Limits of Local Algorithms over Sparse Random Graphs

Transcription

1 Limits of Local Algorithms over Sparse Ranom Graphs [Extene Abstract] ABSTRACT Davi Gamarnik MIT Local algorithms on graphs are algorithms that run in parallel on the noes of a graph to compute some global structural feature of the graph. Such algorithms use only local information available at noes to etermine local aspects of the global structure, while also potentially using some ranomness. Research over the years has shown that such algorithms can be surprisingly powerful in terms of computing structures like large inepenent sets in graphs locally. These algorithms have also been implicitly consiere in the work on graph limits, where a conjecture ue to Hatami, Lovász an Szegey [17] implie that local algorithms may be able to compute near-maximum inepenent sets in sparse) ranom -regular graphs. In this paper we refute this conjecture an show that every inepenent set prouce by local algorithms is smaller that the largest one by a multiplicative factor of at least 1/ + 1/ ).853, asymptotically as. Our result is base on an important clustering phenomena preicte first in the literature on spin glasses, an recently prove rigorously for a variety of constraint satisfaction problems on ranom graphs. Such properties suggest that the geometry of the solution space can be quite intricate. The specific clustering property, that we prove an apply in this paper shows that typically every two large inepenent sets in a ranom graph either have a significant intersection, or have a nearly empty intersection. As a result, large inepenent sets are clustere accoring to the proximity to each other. While the clustering property was postulate earlier as an obstruction for the success of local algorithms, such as for example, the Belief Propagation algorithm, our result is the first one where the clustering property is use to formally prove limits on local algorithms. A full version of this paper is available as Electronic Colloquium on Computational Complexity, Technical Report TR , February 0, 013. Research supporte by the NSF grants CMMI Permission to make igital or har copies of all or part of this work for personal or classroom use is grante without fee provie that copies are not mae or istribute for profit or commercial avantage an that copies bear this notice an the full citation on the first page. Copyrights for components of this work owne by others than ACM must be honore. Abstracting with creit is permitte. To copy otherwise, or republish, to post on servers or to reistribute to lists, requires prior specific permission an/or a fee. Request permissions from permissions@acm.org. ITCS 14, January 1 14, 014, Princeton, New Jersey, USA. Copyright 014 ACM /14/01...$ Mahu Suan Microsoft Research mahu@mit.eu Categories an Subject Descriptors F.1.3 [Complexity Measures an Classes]: Lower Bouns General Terms Theory Keywors Local algorithms, lower bouns, ranom graphs 1. INTRODUCTION Local algorithms are ecentralize algorithms that run in parallel on noes in a network using only information available from local neighborhoos to compute some global function of ata that is sprea over the network. Local algorithms have been stuie in the past in various communities. They arise as natural solution concepts in parallel an istribute computing see, e.g., [1, 19]). They also lea to efficient sub-linear algorithms algorithms that run in time significantly less than the length of the input an [6, 5, 16, 7] illustrate some of the progress in this irection. Finally local algorithms have also been propose as natural heuristics for solving har optimization problems with the popular Belief Propagation algorithm see for instance [9, 3]) being one such example. In this work we stuy the performance of a natural class of local algorithms on ranom regular graphs an show limits on the performance of these algorithms. 1.1 Motivation The motivation for our work comes from the a notion of local algorithms that has appeare in a completely ifferent mathematical context, namely that of the theory of graph limits, evelope in several papers, incluing [8], [7], [0], [6], [5], [1], [17]. In the realms of this theory it was conjecture that every reasonable combinatorial optimization problem on ranom graphs can be solve by means of some local algorithms. To the best of our knowlege this conjecture for the first time was formally state in Hatami, Lovász an Szegey in [17, Conjecture 7.13], an thus, from now on, we will refer to it as Hatami-Lovász-Szegey or HLS) conjecture, though informally it was pose by Szegey earlier, an was reference in several papers, incluing Lyons an Nazarov [], an Csoka an Lippner [11]. In the concrete context of the problem of fining largest inepenent sets in sparse ranom regular graphs, the conjecture is state as follows. Let T,r be a roote -regular tree with epth r.

