ,we will write G en instead of G { e } n , and we will write G F ∪ en instead of G F ∪{ e } n .Recalling the definition of ∆ e f := f ( G n ) − f ( G en ), for F ⊆ E n \{ e } similarlylet ∆ e f F := f ( G Fn ) − f ( G F ∪ en ) . Recall σ n := Var( f ( G n )), λ n := np n . The following is Corollary 3.2 of [10],adapted to our situation. Lemma . For each e = ( v, u ) , e ′ = ( v ′ , u ′ ) , let c ( e, e ′ ) be such that forall F ⊆ E n \{ e } , F ′ ⊆ E n \{ e ′ } , we have σ n Cov(∆ e f ∆ e f F , ∆ e ′ f ∆ e ′ f F ′ ) ≤ c ( e, e ′ ) . Then sup t ∈ R | P ( Z n ≤ t ) − Φ( t ) | ≤ √ X e,e ′ ∈ E n c ( e, e ′ ) ! / + σ n X e ∈ E n E | ∆ e f | ! / . Remark.
As we will see, the only terms that are nontrivial to boundare c ( e, e ′ ) for edges e, e ′ with distinct vertices. This is Lemma 5.7, but themain work is done by Lemmas 5.5 and 5.6. All other terms will be boundedby applying the assumed regularity conditions and using Cauchy-Schwarzor related inequalities.We begin by bounding the second term. Lemma . We have X e ∈ E n E | ∆ e f | ≤ J / n p n = J / nλ n . Proof.
This follows by conditions (2.2) and (2.1), the independence of
W, B , and a calculation.
Lemma . We may take c ( e, e ) = 1 σ n CJ / p n = 1 σ n CJ / λ n n . PARSE OPTIMIZATION CLTS Proof.
For
F, F ′ ⊆ E n \{ e } , we want to boundCov(∆ e f ∆ e f F , ∆ e f ∆ e f F ′ ) . This may be done by applying (2.2) and (2.1).
Lemma . Let e = ( v, u ) , e ′ = ( v ′ , u ′ ) ∈ E n be edges which shareexactly one vertex. We may take c ( e, e ′ ) = 1 σ n CJ / p n = 1 σ n CJ / λ n n . Proof.
Again, we apply (2.2) and (2.1), and proceed.The main work will be in bounding c ( e, e ′ ) for edges e = ( v, u ), e ′ = ( v ′ , u ′ )with all distinct vertices. Let E be the event that both B k ( v, G n ) , B k ( v, G en )are trees. Let A e := { ( b e , b ′ e ) = (1 ,
0) or (0 , } . Define˜ L ek := 1 E A e LA Lk ( B k ( v, G n ) , B k ( v, G en )) . Let Q ( x, y ) := max(min( x, y ) , − y ) (in words, Q is truncation of x at level y ). Define L ek := Q ( ˜ L ek , H ( w e , w ′ e )) , (“ L ” is for “local”), so that(5.1) | L ek | ≤ A e H ( w e , w ′ e ) . Let R ek := ∆ e f − L ek . Here “ R ” is for “remainder”. Let ˜ A e := { max( b e , b ′ e ) = 1 } , so that ˜ A e = A e ∪ { b e , b ′ e = 1 } . Observe by (2.2), (5.1), we have(5.2) | R ek | ≤ ˜ A e H ( w e , w ′ e ) . For F ⊆ E n \{ e } , we may define L F ∪ ek by using B k ( v, G Fn ) , B k ( v, G F ∪ en )in place of B k ( v, G n ), B k ( v, G en ). Then let R F ∪ ek := ∆ e f F − L F ∪ ek . To bound c ( e, e ′ ), we need to upper bound(5.3) Cov (cid:18) ( R ek + L ek )( R F ∪ ek + R F ∪ ek ) , ( R e ′ k + L e ′ k )( R F ′ ∪ e ′ k + L F ′ ∪ e ′ k ) (cid:19) . Upon expanding this covariance, we obtain 16 terms, one of which onlyinvolves local approximation quantities. We should think of local quantitiesas essentially independent, and so their covariance should be essentially 0.The following lemma makes this precise. SKY CAO
Lemma . With ρ k ( n ) as in Theorem 2.4, we have Cov( L ek L F ∪ ek , L e ′ k L F ′ ∪ e ′ k ) ≤ CJ / p n ρ k ( n ) . To not distract too much from the main thrust of the argument, we willdefer the proof of this lemma to Section 5.3. The other 15 terms which comefrom expanding (5.3) all involve at least one remainder term.
Lemma . With δ k as in Property GLA and ε k ( n ) , ρ k ( n ) as in Theo-rem 2.4, any of the other 15 terms which come from expanding (5.3) maybe bounded by CJ p n ( δ / k + ε k ( n ) / + ρ k ( n ) / ) . Proof.
