Towards Computing the Grothendieck Constant

Prasad Raghavendra∗ David Steurer†

Abstract 1 Introduction

The Grothendieck constant KG is the smallest constant such The Grothendieck inequality states that for every m × n that for every d ∈ N and every matrix A = (aij ), matrix A = (aij) and every choice of unit vectors u ,..., u and v ,..., v , there exists a choice of signs X X 1 m 1 n sup aij hui, vj i 6 KG · sup aij xiyj , x1, . . . , xm, y1, . . . , yn ∈ {1, −1} such that (d) ui,vj ∈B ij xi,yj ∈[−1,1] ij m n m n (d) d X X X X where B is the unit ball in R . Despite several efforts aijhui, vji 6 K aijxiyj , [15, 23], the value of the constant KG remains unknown. i=1 j=1 i=1 j=1 The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the K M,N -Quadratic where K is a universal constant. The smallest value of Programming problem. The input to this problem is a K for which the inequality holds, is referred to as the matrix A = (aij ) and the objective is to maximize the P Grothendieck constant KG. Since the inequality was first quadratic form ij aij xiyj over xi, yj ∈ [−1, 1]. In this work, we apply techniques from [22] to the KM,N - discovered [8], the inequality has not only undergone Quadratic Programming problem. Using some standard various restatements under different frameworks of anal- but non-trivial modifications, the reduction in [22] yields ysis (see [16]), it has also found numerous applications the following hardness result: Assuming the Unique Games in functional analysis. Conjecture [9], it is NP-hard to approximate the KM,N - In recent years, the Grothendieck’s inequality has Quadratic Programming problem to any factor better found algorithmic applications in efficient construction than the Grothendieck constant KG. of Szemerédipartitions of graphs and estimation of By adapting a “bootstrapping” argument used in a proof cut norms of matrices [2], in turn leading to efficient of Grothendieck inequality [5], we are able to perform a tighter approximation algorithms for problems in dense and analysis than [22]. Through this careful analysis, we obtain the following new results: quasi-random graphs [7, 4]. The inequality has also proved useful in certain lower bound techniques for

◦ An approximation algorithm for KM,N -Quadratic communication complexity [17]. Among its various Programming that is guaranteed to achieve an ap- applications, we shall elaborate here on the KM,N - proximation ratio arbitrarily close to the Grothendieck Quadratic Programming problem. In this problem, constant KG (optimal approximation ratio assuming the objective is to maximize the following quadratic the Unique Games Conjecture). program with the matrix A = (aij) given as input.

◦ We show that the Grothendieck constant KG can be computed within an error η, in time depending only X Maximize aijxiyj on η. Speciﬁcally, for each η, we formulate an explicit i,j ﬁnite linear program, whose optimum is η-close to the Grothendieck constant. subject to xi, yj ∈ {1, −1} .

We also exhibit a simple family of operators on the Alternatively, the problem amounts to computing the Gaussian Hilbert space that is guaranteed to contain tight norm kAk∞→1 of the matrix A, which is deﬁned as examples for the Grothendieck inequality.

kAxk1 kAk∞→1 := max . n x∈R kxk∞ ∗Department of Computer Science & Engineering, University of Washington, Seattle, WA 98195. Work done while the author The KM,N -Quadratic Programming problem is a was visiting Princeton University. Supported by NSF grant CCF- 0343672. formulation of the correlation clustering problem for two †Princeton University, Princeton, NJ. Supported by NSF grants clusters on a bipartite graph [6]. The following natural MSPA-MCS 0528414, ITR-0205594, and CCF-0832797. SDP relaxation to the problem is obtained by relaxing the variables xi, yj to unit vectors. is achieved using a standard truncation argument as outlined in [19]. X Maximize aijhui, vji Even with the above modiﬁcation, the optimal algo- i,j rithm for CSPs in [22] does not directly translate to an