2 Namely, every noe incluing the root, has egree r, except for the leaves, an the istance from the root to every leaf is r. Consier a function f r : [0, 1] T,r {0, 1} which maps every such tree whose noes are associate with real values from [0, 1] to a ecision encoe by 0 an 1, where the ecision is a function of the values associate with the noes of the tree. In light of the fact that in a ranom -regular graph G r) the typical noe has epth-r neighborhoo isomorphic to T,r, for any constant r, such a function f r can be use to generate ranom) subsets I of G r) as follows: associate with every noe of G r) a uniform ranom values from [0, 1] inepenently for each noe) an apply function f r to each noe. The set of noes for which f r prouces value 1 efines I, an is calle i.i.. factor. It is clear that f r essentially escribes a local algorithm for proucing sets I sweeping issue of computability of f r uner the rug). The HLS conjecture postulates the existence of a sequence of f r, r = 1,,..., such that the set I thus prouce is an inepenent subset of G r) an asymptotically achieves the largest possible value as r. Namely, largest inepenent subsets of ranom regular graphs are i.i.. factors. The precise connection between this conjecture an the theory of graph limits is beyon the scope of this paper. Instea we refer the reaer to the relevant papers [17],[1] 1. The concept of i.i.. factors appears also in one of the open problem by Davi Alous [] in the context of coing invariant processes on infinite trees. It turns out that an analogue for the HLS conjecture is inee vali for another important combinatorial optimization problem - the matching problem. Lyons an Nazarov [] establishe it for the case of bi-partite locally T,r -tree-like graphs, an Csoka an Lippner establishe this result for general locally T,r -tree-like graphs. Further, one can moify the framework of i.i.. factors by encapsulating non-t,r type neighborhoos, for example by making f r epen not only on the realization of ranom uniform in [0, 1] values, but also on the realization of the graph-theoretic neighborhoos aroun the noes. Some probabilistic boun on a egree might be neee to make this efinition rigorous though we will not attempt this formalization in this paper). In this case one can consier, for example, i.i.. factors when neighborhoos are istribute as r generations of a branching process with Poisson istribution, an then ask which combinatorial optimization problems efine now on sparse Erös-Rényi graphs Gn, /n) can be solve as i.i.. factors. Here Gn, /n) is a ranom graph on n noes with each of the n ) eges selecte with probability /n, inepenently for all eges, an > 0 is a fixe constant. In this case it is possible to show that when c e, the maximum inepenent set problem on Gn, /n) can be solve nearly optimally by the well known Belief Propagation BP) algorithm with constantly many rouns. Since the BP is a local algorithm, then the maximum inepenent set on Gn, /n) is an i.i.. factor, in the extene framework efine above. We shoul note that the original proof of Karp an Sipser [18] of the very similar result, relie on a ifferent metho.) Thus, the framework of local algorithms viewe as i.i.. factors is rich enough to solve several interesting combinatorial optimization problems. 1 We also remark that the language use in [17] is quite ifferent from ours. Other works, e.g., [] however o use langauge similar to ours in escribing this conjecture. Nevertheless, in this paper we refute the HLS conjecture in the context of maximum inepenent set problem on ranom regular graphs G n). Specifically, we show that for large enough, with high probability as n, every inepenent set proucible as an i.i.. factor is a multiplicative factor γ < 1 smaller than the largest inepenent subset of G n). We establish that γ is asymptotically at most though we conjecture that the result hols simply for γ = 1/, as we iscuss in the boy of the paper). 