Note J ≥ J r ≤ J , for r ≤ R ek X , X , X ) , where X is either R F ∪ ek or L F ∪ ek , X is either R e ′ k or L e ′ k , and X is either R F ′ ∪ e ′ k or L F ′ ∪ e ′ k . The other terms may be bounded in a similar manner.Define ˜ H := max( H ( w e , w ′ e ) , H ( w e ′ , w ′ e ′ )) . By (5.1), (5.2), we have | R ek X X X | ≤ C ˜ A e ˜ A e ′ ˜ H | R ek |≤ C A e ˜ A e ′ ˜ H | R ek | + C b e , b ′ e = 1)1 ˜ A e ′ ˜ H . By the independence of
W, B, B ′ , we have E b e , b ′ e = 1)1 ˜ A e ′ ˜ H ≤ C E ˜ H p n ≤ CJ / p n . As p n = λ n /n ≤ ρ k ( n ), we are done with this term. Moving on to the otherterm, let Y e := ( b e , b ′ e ). By Cauchy-Schwarz and the independence of W, B ,we have E [ ˜ H | R ek | | Y e , Y e ′ ] ≤ CJ / ( E [( R ek ) | Y e , Y e ′ ]) / . For brevity, write B k , B ′ k instead of B k ( v, G n ), B k ( v, G en ). By (A1), (5.1),we have 1 A e | R ek | ≤ E A e ( LA Uk ( B k , B ′ k ) − LA Lk ( B k , B ′ k )) + C ˜ H E c . PARSE OPTIMIZATION CLTS Thus1 A e E [( R ek ) | Y e , Y e ′ ] ≤ A e E [1 E ( LA Uk ( B k , B ′ k ) − LA Lk ( B k , B ′ k )) | Y e , Y e ′ ]+ CJ / ( P ( E c | Y e , Y e ′ )) / . (5.4)We may bound P ( E c | Y e , Y e ′ ) ≤ P ( B k not a tree | b e , b e ′ ) + P ( B ′ k not a tree | b ′ e , b e ′ ) . We may bound P ( B k not a tree | b e , b e ′ ) ≤ min (cid:18) ( λ n + C ) k + C n , (cid:19) , and similarly for B ′ k . This may be done by noting that if B k is not a tree,then either B k ( v, G n − e ) is not a tree, or B k − ( u, G n − e ) is not a tree,or B k ( v, G n − e ) , B k − ( u, G n − e ) intersect. To remove the conditioning, wemay show that B k ( v, G n − e ) = B k ( v, G n − { e, e ′ } ) and B k − ( u, G n − e ) = B k − ( u, G n − { e, e ′ } ) with very high probability, even conditional on b e , b e ′ .Then finish by Lemma 6.7.Moving to bound the other term in (5.4), we apply Lemma 6.2 to couple( B k , B ′ k , T , T ′ ). Observe by Lemma 6.1, we may take ε k ( n ) defined in Lemma6.2 to be exactly the ε k ( n ) that is given in the statement of Theorem 2.4.Let E := { ( B k , B ′ k ) ∼ = ( T , T ′ ) } . By (A2), (A3) of Property GLA, we have1 A e E [1 E ( LA Uk ( B k , B ′ k ) − LA Lk ( B k , B ′ k )) | Y e , Y e ′ ] ≤ δ k + CJ / (cid:18) ε k ( n ) + C ( λ n + 1) k n + 2 d T V ( F w n , F w ) (cid:19) / . We have thus bounded E | R ek X X X | . Note the term d T V ( F w n , F w ) may beabsorbed into ε k ( n ), and ( λ n + 1) k /n may be bounded by ρ k ( n ) (we maytake min of ( λ n + 1) k /n with 1 since this term comes from bounding aprobability). The term E R ek X E X X may be similarly bounded.We now collect Lemmas 5.5, 5.6 into the following lemma. Here we alsouse the fact that by definition, ρ k ( n ) ≤
1, so that ρ k ( n ) ≤ ρ k ( n ) / . Lemma . For edges e, e ′ using all distinct vertices, we may take c ( e, e ′ ) = CJ p n σ n (cid:18) δ / k + ε k ( n ) / + ρ k ( n ) / (cid:19) . Proof of Theorem (2.4) . Combine Lemmas (5.1), (5.2), (5.3), (5.4),(5.7). SKY CAO
Proof of Lemma 5.5.
First, we set some notation. Define S e :=( w e , b e , w ′ e , b ′ e ), A e := { ( b e , b ′ e ) = (1 ,
0) or (0 , } . Let X e := L ek L F ∪ ek , and X e ′ := L e ′ k L F ′ ∪ e ′ k . By definition of L ek , note that X e = 1 A e X e , and similarlyfor X e ′ . We may writeCov( X e , X e ′ ) = Cov(1 A e E ( X e | S e , S e ′ ) , A e ′ E ( X e ′ | S e , S e ′ )) + E A e A e ′ Cov( X e , X e ′ | S e , S e ′ ) . To prove Lemma 5.5, we prove the following two lemmas.
Lemma . We have
Cov(1 A e E [ X e | S e , S e ′ ] , A e ′ E [ X e ′ | S e , S e ′ ]) ≤ CJ / p n ( λ n + 1) k n . Lemma . We have E A e A e ′ Cov( X e , X e ′ | S e , S e ′ ) ≤ CJ / p n ρ k ( n ) . Proof of Lemma 5.8.