subject to kuik = kvjk = 1 . algorithm for KM,N -Quadratic Programming. The main issue is the additive error of constant magnitude The Grothendieck constant KG is precisely the integral- incurred in all the reductions of [22]. For a CSP, the ity gap of this SDP relaxation for the KM,N -Quadratic objective function is guaranteed to be at least a fixed Programming problem. constant fraction (say 0.5). Hence, it is sufficient if Despite several proofs and reformulations, the value the additive error term(say η) in the reduction can be of the Grothendieck constant KG still remains unknown. bounded by an arbitrarily small constant. In case of π In his original work, Grothendieck showed that 2 6 KM,N -Quadratic Programming, the value of the KG 2.3. The upper bound has been later improved to optimum solution could be as small as 1/log n. Here an 6 √ π/2 log(1+ 2) ≈ 1.78 by Krivine [15], while the best known additive constant error would completely change the lower bound is roughly 1.67 [23]. More importantly, very approximation ratio. little seems to be known about the matrices A for which To obtain better bounds on the error, we use a the inequality is tight [14]. Computing the Grothendieck bootstrapping argument similar to the Gaussian Hilbert constant approximatively and characterizing the tight space approach to the Grothendieck inequality [5] (this examples for the inequality form the original motivation approach is used for algorithmic purposes in [2, 1, 13]). for this work. Towards this goal, we will harness the Using ideas from the proof of the Grothendieck inequality, emerging connections between semidefinite programming we perform a tighter analysis of the reduction in [22] for (SDP) and hardness of approximation based on the the special case of KM,N -Quadratic Programming. Unique Games Conjecture (UGC) [9]. This tight analysis yields the following new results: In a recent work [22], the first author obtained general results connecting SDP integrality gaps to Theorem 1.2. For every η > 0, there is an efficient UGC-based hardness results for arbitrary constraint algorithm that achieves an approximation ratio KG − η satisfaction problems (CSP). These connections yielded for KM,N -Quadratic Programming running in time optimal algorithms and inapproximability for every CSP F (η) · poly(n) where F (η) = exp(exp(O(1/η3))). assuming the Unique Games Conjecture. Further, for the special case of 2-CSPs, it yielded an algorithm to Theorem 1.3. For every η > 0, the Grothendieck compute the value of the integrality gap of a natural constant KG can be computed within an error η in time SDP. proportional to exp(exp(O(1/η3))). Recall that the Grothendieck constant is precisely the integrality gap of the SDP for K M,N -Quadratic A more careful analysis could lower the degree of the Programming. In this light, the current work applies polynomial O(1/η3) in the above bounds, but reducing the techniques of Raghavendra [22] to the K M,N - the number of exponentiations seems to require new . Quadratic Programming ideas. With the intent of characterizing the tight cases for 1.1 Results We obtain the following UGC-based the Grothendieck inequality, we perform a non-standard hardness result for K . M,N -Quadratic Programming reduction from dictatorship tests to integrality gaps. Theorem 1.1. Assuming the Unique Games Conjec- Unlike the reduction in [22], our reduction does not use the Khot–Vishnoi [12] integrality gap instance for ture, it is NP-hard to approximate KM,N -Quadratic Programming by any constant factor smaller than the Unique games. This new reduction yields a simple family of operators which are guaranteed to contain the tight Grothendieck constant KG. cases for the Grothendieck inequality. Specifically, we