1. Techniques Our result is prove in two steps. First we show that large inepenent sets in ranom regular graphs are somewhat clustere. Next we show that solutions of natural algorithms are not clustere. We elaborate on the two parts below. Clustering is a powerful though fairly simple to establish in our case) phenomenon associate with the solutions of some combinatorial optimization problems on ranom graphs. Roughly it states that such solutions are clustere in some natural topology associate with the solution space. We will escribe the precise version we care about shortly.) Such phenomena were first conjecture in the theory of spin glasses an later confirme by rigorous means. Initially, this clustering property was iscusse in terms of so-calle overlap structure of the solutions of the Sherrington-Kirkpatrick moel [8]. Later works highlighte this phenomenon in the context of ranom K-SAT problem. In particular, in inepenent works Achlioptas, Coja-Oghlan an Ricci-Tersenghi [1], an Mezar, Mora an Zecchina [4] prove this phenomenon rigorously. We o not efine the ranom K-SAT problem here an instea refer the reaer to the aforementione papers. What these results state is that in certain regimes, the set of satisfying assignments, with high probability, can be clustere into groups such that two solutions within the same cluster agree on a certain minimum number of variables, while two solutions from ifferent clusters have to isagree on a certain minimum number of variables. In particular, one can ientify a certain non-empty interval [z 1, z ] [0, 1] such that no two solutions of the ranom K-SAT problem agree on precisely z fraction of variables for all z [z 1, z ]. One can further show that the onset of clustering property occurs when the ensity of clauses to variables becomes at least K /K, while at the same time the formula remains satisfiable with high probability, when the ensity is below K log. Interestingly, the known algorithms for fining solutions of ranom instances of K-SAT problem also stop working aroun the K /K threshol. It was wiely conjecture that the onset of the clustering phase is the main obstruction for fining such algorithms. In fact, Coja Oghlan [9] showe that the BP algorithm, which was earlier conjecture to be a goo contener for solving the ranom instances of K-SAT problems, also fails when the ensity of clauses to variables is at least K log K/K, though Coja-Oghlan s approach oes not explicitly rely on the clustering property, an one coul argue that the connection between the clustering property an the failure of the BP algorithm is coinciental. Closer to the topic of this paper, the clustering property was also recently establishe for inepenent sets in Erös- Rényi graphs by Coja-Oghlan an Efthymiou [10]. To escribe their result, we first note a well-known fact that the largest inepenent subset of Gn, /n) has size approximately log /)n, when is large see the next section for

3 precise etails). In [10] it is shown that the set of inepenent sets of size at least approximately log /)n namely those within factor 1/ of the optimal), are also clustere. Namely, one can split them into groups such that intersection of two inepenent sets within a group has a large carinality, while intersection of two inepenent sets from ifferent groups has a small carinality. One shoul note that algorithms for proucing large inepenent subsets of ranom graphs also stop short factor 1/ of the optimal, both in the case of sparse an in the ense ranom graph cases, as exhibite by the well-known Karp s open problem regaring inepenent subsets of Gn, 1/) [3]. This is almost the result we nee for our analysis with two exceptions. First, we nee to establish this clustering property for ranom regular as oppose Erös-Rényi graphs. Secon, an more signficanlty, the result in [10] applies to typical inepenent sets an oes not rule out the possibility that there exist two inepenent sets with some intermeiate intersection carinality, though the number of such pairs is insignificant compare to the total number of inepenent sets. For our result we nee to show that, without exception, every pair of large inepenent sets has either large or small intersection. We inee establish this, but at the cost of losing aitional factor 1/ ). In particular, we show see Theorem.6) that for large enough, with high probability as n, every two inepenent subsets of G n) with carinality asymptotically 1 + β)log /)n, where 1 β > either have intersection size at least 1 + z)log /)n or at most 1 z)log /)n, for some z < β. The result is establishe using a straightforwar first moment argument: we compute the expecte number of pairs of inepenent sets with intersection lying in the interval [1 z)log /)n, 1 + z)log /)n], an show that this expectation converges to zero exponentially fast. We remark that even though our conclusion is somewhat stronger than that of previous clustering results, our proof of the clustering phenomenon is extremely simple. With this result at han, the refutation of the HLS conjecture is fairly simple to erive. We prove see Theorem.7) that if local algorithms can construct inepenent sets of size asymptotically 1 + β)log /)n, then, by means of a simple coupling construction, we can construct two inepenent sets with intersection size z for