The starting point is that X e is essentially inde-pendent of S e ′ , so we should be able to write E [ X e | S e , S e ′ ] ≈ E [ X e | S e ],and analogously for X e ′ . Towards this end, recall that X e is a function of (cid:18) B k ( v, G n ) , B k ( v, G en ) , B k ( v, G Fn ) , B k ( v, G F ∪ en ) (cid:19) . Let us define the approximation ˜ X e to be the same function, applied to (cid:18) B k ( v, G n − e ′ ) , B k ( v, G en − e ′ ) , B k ( v, G Fn − e ′ ) , B k ( v, G F ∪ en − e ′ ) (cid:19) . Observe that ˜ X e is independent of S e ′ . In the same manner, we way define theapproximation ˜ X e ′ which is independent of S e . Now let Z e := E [ X e | S e , S e ′ ],˜ Z e := E [ ˜ X e | S e , S e ′ ] = E [ ˜ X e | S e ], and similarly define Z e ′ , ˜ Z e ′ . Let H e := H ( w e , w ′ e ). Observe that by 5.1, we have | X e | , | ˜ X e | ≤ A e H e , which implies Z e , ˜ Z e ≤ A e H e , and similarly for Z e ′ , ˜ Z e ′ . We may writeCov(1 A e Z e , A e ′ Z e ′ ) = Cov(1 A e ( Z e − ˜ Z e ) , A e ′ ˜ Z e ′ ) +Cov(1 A e ˜ Z e , A e ′ ( Z e ′ − ˜ Z e ′ )) + Cov(1 A e ( Z e − ˜ Z e ) , A e ′ ( Z e ′ − ˜ Z e ′ )) . PARSE OPTIMIZATION CLTS To finish, we will bound the three terms on the right hand side. I will onlywrite out how to bound the first term, as the other two terms are boundedsimilarly. First, observe | A e A e ′ ( Z e − ˜ Z e ) ˜ Z e ′ | ≤ H e ′ A e A e ′ | Z e − ˜ Z e | , and | Z e − ˜ Z e | = | E ( X e − ˜ X e | S e , S e ′ ) | ≤ H e P ( X e = ˜ X e | S e , S e ′ ) . I claim that P ( X e = ˜ X e | S e , S e ′ ) ≤ Cn k X j =1 λ jn ≤ C ( λ n + 1) k n . Given this claim, putting everything together, we have | E A e A e ′ ( Z e − ˜ Z e ) ˜ Z e ′ | ≤ CJ / p n ( λ n + 1) k n . The term | E A e ( Z e − ˜ Z e ) E A e ′ ˜ Z e ′ | may be bounded in a similar manner.To show the claim, observe that the event { X e = ˜ X e } implies that B k ( v, G − e ′ ) = B k ( v, G ) for at least one of G = G n , G en , G Fn , G F ∪ en . Takingsay G = G n , we have P ( B k ( v, G n − e ′ ) = B k ( v, G n ) | S e , S e ′ ) = P ( e ′ ∈ B k ( v, G n ) | b e , b e ′ ) . This may be bounded using arguments similar to those appearing in theproof of Lemma 6.2.
Proof of Lemma 5.9.
Define B k ( − e ) := ( B k ( v, G n − e ) , B k ( v, G Fn − e ) , B k ( u, G n − e ) , B k ( u, G Fn − e )) , B ′ k ( − e ′ ) := ( B k ( v ′ , G n − e ′ ) , B k ( v ′ , G F ′ n − e ′ ) , B k ( u ′ , G n − e ′ ) , B k ( u ′ , G F ′ n − e ′ )) . Observe that X e is a function of ( B k ( − e ) , S e ), and X e ′ is a function of( B k ( − e ′ ), S e ′ ). Now define the approximation ˜ X e as the same function ap-plied to ( B k ( − ) , S e ), where B k ( − ) := ( B k ( v, G n − ∆) , B k ( v, G Fn − ∆) , B k ( u, G n − ∆) , B k ( u, G Fn − ∆)) , and ∆ := { e, e ′ } . Similarly define the approximation ˜ X e ′ of X e ′ as a functionof ( B ′ k ( − ) , S e ′ ), where B ′ k ( − ) := ( B k ( v ′ , G n − ∆) , B k ( v ′ , G F ′ n − ∆) , B k ( u ′ , G n − ∆) , B k ( u ′ , G F ′ n − ∆)) . SKY CAO
As before, we may bound P ( B k ( − ) = B k ( − e ) | S e , S e ′ ) ≤ C ( λ n + 1) k n , and similarly for B ′ k ( − ). Letting ˜ H := max( H e , H e ′ ), we obtain (cid:12)(cid:12)(cid:12) Cov( X e , X e ′ | S e , S e ′ ) − Cov( ˜ X e , ˜ X e ′ | S e , S e ′ ) (cid:12)(cid:12)(cid:12) ≤ C ˜ H ( λ n + 1) k n . Thus it suffices to focus on ˜ X e , ˜ X e ′ . Observe that by construction, we havethat B k ( − ) , B ′ k ( − ) are independent of S e , S e ′ . We may thus expressCov( ˜ X e , ˜ X e ′ | S e , S e ′ ) = Cov(Ψ( B k ( − )) , Ψ ′ ( B ′ k ( − ))) , where the functions Ψ , Ψ ′ depend on S e , S e ′ , but we hide this dependence.Moreover, we have that | Ψ | , | Ψ ′ | ≤ ˜ H , which we now think of as constantwhen we take the covariance between Ψ( B k ( − )) and Ψ ′ ( B ′ k ( − )). Define B k := ( B k ( v, G n ) , B k ( v, G Fn ) , B k ( u, G n ) , B k ( u, G Fn )) , B ′ k := ( B k ( v ′ , G n ) , B k ( v ′ , G F ′ n ) , B k ( u ′ , G n ) , B k ( u ′ , G F ′ n )) . We may show P ( B k = B k ( − )) ≤ C ( λ n + 1) k n , and similarly for B ′ k . This allows us to obtain (cid:12)(cid:12) Cov(Ψ( B k ( − )) , Ψ ′ ( B ′ k ( − ))) − Cov(Ψ( B k ) , Ψ ′ ( B ′ k )) (cid:12)(cid:12) ≤ C ˜ H ( λ n + 1) k n . Now by Lemma 6.9, we haveCov(Ψ( B k ) , Ψ ′ ( B ′ k )) ≤ ˜ H min (cid:18) ( λ n + C ) k + C n , (cid:19) . The desired result now follows by putting everything together.