Although KM,N -Quadratic Programming falls show the following result: in the “generalized constraint satisfaction problem” framework of Raghavendra [22], the above result does Theorem 1.4. Let Q(k) be the set of linear operators k P∞ not immediately follow from [22] since the reduction A on functions f : R → R of the form A = d=0 λdQd, does not preserve bipartiteness. The main technical where Qd is the orthogonal projector on the span of k- hurdle in obtaining a bipartiteness-preserving reduction, multivariate Hermite polynomials of degree d. There is to give a stronger analysis of the dictatorship test exists operators in Q(k) for which the Grothendieck so as to guarantee a common influential variable. This inequality is near tight. More precisely, for every η, there exists an operator A ∈ Q(k) for some k, such that the notions of Noise Operators, Hermite polynomials, Z Multilinear extensions and influences. The overall sup kAf(x)k dγ(x) > structure of the reductions, along with key definitions k (d) f : R →B and lemmas are described in Section3. This overview Z includes reductions from integrality gaps to dictatorships (KG − η) · sup |Af(x)| dγ(x) . k (Subsection 3.1) and vice versa (Subsection 3.2). Using f : R →[−1,1] these reductions, we outline the proofs of Theorems 1.1, Here γ denotes the k-dimensional Gaussian probability 1.2, 1.3 and 1.4 in Section3. Finally, in Sections4, measure, and for a function f : Rk → Rd, we denote by 5 we present the proof details for the reduction from Af(x) the vector (Af1(x), . . . , Afd(x)) where f1, . . . , fd integrality gaps to dictatorship tests and vice versa. are the coordinates of f. 2 Preliminaries We remark that Theorem 1.4 can also be shown in a direct way without using dictatorship tests (details in Problem 1. (KM,N -Quadratic Programming) the full version). We can strengthen the statement of Given an m × n matrix A = (aij), compute the optimal Theorem 1.4 in the following way: For every η > 0, there value of the following optimization problem, exists a linear operator A on functions f : Rk → R of the X ∞ opt(A) := max aijxiyj , form A = P λ Q such that (1) k = poly(1/η), i=0 2i+1 2i+1 ij (2) λ1 = maxd |λd|, and (3) KG = λ1/kAk∞→1 ± η. In [14], some evidence is given that the operator A = where the maximum is over all x1, . . . , xm ∈ [−1, 1] and ∞ P i y1, . . . , yn ∈ [−1, 1]. Note that the optimum value opt(A) i=0(−1) Q2i+1 is a tight instance for Grothendieck’s inequality when k tends to ∞. is attained for numbers with |xi| = |yj| = 1.

Problem 2. (KM,N -SemidefiniteProgramming) 1.2 Prior Work The general Grothendieck problem Given an m × n matrix A = (aij), compute the optimal on a graph G amounts to maximizing a quadratic value of the following optimization problem, polynomial P a x x over {1, −1} values, where a ij ij i j ij X is non zero only for edges (i, j) in G. The KM,N - sdp(A) := max aijhui, vji , Quadratic Programming is the special case where ij G is a complete bipartite graph K . M,N where the maximum is over all vectors u ,..., u ∈ B(d) The Grothendieck problem on a complete graph 1 m and all vectors v ,..., v ∈ B(d). Here B(d) denotes the admits a O(log n) approximation [21, 18, 6] and has 1 n unit ball in d and we choose d m + n. Note that applications in correlation clustering [6]. For the R > the optimum value sdp(A) is always attained for vectors Grothendieck problem on general graphs, [1] obtain an with ku k = kv k = 1. approximation that depends on the Lovásznumber of i j the graph. For every matrix A, we have opt(A) 6 sdp(A). In an alternate direction, the Grothendieck problem Hence, KM,N -SemidefiniteProgramming is a relax- has been generalized to the Lp-Grothendieck problem ation of KM,N -Quadratic Programming. The value where the Lp-norm of the assignment is bounded by 1. sdp(A) can be computed in polynomial time (up to ar- The traditional Grothendieck corresponds to the case bitrarily small numerical error). when p = ∞. In a recent work, [13] obtain UGC-based hardness results and approximation algorithms for the Definition 2.1. The Grothendieck constant KG is the supremum of sdp(A)/opt(A) over all matrices A. Lp-Grothendieck problem. On the hardness side, [3] show a O(logγ n) hardness for the Grothendieck problem on the complete graph 2.1 Notation. For a probability space Ω, let L2(Ω) for some fixed constant γ > 0. Tight integrality denote the Hilbert space of real-valued random variables gaps for the Grothendieck problem on complete graphs over Ω with finite second moment, were exhibited in [11, 1]. For the K 2 N,N -Quadratic L2(Ω) := {f :Ω → R | E f(ω) < ∞} Programming problem, a UGC-based hardness of ω←Ω roughly 1.67 was shown in [11]. The reduction uses Here, we will consider two kinds of probability spaces. the explicit operator constructed in the proof of a lower One is the uniform distribution over the Hamming cube bound [23] for the Grothendieck constant. {1, −1}k, denoted Ω = Hk. The other one is the k Gaussian distribution over Rk, denoted Ω = G . For p 2 1.3 Organization of the Paper In Section2, we f, g ∈ L2(Ω), we denote hf, gi := E fg, kfk := E f , formally define the Grothendieck constant, and review and kfk∞ := supx∈Ω f(x). We have kfk 6 kfk∞. Lemma 2.1. (Bootstrapping Lemma) Given an 2.3 Variable Influences. For a function f ∈ k P ˆ2 ˆ m × n matrix A = (aij), and vectors u1,..., um and L2(H ), we define Infif = S3i fS, where fS are the v1,..., vn, then Fourier coefficients of f,