all z in the interval [1 + β) log /) n, 1 + β)log /)n], clearly violating the clustering property. The aitional factor 1/ ) is an artifact of the analysis, an hence we believe that our result hols for all β 0, 1]. Namely, no local algorithm is capable of proucing inepenent sets with size larger than factor 1/ of the optimal, asymptotically in. We note again that this coincies with the barrier for known algorithms. It is noteworthy that our result is the first one where algorithmic harness erivation relies irectly on the the geometry of the solution space, viz a vi the clustering phenomena, an thus the connection between algorithmic harness an clustering property is not coinciental. The remainer of the paper is structure as follows. We introuce some basic material an the HLS conjecture in the next section. In the same section we state our main theorem non-valiity of the conjecture Theorem.5). We also state two seconary theorems, the first escribing the overlap structure of inepenent sets in ranom graphs Theorem.6) - the main tool in the proof of our result, an the secon escribing overlaps that can be foun if local algorithms work well Theorem.7). We prove our main theorem easily from the two seconary theorems in Section 3. We prove Theorem.7 in Section 4. We omit the proof of Theorem.6 from this version.. PRELIMINARIES AND MAIN RESULT For convenience, we repeat here some of the notions an efinitions alreay introuce in the first section. Basic graph terminology. All graphs in this paper are unerstoo to be simple unirecte graphs. Given a graph G with noe set V G) an ege set EG), a subset of noes I V G) is an inepenent set if u, v) / EG) for all u, v I. A path between noes u an v with length r is a sequence of noes u 1,..., u r 1 such that u, u 1), u 1, u ),..., u r 1, v) EG). The istance between noes u an v is the length of the shortest path between them. For every positive integer value r an every noe u V G), B G u, r) enotes the epth-r neighborhoo of u in G. Namely, B G u, r) is the subgraph of G inuce by noes v with istance at most r from u. When G is clear from context we rop the subscript. The egree of a vertex u V G) is the number of vertices v such that u, v) EG). The egree of a graph G is the maximum egree of a vertex of G. A graph G is -regular if the egree of every noe is. Ranom graph preliminaries. Given a positive real, Gn, /n) enotes the Erös-Rényi graph on n noes [n] {1,,..., n}, with ege probability /n. Namely each of the n ) eges of a complete graph on n noes belongs to EGn, /n)) with probability /n, inepenently for all eges. Given a positive integer, G n) enotes a graph chosen uniformly at ranom from the space of all -regular graphs on n noes. This efinition is meaningful only when n is an even number, which we assume from now on. Given a positive integer m, let In,, m) enote the set of all inepenent sets in Gn, /n) with carinality m. I n, m) stans for a similar set for the case of ranom regular graphs. Given integers 0 k m, let On,, m, k) enote the set of pairs I, J In,, m) such that I J = k. The efinition of the set O n, m, k) is similar. The sizes of the sets On,, m, k) an O n, m, k), an in particular whether these sets are empty or not, is one of our focuses. Denote by αn, ) the size of a largest in carinality inepenent subset of Gn, /n), normalize by n. Namely, αn, ) = n 1 max{m : In,, m) }. α n) stans for the similar quantity for ranom regular graphs. It is known that αn, ) an α n) have eterministic limits as n. Theorem.1. For every R + there exists α) such that w.h.p. as n, αn, ) α). 1) Similarly, for every positive integer there exists α such that w.h.p. as n α n) α. )

4 Furthermore as. α) = log 1 o1)), 3) α = log 1 o1)), 4) The convergence 1) an ) were establishe in Bayati, Gamarnik an Tetali [4]. The limits 3) an 4) follow from much oler results by Frieze [13] for the case of Erös-Rényi graphs an by Frieze an Luczak [14] for the case of ranom regular graphs, which establishe these limits in the lim sup n an lim inf n sense. The fallout of these results is that graphs Gn, /n) an G n) have inepenent sets of size up to approximately log /)n, when n an are large, namely in the oubly asymptotic sense when we first take n to infinity an then to infinity. Local graph terminology. A ecision function is a measurable function f = fu, G, x) where G is a graph on vertex set [n] for some positive integer n, u [n] is a vertex an x [0, 1] N is a sequence of real numbers for some N n an returns a Boolean value {0, 1}. A ecision function f is sai to compute an inepenent set if for every graph G an every sequence x an for every pair