6. Facts about neighborhoods of sparse Erd˝os-R´enyi graphs.
This section collects the key facts about neighborhoods of sparse Erd˝os-R´enyi graphs which are needed. Throughout this section, write λ n := np n .The following lemma shows that not only are neighborhoods essentiallyGalton-Watson trees, but pairs of neighborhoods are essentially independentGalton-Watson trees, in a very quantitative manner. This result seems tobe well known and has been proven in [32] for the case p n = 1 − e − λ/n , but Ihaven’t found a reference which provides a proof for general np n → λ . Thusfor completeness, I prove it. PARSE OPTIMIZATION CLTS Lemma . Suppose np n → λ and d T V ( F w n , F w ) → . Fix k > .For distinct vertices v, u , we have a coupling ( B k ( v, G n ) , B k ( u, G n ) , T v , T u ) such that T v , T u i.i.d. ∼ T ( k, λ, F w ) , and P ( B k ( v, G n ) ∼ = T v , B k ( u, G n ) ∼ = T u ) ≥ − (2 λ + 3) k n / − C ( λ n + 1) k min( λ, (cid:18) | λ n − λ | + d T V ( F w n , F w ) + λ n (cid:19) . Proof.
We may assume λ ≤ n , otherwise the bound is trivial. We firstwork without edge weights. Let ˜ p n := 1 − e − λ/n , and let ˜ G n be the Erd˝os-R´enyi graph with edge probability ˜ p n . For brevity, let ˜ B vk := B k ( v, ˜ G n ),˜ B uk := B k ( u, ˜ G n ). It follows by Lemma 2.4 of [32] that we may couple( ˜ B vk , ˜ B uk , T v , T u ) such that T v , T u i.i.d. ∼ T ( k, λ ), and P ( ˜ B vk ∼ = T v , ˜ B uk ∼ = T u ) ≥ − (2 λ + 3) k n / . Now suppose p n ≥ ˜ p n . We may couple G n , ˜ G n in the following manner. If˜ G n is defined by the edges ˜ B = (˜ b e , e = ( i, j )), with ˜ b e ∼ Bernoulli(˜ p n ),then define b e = max(˜ b e , ε e ), with ε e ∼ Bernoulli(( p n − ˜ p n ) / (1 − ˜ p n )). Then b e ∼ Bernoulli( p n ) as required. Let B vk := B k ( v, G n ), B uk := B k ( u, G n ).Observe that the event { B vk = ˜ B vk } is the event that there exists vertices u ∈ ˜ B vk − , u / ∈ ˜ B vk , such that ˜ b ( u ,u ) = 0, ε ( u ,u ) = 1. We thus have P ( B vk = ˜ B vk | ˜ B vk ) ≤ n | ˜ B vk − | p n − ˜ p n − ˜ p n , where | ˜ B vk − | is the number of vertices in ˜ B vk − . Thus P ( B vk = ˜ B vk ) ≤ n p n − ˜ p n − ˜ p n E | ˜ B vk − | . By comparison with a branching process, we may bound E | ˜ B vk − | ≤ k − X j =0 λ jn ≤ ( λ n + 1) k − . We have ˜ p n ≥ λn − λ n , and thus (recalling λ ≤ n ) n p n − ˜ p n − ˜ p n ≤ e λ/n (cid:18) λ n − λ + λ n (cid:19) ≤ C (cid:18) | λ n − λ | + λ n (cid:19) . SKY CAO
We may thus couple ( B vk , B uk , T v , T u ) such that P ( B vk ∼ = T v , B uk ∼ = T u ) ≥ − (2 λ + 3) k n / − C ( λ n + 1) k − (cid:18) | λ n − λ | + λ n (cid:19) . Now for each e introduce a coupling ( w e , ℓ e ), such that P ( w e = ℓ e ) = d T V ( F w n , F w ). Let E be the event that there is an e in B vk or B uk suchthat w e = ℓ e . We may naturally couple ( B k ( v, G n ), B k ( u, G n ) , T v , T u ),such that T v , T u i.i.d. ∼ T ( k, λ, F w ), and P ( B k ( v, G n ) ∼ = T v , B k ( u, G n ) ∼ = T u ) ≥ P ( B vk ∼ = T v , B uk ∼ = T u ) − P ( E ) . By comparison with a branching process, the expected number of edges in B vk is at most P kj =1 λ jn ≤ ( λ n + 1) k . Thus we have P ( E ) ≤ λ n + 1) k d T V ( F w n , F w ) . To finish, combine the previous results. The case p n < ˜ p n is handled simi-larly.The following lemma says that if we can couple unconditionally with highprobability, then we can couple conditionally with high probability. It wasneeded in Lemma 5.6, in combination with Property GLA, to show that theremainder terms were small. Recall in (A3) of Property GLA the definitionsof T k , ˜ T k . Lemma . Suppose we have a coupling ( B k ( v, G n ) , B k ( u, G n ) , T v , T u ) , with T v , T u i.i.d. ∼ T ( k, λ, F w ) . Suppose ε k ( n ) is such that ε k ( n ) ≥ − P ( B k ( v, G n ) ∼ = T v , B k ( u, G n ) ∼ = T u ) . Let e = ( v, u ) , and let e ′ = ( v ′ , u ′ ) be another edge with vertices distinct from v, u . Let Y e := ( b e , b ′ e ) . It is possible to couple ( B k ( v, G n ) , B k ( v, G en ) , T , T ′ ) such that ( T , T ′ ) | ( Y e = (1 , , Y e ′ ) d = ( ˜ T k , T k ) , ( T , T ′ ) | ( Y e = (0 , , Y e ′ ) d = ( T k , ˜ T k ) , and furthermore, P (( B k ( v, G n ) , B k ( v, G en )) ∼ = ( T , T ′ ) | Y e , Y e ′ ) ≥ − (cid:18) ε k ( n ) + C ( λ n + 1) k n + 2 d T V ( F w n , F w ) (cid:19) . PARSE OPTIMIZATION CLTS Proof.
To construct T , T ′ , first take an i.i.d. copy ( W ′′ , B ′′ ) of ( W, B ),independent of everything else. Define G ′′ n to be the weighted graph ob-tained by using w ′′ e , b ′′ e in place of w e , b e , and w ′′ e ′ , b ′′ e ′ in place of w e ′ , b e ′ .Now obtain a coupling ( B k ( v, G ′′ n ) , B k − ( u, G ′′ n ) , T k , T ′ k − ), with T k , T ′ k − independent, and T k d = T ( k, λ, F w ), and T ′ k − d = T ( k − , λ, F w ). Observethat ( B k ( v, G ′′ n ) , B k − ( u, G ′′ n )) is independent of Y e , Y e ′ , w e , w ′ e , and so wemay also assume that ( T k , T ′ k − ) is independent of Y e , Y e ′ , w e , w ′ e . Write B k := B k ( v, G n ), B ′ k := B k ( v, G en ). There is some function Ψ such that B k = Ψ( B k ( v, G n − e ) , B k − ( u, G n − e ) , w e , b e ) ,B ′ k = Ψ( B k ( v, G n − e ) , B k − ( u, G n − e ) , w ′ e , b ′ e ) . Take a coupling ( w e , w ′ e , ℓ, ℓ ′ ) independent of everything else such that wehave w e , w ′ e i.i.d. ∼ F w n , ℓ, ℓ ′ i.i.d. ∼ F w , and P ( w e = ℓ ) = P ( w ′ e = ℓ ′ ) = d T V ( F w n , F w ) . Define T := Ψ( T k , T ′ k − , ℓ, b e ) , T ′ := Ψ( T k , T ′ k − , ℓ ′ , b ′ e ) . Observe that ( T , T ′ ) has the desired conditional distribution. Let E be theevent that B k ( v, G ′′ n ) , B k − ( u, G ′′ n ) share a vertex. We have P (( B k , B ′ k ) ∼ = ( T , T ′ ) | Y e , Y e ′ ) ≥ P ( B k ( v, G ′′ n ) ∼ = T k , B k − ( u, G ′′ n ) ∼ = T ′ k − ) − P ( E ) − P ( B k ( v, G ′′ n ) = B k ( v, G n − e ) | Y e , Y e ′ ) − P ( B k − ( u, G ′′ n ) = B k ( v, G n − e ) | Y e , Y e ′ ) − P ( ℓ = w e ) . By assumption, we have P ( B k ( v, G ′′ n ) ∼ = T k , B k − ( u, G ′′ n ) ∼ = T ′ k − ) ≥ − ε k ( n ) . By the union bound, we have P ( E ) ≤ n k X j =1 k − X j =1 n j − n j − p j + j n ≤ ( λ n + 1) k − n . Proceeding, we may bound P ( B k ( v, G ′′ n ) = B k ( v, G n − e ) | Y e , Y e ′ ) ≤ P ( e ∈ B k ( v, G ′′ n )) + P ( e ′ ∈ B k ( v, G ′′ n )) + P ( e ′ ∈ B k ( v, G n − e ) | b e ′ ) . SKY CAO
We have P ( e ′ ∈ B k ( v, G n − e ) | b e ′ ) ≤ P ( v ′ ∈ B k ( v, G n − { e, e ′ } )) + P ( u ′ ∈ B k ( v, G n − { e, e ′ } )) . We have P ( v ′ ∈ B k ( v, G n − { e, e ′ } )) ≤ P ( v ′ ∈ B k ( v, G n )) ≤ n k X j =1 λ jn ≤ ( λ n + 1) k n . All other terms may be handled similarly.In the rest of the section, we will work towards Lemma 6.9, which wasneeded to bound the covariance between local quantities (Lemma 5.5, ormore specifically, Lemma 5.9). The main work is done by Lemma 6.8. Insteadof proving this straight away, we will first prove the simpler Lemma 6.3,where the main idea becomes easier to describe.With n implicit, we write B vk instead of B k ( v, G n ) for brevity. We mayexplore B vk by breadth first search. I.e., from the root v , find all neighbors of v , and call these the depth 1 vertices. Then find all neighbors of the depth1 vertices, and call these the depth 2 vertices. Here we specify that if aneighbor of a depth 1 