X X ˆ aijhui, vji maxkuik maxkvjk · sdp(A) f = fSχS . 6 i j ij S⊆[k] 6 2 maxkuik maxkvjk · opt(A) i j Let us denote MaxInf f := maxi∈[k] Infif. For a pair of k functions f, g ∈ L2(H ), we define MaxComInf(f, g) := Definition 2.2. (Noise Operator) For Ω = Hk or k maxi∈[k] min{Infif, Infig} to be the maximum common Ω = G , let Tρ denote the linear operator on L2(Ω) influence. defined as k Similarly, for f ∈ L2(G ), we denote by Infif = k P ˆ2 ˆ X d fσ the influence of coordinate i. Here, fσ are Tρ := ρ Pd , σ;σi>0 the Hermite coefficients of f, d=0 where Pd denotes the orthogonal projector on the sub- X ˆ f = fσHσ . space of L2(Ω) spanned by the (multilinear) degree-d σ∈ k Q N0 monomials χS(x) := i∈S xi | S ⊆ [k], |S| = d . Fact 2.3. For f ∈ L (Hk) and γ ∈ [0, 1], we have For every function f ∈ L (Hk). 2 Fact 2.1. 2 Pk kfk2 k i=1 InfiT1−γ f 6 /γ. Similarly, for f ∈ L2(G ) Pk kfk2 (Tρf)(x) = E f(y) . and γ ∈ [0, 1], i=1 InfiU1−γ f 6 /γ. y∼ρx k Here, y ∼ x means that y is obtained from x by 2.4 Multilinear Extensions. For f ∈ L2(H ), let ρ ¯ k replacing each coordinate i independently with probability f ∈ L2(G ) denotes the (unique) multilinear extension of f to Rk. 1 − ρ by a random sign. In particular, Ey∼ρxhx, yi = ρ. Let u, v ∈ d be two unit vectors, and 2.2 Hermite Polynomials and Gaussian Noise Lemma 2.2. R f, g ∈ L (Hk). Then, operator. Let G be the probability space over R with 2 Gaussian probability measure. The set of (univariate) ¯ (2.1) E f(Φu)¯g(Φv) = hf, Thu,vigi Hermite polynomials {Hd | d ∈ N} forms an orthonormal Φ basis for L2(G). The degree of Hd ∈ R[x] is equal to d. The first Hermite polynomials are 1, x, x2 − 1, and where Φ is a k × d Gaussian matrix, that is, the entries 3 k of Φ are mutually independent normal variables with x − 3x. An orthogonal basis for L2(G ) is given by the standard deviation √1 . Qk k d set of functions {Hσ(x) := i=1 Hσi (xi) | σ ∈ N0 }. Let Proof. Note that Φu ∼ρ Φv for ρ = hu, vi. Hence, Definition 2.3. (Gaussian Noise Operator) ¯ k the left-hand side of equation (2.1) is equal to hf, Uρg¯i. Uρ denote the linear operator on L2(G ) defined as Since g¯ is multilinear, we have Qdg¯ = Pdg¯. Therefore, ¯ k hf, Uρg¯i = hf, Tρgi, as desired. X d Uρ := ρ Qd , d=0 2.5 Truncation of Low-influence Functions. For f : Rk → R, let trunc f : Rk → [−1, 1] denote the where Qd denotes the orthogonal projector on the sub- k function space of L2(H ) spanned by the set of k-variate degree-d k P  Hermite polynomials Hσ(x) | σ ∈ N0 , σi = d . 1 if f(x) > 1 ,  trunc f(x) := f(x) if −1 < f(x) < 1 , Fact 2.2. For every function f ∈ L (Gk). 2 −1 if f(x) < −1 .