u, v) EG) it is the case that either fu, G, x) = 0 or fv, G, x) = 0, or both. We refer to such an f as an inepenence function. For an inepenence function f, graph G on vertex set [n] an x [0, 1] N for N n, we let I G f, x) enote the inepenent set of G returne by f, i.e., I G f, x) = {u [n] fu, G, x) = 1}. We will assume later that X is chosen ranomly accoring to some probability istribution. In this case I G f, x) is a ranomly chosen inepenent set in G. We now efine the notion of a local ecision function, i.e., one whose actions epen only on the local structure of a graph an the local ranomness. The efinition is a natural one, but we formalize it below for completeness. Let G 1 an G be graphs on vertex sets [n 1] an [n ] respectively. Let u 1 [n 1] an u [n ]. We say that π : [n 1] [n ] is an r-local isomorphism mapping u 1 to u if π is a graph isomorphism from B G1 u 1, r) to B G u, r) so in particular it is a bijection from B G1 u 1, r) to B G u, r), an further it preserves ajacency within B G1 u 1, r) an B G u, r)). For G 1, G, u 1, u an an r-local isomorphism π, we say sequences x 1) [0, 1] N 1 an x ) [0, 1] N are r-locally equivalent if for every v B G1 u 1, r) we have x 1) v = x ) πv). Finally we say fu, G, x) is an r-local function if for every pair of graphs G 1, G, for every pair of vertices u 1 V G 1) an u V G ), for every r-local isomorphism π mapping u 1 to u an r-locally equivalent sequences x 1) an x ) we have fu 1, G 1, x 1) ) = fu, G, x ) ). We often use the notation f r to enote an r-local function. Let n,r 1 + 1) r 1)/ ) enote the number of vertices in a roote tree of egree an epth r. We let T,r enote a canonical roote tree on vertex set [n,r ] with root being 1. For n n,r, x [0, 1] n an an r-local function f r, we let f rx) enote the quantity f r1, T,r, x). Let X be chosen accoring to a uniform istribution on [0, 1] n. The set subset of noes I G n)f r, X) is calle i.i.. factor prouce by the r-local function f r. As we will see below the αf r) 1 E n X[f rx)] accurately captures to within an aitive o1) factor) the ensity of an inepenent returne by an r-local inepenence function f r on G n). First we recall the following folklore proposition which we will also use often in this paper. Proposition.. As n, with probability tening to 1 almost all local neighborhoos in G n) look like a tree. Formally, for every, r an ɛ, for sufficiently large n, P G n) {u [n] BG n)u, r) = T,r } ɛn ) ɛ. This immeiately implies that the expecte value of the inepenent set I G n)f r, X) prouce by f r is αf r)n+on). In fact the following concentration result hols. Proposition.3. As n, with probability tening to 1 the inepenent set prouce by a r-local function f on G n) is of size αf) n + on). Formally, for every, r, ɛ an every r-local function f, for sufficiently large n, P G n),x [0,1] N IG n)f r, X) αf r)n ɛn ) ɛ. Proof. The proof follows from by the fact that the variance of I G n),x is On) an its expectation is αf r)n+on), an so the concentration follows by Chebychev s inequality. The boun on the variance in turn follows from the fact that for every graph G, there are at most On) pairs of vertices u an v for which the events fu, G, X) an fv, G, X) are not inepenent for ranom X. Details omitte. The Hatami-Lovász-Szegey Conjecture an our result. We now turn to escribing the Hatami-Lovász-Szegey HLS) conjecture an our result. Recall α efine by ). The HLS conjecture can be state as follows. Conjecture.4. There exists a sequence of r-local inepenence functions f r, r 1 such that almost surely If r, n) is an inepenent set in G n) an αf r) α as r. Namely, the conjecture asserts the existence of a local algorithm r-local inepenence function f r) which is capable of proucing inepenent sets in G r) of carinality close to the largest that exist. For such an algorithm to be efficient the function f ru, G, x) shoul also be efficiently computable uniformly. Even setting this issue asie, we show that there is a limit on the power of local algorithms to fin large inepenent sets in G n) an in particular the HLS conjecture oes not hol. Let ˆα = sup r sup fr αf r), where the secon supremum is taken over all r-local inepenence functions f r. Theorem.5. [Main] For every ɛ > 0 an all sufficiently large, ˆα 1 α ɛ. That is, for every ɛ > 0 an for all sufficiently large, a largest inepenent set obtainable by r-local functions is at most ɛ for all r. Thus for all large enough there is a multiplicative gap between ˆα an the inepenence ratio α. That being sai, our result oes not rule out that for small, ˆα in fact equals α, thus leaving the HLS conjecture open in this regime.