vertex has already been found, then we don’t call ita depth 2 vertex. If we can keep exploring in this manner, we obtain aniterative description of B vk as follows.Let S vk be the vertex set of B vk , and let D vk := S vk − S vk − be the set ofdepth k vertices of B vk . Given subsets S , S of the vertex set V n of G n , let X ( S , S ) := { ( e, w e , b e ) : e = ( v , v ) , v ∈ S , v ∈ S } . There is some function Ψ such that for each k , we have B vk +1 = Ψ( B vk , X ( D vk , V n − S vk − )) . In words, this says that in the ( k +1)st iteration, breadth first search exploresall edges incident to a vertex in D vk , i.e. a depth k vertex. Moreover, we onlyneed to look at edges which connect D vk and V n − S vk − . This is because edgesbetween D vk and S vk − have already been explored by previous iterations. Wemay use this iterative description of B vk to obtain the following lemma. Lemma . For each k , there is a coupling of ( B vk +1 , B uk +1 , ˜ B vk +1 , ˜ B uk +1 ) that satisfies the following properties. Let I k := { S vk ∩ S uk = ∅ } . Then on I k , ˜ B vk +1 , ˜ B uk +1 are conditionally independent given B vk , B uk . Moreover, on PARSE OPTIMIZATION CLTS the event I k , the conditional law of ˜ B vk +1 given B vk , B uk is the conditionallaw of B vk +1 given B vk , and the conditional law of ˜ B uk +1 given B vk , B uk is theconditional law of B uk +1 given B uk . I.e., for bounded measurable functions g v , g u , we have I k Cov( g v ( ˜ B vk +1 ) , g u ( ˜ B uk +1 ) | B vk , B uk ) = 0 , I k E ( g v ( ˜ B vk +1 ) | B vk , B uk ) = 1 I k E ( g v ( B vk +1 ) | B vk ) , I k E ( g u ( ˜ B uk +1 ) | B vk , B uk ) = 1 I k E ( g u ( B uk +1 ) | B uk ) . Finally, we have I k P ( ˜ B vk +1 = B vk +1 | B vk , B uk ) ≤ I k C | S vk || S uk | p n , I k P ( ˜ B uk +1 = B uk +1 | B vk , B uk ) ≤ I k C | S vk || S uk | p n . Remark.
The main idea of the proof is noting that as long as B vk , B uk don’t intersect, the objects in next iteration B vk +1 , B uk +1 are very weaklyinteracting with each other. Moreover, the amount of interaction is governedby the size of B vk , B uk . We can remove these interactions by re-randomization,and if the sizes of B vk , B uk are not too large, then this re-randomization isunlikely to cause changes. Proof.
We first show how to generate the pair ( B vk +1 , B uk +1 ) startingfrom B vk , B uk , on the event I k . Let X := X ( D vk , V n − S vk − − S uk ) ,X := X ( D uk , V n − S uk − − S vk ) ,X := X ( D vk , D uk ) . Then on I k , we have B vk +1 = Ψ( B vk , X ∪ X ) ,B uk +1 = Ψ( B uk , X ∪ X ) . Note X ∪ X = X ( D vk , V n − S vk − − S uk − ), and X ∪ X = X ( D uk , V n − S uk − − S vk − ). The point is that on I k , there are some further restrictions on whichedges can be present. I.e., there can be no edges between D vk and S uk − , andthere can be no edges between D uk and S vk − . Note that on I k , the objects X , X are conditionally independent given B vk , B uk .We now construct ( ˜ B vk +1 , ˜ B uk +1 ). First, let ˜ X be an i.i.d. copy of X ,conditional on B vk , B uk . Observe then that Ψ( B vk , X ∪ X ), Ψ( B uk , X ∪ ˜ X ) SKY CAO are conditionally independent (at least on I k ). However, the conditional lawof Ψ( B vk , X ∪ X ) is not as desired. To correct this, we re-randomize theedges between D vk and S uk − . I.e., take ( W ′ , B ′ ) an i.i.d. copy of ( W, B ).Define X ′ := { ( e, w ′ e , b ′ e ) : e = ( v , v ) , v ∈ D vk , v ∈ S uk − } . Similarly, let X ′ := { ( e, w ′ e , b ′ e ) : e = ( v , v ) , v ∈ D uk , v ∈ S vk − } . We now set ˜ B vk +1 := Ψ( B vk , X ∪ X ′ ∪ X ) , ˜ B uk +1 := Ψ( B uk , X ∪ X ′ ∪ ˜ X ) . The