(Uρf)(x) = E f(y) . y∼ρx In our context, the invariance principle [20] roughly says that if f is a bounded function on Hk with no Here, y ∼ρ x means that y can be written as y = influential coordinate, then the multilinear extension of p 2 ρx + 1 − ρ z for a random Gaussian vector z. In f as function on Gk is close to a bounded function (its particular, hx, yi = ρ. Ey∼ρx truncation). Theorem 2.1. (Invariance Principle, [20]) will however assume that for every instance A we can There is a universal constant C such that, for all uniquely associate an optimal SDP solution, e.g., the k ρ = 1 − γ ∈ (0, 1) the following holds: Let f ∈ L2(H ) one computed by a given implementation of the ellipsoid with kfk∞ 6 1 and Infi(Tρf) 6 τ for all i ∈ [k]. Then, method. With this notation, we are ready to define the ¯ ¯ C·γ Tρf − trunc Tρf 6 τ dictatorship test D(A). ¯ k where f ∈ L2(G ) denotes the (unique) multilinear Definition 3.2. For d ∈ N, let us define coefficients k extension of f to R . λd ∈ R, X d λd := aijhui, vji . 3 Proof Overview ij In this section, we will outline the overall structure of Define linear operators D(A), D (A) on L (Hk), the reductions, state the key definitions and lemmas, η 2 and show how they connect with each other. In the k X X subsequent sections, we will present the proofs of the D(A) := λdPd = aijThui,vj i lemmas used. The overall structure of the reduction is d=0 ij along the lines of [22]. We begin by defining dictatorship k X 2d tests in the current context. Dη(A) := TρD(A)Tρ = ρ λdPd , d=0 Definition 3.1. A dictatorship test B is an operator k on L2(H ) of the following form: where ρ = 1 − η.

k By the definition of Completeness(Dη(A)), we have: X B = λdPd (λ1 |λd| for all d) > Lemma 3.1. For all matrices A, d=0 2 where Pd is the projection operator on to the degree-d Completeness(Dη(A)) = λ1ρ > sdp(A)(1 − 2η) . part. We define two parameters of B: Towards bounding Soundnessη,τ (Dη(A)), we define Completeness(B) := infhχi, Bχii = λ1 , a rounding scheme Roundη,f,g for every pair of func- i k tions f, g ∈ L2(H ) and η > 0. The rounding scheme th Roundη,f,g is an efficient randomized procedure that where χi(x) = xi is the i dictator function, and takes as input the optimal SDP solution for A, and outputs a solution x , . . . , x , y , . . . , y ∈ [−1, 1]. The Soundnessη,τ (B) := sup hTρf, BTρgi , 1 m 1 n k f,g∈L2(H ), details of the randomized rounding procedure are de- kfk∞,kgk∞61, scribed in Section4. MaxComInf(Tρf,Tρg)6τ Definition 3.3. Round (A) is the expected value where ρ = 1 − η. η,f,g of the solution returned by the randomized rounding procedure Roundη,f,g on the input A. The following relationship between performance of 3.1 From Integrality Gaps to Dictatorship rounding schemes and soundness of the dictatorship test Tests: In the first step, we describe a reduction from a is proven using Theorem 2.1 (invariance principle [20]). matrix A of arbitrary size, to a dictatorship test D(A) k on L2(H ) for a constant k independent of the size of Theorem 3.1. Let A be a matrix. For functions k A. f, g ∈ L2(H ) satisfying kfk∞, kgk∞ 6 1 and Towards this, let us set up some notation. Let MaxComInf(Tρf, Tρg) 6 τ for ρ = 1 − η, there exists 0 0 k A = (aij) be an m × n matrix with SDP value sdp(A). functions f , g ∈ L2(H ) such that (d) (d) Let u1,..., um ∈ B and v1,..., vn ∈ B be an SDP 10τ Cη/8 √ solution such that hf, Dη(A)gi 6 Roundη,f 0,g0 (A) + / η · sdp(A) . X 0 0 aijhui, vji = sdp(A) . Further, the functions f , g satisfy kfk∞, kgk∞ 6 1. ij By taking the supremum on both sides of the above In general, an optimal SDP solution u1,..., um and inequality over all low influence functions, one obtains v1,..., vn might not be unique. In the following, we the following corollary. Corollary 3.1. For every matrix A and η > 0, Corollary 3.3. For all η > 0, there exists k, τ such k that for any dictatorship test B on L2(H ), Soundnessη,τ (Dη(A)) Cη/8 Soundnessη,τ (B) 1 10τ sdp(A) (3.4) > − η . 6 sup Roundη,f,g(A) + √ , Completeness(B) KG k η f,g∈L2(H ), kfk∞,kgk∞61 From the above corollary, we know that Soundness (B) and Completeness(B) are within As Round is the expected value of a [−1, 1] η,τ η,f,g constant factors of each other. Consequently, we have solution, it is necessarily at most opt(A). Further by Grothendieck’s inequality, sdp(A) and opt(A) are within Corollary 3.4. The equation (3.3) can be replaced by constant factor of each other. Together, these facts immediately imply the following corollary: opt(G(B)) 6 Soundnessη,τ (B) (1 + 5η) .