5 The two main ingreients in our proof of Theorem.5 both eal with the overlaps between inepenent sets in ranom regular graphs. Informally, our first result on the size of the overlaps shows that in ranom graphs the overlaps are not of intermeiate size this is formalize in Theorem.6. We then show that we can apply any r-local function f r twice, with couple ranomness, to prouce two inepenent sets of intermeiate overlap where the size of the overlap epens on the size of the inepenent sets foun by f r an the level of coupling. This is formalize in Theorem.7 Theorem.5 follows immeiately by combinig the two theorems an appropriate setting of parameters). Overlaps in ranom graphs. We now state our main theorem about the overlap of large inepenent sets. We interpret the statement after we make the formal statement. Theorem.6. For β 1/, 1) an 0 < z < β 1 < β an, let s = 1 + β) 1 log an let Kz) enote the set 1 z)n log 1+z)n log of integers between an. Then, for all large enough, we have ) lim P k Kz) On,, sn, k) = 0, 5) n an ) lim P k Kz) O n, sn, k) = 0. 6) n In other wors, both in the Erös-Rényi an in the ranom regular graph moels, when β > 1/, an is large enough, with probability approaching unity as n, one cannot fin a pair of inepenent sets I an J with size ns, such that their overlap intersection) has carinality n1 z) log n1+z) log. at least an at most Note that for all β > 1/, there exists z satisfying 0 < z < β 1 an so the theorem is not vacuous in this setting. Furthermore as β 1, z can be chosen arbitrarily close to 1 making the forbien overlap region extremely broa. That is, as the size of the inepenent sets in consieration approaches the maximum possible namely as β 1), an as, we can take z 1. In other wors, with probability approaching one, two nearly largest inepenent sets either overlap almost entirely or almost o not have an intersection. This is the key result for establishing our harness bouns for existence of local algorithms. A slightly ifferent version of the first of these results can be foun as Lemma 1 in [10]. The latter paper shows that if an inepenent set I is chosen uniformly at ranom from the set with size nearly 1 + β)n log /, then with high probability with respect to the choice of I), there exists an empty overlap region in the sense escribe above. In fact, this empty overlap region exists for every β 0, 1), as oppose to just 1 > β > 1/ + 1/ ) as in our case. Unfortunately, this result cannot be use for our purposes, since this result oes not rule out the existence of rare sets I for which no empty overlap exists. Overlapping from local algorithms. Next we turn to the formalizing the notion of using a local function f r twice on couple ranomness to prouce overlapping inepenent sets. Fix an r-local inepenence function f r. Given a vector X = X u, 1 u n) of variables X u [0, 1], recall that I G f r, X) enotes the inepenent set of G given by u I G f r, X) if an only if f ru, G, X) = 1. Recall that X is chosen accoring to the uniform istribution on [0, 1] n. Namely, X u are inepenent an uniformly istribute over [0, 1]. In what follows we consier some joint istributions on pairs of vectors X, Y) such that marginal istributions on the vector X an Y are uniform on [0, 1] n, though X an Y are epenent on each other. The intuition behin the proof of Theorem.5 is as follows. Note that if X = Y then I G f r, X) = I G f r, Y). As a result the overlap I G f r, X) I G f r, Y) between I G f r, X) an I G f r, Y) is αf r)n + on) in expectation. On the other han, if X an Y are inepenent, then