point is that to obtain ˜ B vk +1 from B vk , we no longer include the restric-tions on the edges between D vk and S uk − that are induced by the event I k ,and thus the law of ˜ B vk +1 given B vk , B uk is exactly the law of B vk +1 given B vk .The analogous is true for ˜ B uk +1 .To finish, we need to show that ˜ B vk +1 = B vk +1 with very low probability.This event only happens if one of the re-randomized edges between D vk and S uk − is present, i.e. there must exist some e = ( v , v ), with v ∈ D vk , v ∈ S uk − , such that b ′ e = 1. A union bound now does the trick. A similarargument works for ˜ B uk +1 .By applying Lemma 6.3, we can obtain the following lemma. Lemma . Let g v , g u be measurable functions which are bounded inabsolute value by 1. Then Cov( g v ( B vk +1 ) , g u ( B uk +1 )) ≤ C ( P ( I ck ) + p n E ( | S vk || S uk | )) +Cov( E ( g v ( B vk +1 ) | B vk ) , E ( g u ( B uk +1 ) | B uk )) . The following lemma says that the event I k happens with very high prob-ability, and that the sizes of | S vk | , | S uk | are not too large. Lemma . For any k > , we have P ( I ck ) ≤ n k X l =1 λ ln ≤ ( λ n + 1) k n , E ( | S vk || S uk | ) ≤ Ck ( λ n + C ) k + C ≤ ( λ n + C ) k + C . PARSE OPTIMIZATION CLTS Proof.
The first assertion follows by a union bound over all possiblepaths from v to u that use at most 2 k edges. The second assertion followsfirst by Cauchy-Schwarz, and then noting that | S vk | , | S uk | are stochasticallydominated by the total number of vertices in a depth k Galton-Watson tree,with offspring distribution Binomial( n − , p n ), and then concluding by usingstandard formulas for Galton-Watson trees.By iterating Lemma 6.4 and applying Lemma 6.5, we obtain the following. Lemma . For any k > , we have sup g v ,g u Cov( g v ( B vk ) , g u ( B uk )) ≤ Ck ( λ n + C ) k + C n ≤ ( λ n + C ) k + C n . Here the supremum is taken over all pairs of measurable functions with ab-solute value bounded by 1.
With all the notation set, we now go on a slight diversion and quicklyprove the following lemma. It is needed for Lemma 5.6.
Lemma . We have P ( B vk is not a tree ) ≤ ( λ n + C ) k + C n . Proof.
Let A k be the event that B vk is a tree. We have1 A k − P ( A ck | B vk − ) ≤ A k − (cid:18) | D vk − | (cid:19) p n ≤ A k − | D vk − | p n . Upon taking expectations and iterating, we obtain (note P ( A c ) = 0) P ( A ck ) ≤ p n E k − X j =1 | D vj | . Observe k − X j =1 | D vj | ≤ | S vk − | , and finish by observing that E | S vk − | ≤ ( λ n + C ) k + C , as noted in the proofof Lemma 6.5.By rewriting the proof of Lemma 6.3 in a more general form, we mayobtain the following. SKY CAO
Lemma . Let E n := { ( i, j ) , ≤ i < j ≤ n } . Let e = ( v , u ) , e ′ = ( v ′ , u ′ ) have distinct vertices. Let F ⊆ E n − { e } , F ′ ⊆ E n − { e ′ } .Define B k := ( B k ( v , G n ) , B k ( v , G Fn ) , B k ( u , G n ) , B k ( u , G Fn )) , B ′ k := ( B k ( v ′ , G n ) , B k ( v ′ , G F ′ n ) , B k ( u ′ , G n ) , B k ( u ′ , G F ′ n )) . Let N k be the number of vertices in B k , and N ′ k the number of vertices in B ′ k .Let I k be the event that the vertex sets of B k and B ′ k intersect. There is acoupling ( B k +1 , B ′ k +1 , ˜ B k +1 , ˜ B ′ k +1 ) such that on I k , we have that ˜ B k +1 , ˜ B ′ k +1 are conditionally independent given B k , B ′ k , and the law of ˜ B k +1 given B k , B ′ k is the law of B k +1 given B k , and the law of ˜ B ′ k +1 given B k , B ′ k is thelaw of B ′ k +1 given B ′ k . Moreover, we have I k P ( ˜ B k +1 = B k +1 | B k , B ′ k ) ≤ I k CN k N ′ k p n , I k P ( ˜ B ′ k +1 = B ′ k +1 | B k , B ′ k ) ≤ I k CN k N ′ k p n . Proof.