1 −100 log( /η)/Cη Corollary 3.2. For η > 0, if τ 6 2 , then We present the proof of the Theorems 1.2 to for all matrices A, illustrate how the two conversions outlined in this section come together. The proofs of the remaining theorems Soundnessη,τ (Dη(A)) 6 opt(A)(1 + η) . are deferred to the full version.

3.2 From Dictatorship Tests to Integrality 3.3 Proof of Theorem 1.2 Consider the following Gaps The next key step is the conversion from arbitrary idealized algorithm for the KN,N -Quadratic Pro- dictatorship tests back to integrality gaps. Unlike many gramming problem previous works [22], we obtain a simple direct conversion without using the unique games hardness reduction or ◦ Find the optimal SDP solution ui, vj the Khot–Vishnoi integrality gap instance. In fact, the 3 3 ◦ Fix k = 2200/η and τ = 2−100/η . For every pair of integrality gap instances produced have the following functions f, g ∈ L (Hk) with kfk, kgk 1, run the simple description: 2 6 rounding scheme Roundη,f,g(A) to obtain a [−1, 1] Definition 3.4. Given an dictatorship test B on solution. Output the solution with the largest value. k Pk L2(H ) of the form B = d=0 λdPd , define the corre- k The value of the solution obtained is given by sponding operator Gη(B) on L2(G ) as k supf,g∈L2(H ) Roundη,f,g(A). From Corollary 3.1, we have X 2d Gη(B) = λdQdρ , d sup Roundη,f,g(A) k f,g∈L2(H ),kfk,kgk61 where ρ = 1 − η. > sup Roundη,f,g(A) k We present the proof of the following theorem in f,g∈L2(H ),kfk∞,kgk∞61 Section5. > Soundnessη,τ (Dη(A))(3.5) Cη/8 √ − 10τ / η · sdp(A) . Theorem 3.2. For all η > 0, there exists k, τ such that k following holds: For any dictatorship test B on L2(H ), we have: From Lemma 3.1, we know Completeness(Dη(A)) > sdp(A)(1 − 2η). By the choice of k, τ, we can apply Corollary 3.3 on D (A) to conclude (3.2) sdp(G(B)) > Completeness(B) (1 − 5η) , η opt(G(B)) 6 Soundnessη,τ (B) (1 + η)(3.3) (3.6) + ηCompleteness(B) . 1 Soundnessη,τ (Dη(A)) > Completeness(Dη(A)) ( /KG − η) 1 sdp(A)( /KG − η) (1 − 2η)(3.7) In particular, the choices τ = 2−100/η3 and k = 2200/η3 > suffice. From equations (3.5) and (3.6), we conclude that the value returned by the idealized algorithm is at least By Grothendieck’s theorem, the ratio of sdp(G(B)) 1 10τ Cη/8 √ and opt(G(B)) is at most KG. Hence as a simple sdp(A) ( /KG − η) (1 − η) − / η , corollary, one obtains the following limit to dictatorship 1 testing: which by the choice of τ is at least sdp(A)( /KG − 4η). In order to implement the idealized algorithm, we for large enough choice of k, the operator G(Dη(A)) is an k discretize the unit ball in space L2(H ) using a η-net in operator with sdp(A)/opt(A) > KG − 10η. Since the op- (k) the L2-norm. As k is a fixed constant depending on η, erator G(Dη(A)) belongs to the set Q , this completes there is a finite η-net that would serve the purpose. To the proof of Theorem 1.4. finish the argument, one needs to show that the value of the solution returned is not affected by the discretization. 3.6 Proof of Theorem 1.3 A naive approach to This follows from the next lemma whose proof is deferred compute the Grothendieck constant, is to iterate over to the full version: all matrices A and compute the largest possible value of sdp(A)/opt(A). However, the set of all matrices is 0 0 k Lemma 3.2. For functions f, g, f , g ∈ L2(H ) with an infinite set, and there is no guarantee on when to 0 0 kfk , kgk , kf k , kg k 6 1, terminate. As there is a conversion from integrality gaps to |Roundη,f,g(A) − Roundη,f 0,g0 (A)| dictatorship tests and vice versa, instead of searching 0 0 6 sdp(A)(kf − f k + kg − g k) . for the matrix with the worst integrality gap, we shall find the dictatorship test with the worst possible ratio 3.4 Proof of Theorem 1.1 As a rule of thumb, every between completeness and soundness. Recall that k dictatorship test yields a UG hardness result using by a dictatorship test is an operator on L2(H ) for a now standard techniques [10, 11, 22]. Specifically, we finite k depending only on η the error incurred in the can show the following : reductions. In principle, this already shows that the Grothendieck constant is computable up to an error η Lemma 3.3. Given a dictatorship test A and a unique in time depending only on η. games instance G, it is possible to efficiently construct Define K as follows an operator G ⊗η A that satisfies the following to two conditions: k 1 X 2d 1. if val(G) 1 − , then = inf sup hf, ρ λdQdgi ,, > K λ1=1, k f,g∈L2(H ), d=0 λd∈[−1,1] MaxComInf(Tρf,Tρg)6τ opt(G ⊗η A) > Completeness(A)(1 − o,η→0(1)), kfk∞,kgk∞61