the overlap between I G f r, X) an I G f r, Y) is α f r)n + on) in expectation, since the ecision to pick a vertex u in I is inepenent for most vertices when X an Y are inepenent. In particular, note that if the local neighborhoo aroun u is a tree, which accoring to Proposition. happens with probability approaching unity, then the two ecisions are inepenent, an u I with probability αf r).) Our main theorem shows that by coupling the variables, the overlap can be arrange to be of any intermeiate size, to within an aitive on) factor. In particular, if αf r) excees we will be able to show that the overlap can be arrange to be between the values 1 z)n log 1+z)n log an, escribe in Theorem.6 which contraicts the statement of this theorem. Theorem.7. Fix a positive integer. For constant r, let f ru, G, x) be an r-local inepenence function an let α = αf r). For every γ [α, α] an ɛ > 0, an for every sufficiently large n, there exists a istribution on variables X, Y) [0, 1] n [0, 1] n such that P G n),x,y) IG n)f r, X) I G n)f r, Y) γ ± ɛ)n ) ɛ. 3. PROOF OF THEOREM.5 We now show how Theorems.6 an.7 immeiately imply Theorem.5. Proof Proof of Theorem.5. Fix an r-local function f r an let α = αf r). Fix 0 < η < 1. We will prove below that for sufficiently large we have α/α 1/+1/ )+ η. The theorem will then follow. Let ɛ = η log. By Proposition.3 we have that almost surely an inepenent set returne by f r on G n) is of size at least α ɛ)n. Furthermore for every γ [α, α] we have, by Theorem.7, that G n) almost surely has two inepenent sets I an J, with I, J α ɛ)n an I J [γ ɛ)n, γ + ɛ)n]. 7) Finally, by Theorem.1, we have that for sufficiently large, I, J 1 log )1 + η)n 4 1 log n an so α 1 log, allowing us to set γ = 1 / log. Now we apply Theorem.6 with z = ɛ/ log an β > 1+z. Note that for this choice we have z < 1 an z < β 1 < β < 1. We will also use later the fact that for this choice we have β 1/ +z = 1/ +ɛ 1 log.) Theorem.6 asserts that almost surely G n) has no inepenent sets of size at least 1 + β) 1 log n with intersection size in [1 z) 1 log n, 1 + z) 1 log n]. Since I J [γ ɛ)n, γ +ɛ)n] = [1 z) 1 log n, 1+z) 1 log n], we conclue that min{ I, J } 1 + β) 1 log n. Combining with Equation 7) we get that α ɛ)n min{ I, J }

6 1 + β) 1 log n an so α 1 + β) 1 log + ɛ, which by the given boun on β yiels α 1 + 1/ ) 1 log + ɛ = 1 + 1/ + η) 1 log. On the other han we also have α η) 1 log. It follows that α/α 1/+1/ +η as esire. 4. PROOF OF THEOREM.7 For parameter p [0, 1], we efine the p-correlate istribution on vectors of ranom variables X, Y) to be the following: Let X, Z be inepenent uniform vectors over [0, 1] n. Now let Z u = X u with probability p an Y u with probability 1 p inepenently for every u V G). Let fu, G, x) an α be as in the theorem statement. Recall that fx) = f1, T,r, x) is the ecision of f on the canonical tree of egree an epth r roote at the vertex 1. Let γp) be the probability that fx) = 1 an fy) = 1, for p-correlate variables X, Y). As with Proposition.3 we have the following. Lemma 4.1. For every, r, ɛ > 0 an r-local function f, for sufficiently large n we have: P I G n)f, X) I G n)f, Y) γp) n ɛn ) ɛ, where X, Y) are p-correlate istributions on [0, 1] n. Proof. By Proposition. we have that almost surely almost all local neighborhoos are trees an so for most vertices u the probability that u is chosen to be in the inepenent sets If, X) an If, Y) is γp). By linearity of expectations we get that E[ If, X) If, Y) ] = γp) n + on). Again observing that most local neighborhoos are isjoint we have that