Let V n be the vertex set of G n . Let S k be the vertex set of B k (more precisely, the union of the vertex sets of the four graphs which makeup B k ), and let S ′ k be the vertex set of B ′ k . For S , S ⊆ V n , define X ( S , S ) := { ( e, w e , b e , w ′ e , b ′ e ) : e = ( v, u ) , v ∈ S , u ∈ S } . There is some function Ψ, which depends on F , such that B k +1 = Ψ( B k , X ( S k , V n )) . Similarly, there is a function Ψ ′ which depends on F ′ such that B ′ k +1 = Ψ( B ′ k , X ( S ′ k , V n )) . Define X := X ( S k , V n − S ′ k ) ,X := X ( S ′ k , V n − S k ) ,X := X ( S k , S ′ k ) . Observe then that X ( S k , V n ) = X ∪ X , X ( S ′ k , V n ) = X ∪ X . PARSE OPTIMIZATION CLTS Moreover, note that on I k , we have that X , X are conditionally indepen-dent given B k , B ′ k . We may thus construct ˜ B k +1 , ˜ B ′ k +1 as follows. Condi-tional on B k , B ′ k , let ˜ X be distributed as X , conditional on B k , and let˜ X ′ be distributed as X , conditional on B ′ k . Moreover, let ˜ X , ˜ X ′ be inde-pendent of each other and everything else, conditional on B k , B ′ k . Then on I k , define ˜ B k +1 := Ψ( B k , X ∪ ˜ X ) , ˜ B ′ k +1 := Ψ ′ ( B ′ k , X ∪ ˜ X ′ ) . By construction, on the event I k , we have that ˜ B k +1 , ˜ B ′ k +1 are conditionallyindependent given B k , B ′ k . Moreover, observe that on the event I k , condi-tional on B k , B ′ k , we have that X ∪ ˜ X has the law of X ( S k , V n ) conditionalonly on B k . Thus on I k , the law of ˜ B k +1 conditional on B k , B ′ k is exactly thelaw of B k +1 conditional on B k . The analogous statement is true for ˜ B ′ k +1 .To finish, we need to show1 I k P ( ˜ B k +1 = B k +1 | B k , B ′ k ) ≤ I k CN k N ′ k p n . The proof for ˜ B ′ k +1 will be the exact same. To set notation, write˜ X = { ( e, ˜ w e , ˜ b e , ˜ w ′ e , ˜ b ′ e ) : e = ( v, u ) , v ∈ S k , u ∈ S ′ k } . Observe that if for all e = ( v, u ), v ∈ S k , u ∈ S ′ k , we have b e , b ′ e , ˜ b e , ˜ b ′ e = 0,then necessarily ˜ B k +1 = B k +1 . Thus it suffices to bound the probability thatthis event doesn’t happen. The point is that on the event I k , for e = ( v, u ), v ∈ S k , u ∈ S ′ k , the conditional distribution of any of the b e , b ′ e , ˜ b e , ˜ b ′ e given B k , B ′ k is either Bernoulli( p n ), or the point mass at 0 (in words, either theedge e is left unrestricted, or it forced to not be present). We now finish bythe union bound, along with the fact that if U is a random variable whosedistribution is either identically 0 or Bernoulli( p n ), then P ( U = 1) ≤ p n . We may now deduce the following lemma from Lemma 6.8 in the same waywe deduced Lemma 6.6 from Lemma 6.3. Here we additionally use the factthat if two random variables lie in the interval [ − , − , Lemma . For any k > , we have sup g,g ′ Cov( g ( B k ) , g ′ ( B ′ k )) ≤ min (cid:18) ( λ n + C ) k + C n , (cid:19) . Here the supremum is taken over all pairs of measurable functions whichhave absolute value bounded by 1. SKY CAO
7. Concluding remarks.
A natural direction for future work is to tryto prove a central limit theorem for minimum matching in the mean fieldsetting, following the same strategy as was used for optimal edge cover. Oneof the main difficulties is in proving the analog of Proposition 4.13 (recallthis proposition allowed us to take λ → ∞ with n ) for minimum matching.To do so, we need to analyze the following operator (see [27, 32, 35]). Let λ >
0. Given F : [ − λ/ , λ/ → [0 , V λ F : [ − λ/ , λ/ → [0 ,
1] bythe following: ( V λ F )( x ) := exp − Z λ/ − x F ( ℓ ) dℓ ! . From some simulations, I don’t think the analog of Proposition 4.13 is actu-ally true for this operator V λ , because it seems that α ( λ ) in fact convergesto 1 as λ → ∞ . The difficulty is then trying to understand the rate of con-vergence of α ( λ ), i.e. does it behave like 1 − λ , or 1 − λ , or somethingelse.Another direction is to consider vertex-weighted graphs instead of edge-weighted graphs. For example, [16] proves the long-range independence prop-erty for the maximum weight independent set problem, when the averagevertex degree (i.e. λ ) is at most 2 e . This problem is a combinatorial optimiza-tion problem on vertex-weighted graphs. All the arguments in proving The-orem 2.4 should carry over with small modifications to the vertex-weightedcase; I decided not to include this in the paper because I couldn’t figure outa good way to have one reasonable set of notation that covers both cases.It is also possible to apply Theorem 2.4 to functions of sparse randomgraphs which are not combinatorial optimization problems. For example,Dembo and Montanari [14] use a form of the Objective method to computelimiting constants for the free energy of Ising models on locally tree-likegraphs (this includes sparse Erd˝os-R´enyi graphs). One may use the resultsof [14] to verify Property GLA for the free energy of the Ising model on asparse Erd˝os-R´enyi graph. In particular, Theorem 3.1 of [14] can be usedto check (A1), and Lemma 4.3 of [14] can be used to check (A3). Thus onemay prove a central limit theorem for the free energy. Acknowledgments.
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