the variance of If, X) If, Y) is On). We conclue, by applying the Chebychev boun, that If, X) If, Y) is concentrate aroun the expectation an the lemma follows. We also note that for p = 1 an p = 0 the quantity γp) follow immeiately from their efinition. Proposition 4.. γ1) = α an γ0) = α. Now to prove Theorem.7 it suffices to prove that for every γ [α, α] there exists a p such that γp) = γ. We show this next by showing that γp) is continuous. Lemma 4.3. For every r, γp) is a continuous function of p. Proof. Let W u, u T,r ) be ranom variables associate with noes in T,r, uniformly istribute over [0, 1], which are inepenent for ifferent u an also inepenent from X u an Z u. We use W u as generators for the events Y u = X u vs Y u = Z u. In particular, given p, set Y u = X u if W u p an Y u = Z u otherwise. This process is exactly the process of setting variables Y u to X u an Z u with probabilities p an 1 p respectively, inepenently for all noes u. Now fix any p 1 < p, an let δ < p p 1)/ r+1. We use the notation f rx u, Z u, W u, p) to enote the value of f r when the see variables realization is W u, u T,r ), an the threshol value p is use. Namely, f rx u, Z u, W u, p) = f r X u1{w u p} + Z u1{w u > p}, u T,r ). Here, for ease of notation, the reference to the tree T,r is roppe. Utilizing this notation we have γp) = P f rx u) = f rx u, Z u, W u, p) = 1). Manipulating the expressions somewhat we fin γp ) γp 1) r+1 p p 1). Since r is fixe, the continuity of γp) is establishe. We are now reay to prove Theorem.7. Proof Proof of Theorem.7. Given γ [α, α] by Lemma 4.3 we have that there exists a p such that γ = γp). For this choice of p, let X, Y) be a pair of p-correlate istributions. Applying Lemma 4.1 to this choice of p, we get that with probability at least 1 ɛ we have I G n)f, X) I G n)f, Y) [γ ɛ)n, γ + ɛ)n] as esire. 5. PROOF OF THEOREM.6 We omit the proof of this theorem from this version. Details can be foun in the full version [15]. Acknowlegements The first author wishes to thank Microsoft Research New Englan for proviing exciting an hospitable environment uring his visit in 01 when this work was conucte. The same author also wishes acknowlege the general support of NSF grant CMMI REFERENCES [1] D. Achlioptas, A. Coja-Oghlan, an F. Ricci-Tersenghi. On the solution space geometry of ranom formulas. Ranom Structures an Algorithms, 38:51 68, 011. [] D. Alous. Some open problems. alous/ Research/OP/inex.html. [3] N. Alon an J. Spencer. Probabilistic Metho. Wiley, 199. [4] M. Bayati, D. Gamarnik, an P. Tetali. Combinatorial approach to the interpolation metho an scaling limits in sparse ranom graphs. Annals of Probability, to appear. Conference version in Proc. 4n Ann. Symposium on the Theory of Computing STOC), 010. [5] C. Borgs, J. Chayes, an D. Gamarnik. Convergent sequences of sparse graphs: A large eviations approach. arxiv: [math.pr], 013. [6] C. Borgs, J. Chayes, L. Lovász, an J. Kahn. Left an right convergence of graphs with boune egree [7] C. Borgs, J. Chayes, L. Lovász, V. Sós, an K. Vesztergombi. Convergent graph sequences II: Multiway cuts an statistical physics. Ann. of Math., 176:151 19, 01. [8] C. Borgs, J. Chayes, L. Lovász, V. Sós, an K. Vesztergombi. Convergent graph sequences I: Subgraph frequencies, metric properties, an testing. Avances in Math., 19: , 08. [9] A. Coja-Oghlan. On belief propagation guie ecimation for ranom k-sat. In Proceeings of the Twenty-Secon Annual ACM-SIAM Symposium on Discrete Algorithms, pages SIAM, 011.

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