2. if val(G) < , then where ρ = 1 − η. Let P denote the space of all pairs of func- opt(G ⊗η A) < Soundnessη,τ (A)(1 + o,η,τ→0(1)) . k tions f, g ∈ L2(H ) with MaxComInf(Tρf, Tρg) 6 τ and kfk∞, kgk∞ 6 1. Since P is a compact Due to space constraints, we omit the proof of the above set, there exists an η-net of pairs of functions F = lemma here. {(f1, g1),..., (fN , gN )} such that: For every point 0 To finish the proof of Theorem 1.1, let A be a (f, g) ∈ P, there exists fi, gi ∈ F satisfying kf − f k + 0 matrix for which the ratio of sdp(A)/opt(A) > KG − η. kg − g k 6 η. The size of the η-net is a constant depend- Consider the dictatorship test Dη(A) obtained from ing only on k and η (note: k depends only on η). the matrix A. By Corollary 3.1, the completeness The constant K can be expressed up to an error of of Dη(A) is sdp(A)(1 − η). Further by Corollary 3.2, O(η) using the following finite linear program: the soundness is at most opt(A)(1 + η) for sufficiently small choice of τ. Plugging this dictatorship test Dη(A) in to the above lemma, we obtain a UG hardness of (K − η)(1 − η)/(1 + η) K − 5η. Since η can be 1 G > G Minimize = µ made arbitrarily small, the proof is complete. K k k X X 2d 3.5 Proof of Theorem 1.4 Let A be an arbi- Subject to µ > λd · hf, ρ Qdgi trary finite matrix for which sdp(A)/opt(A) > KG − d=0 d=0 η. Consider the dictatorship test/operator Dη(A) on for all functions f, g ∈ F , L (Hk). From Lemma 3.1 and Corollary 3.2, the ra- 2 λi ∈ [−1, 1] for all 0 6 i 6 k , tio of Completeness(A) to Soundness (A) is at least η,τ λ = 1 . sdp(A)/opt(A) − 2η for sufficiently small choice of τ. 1 Further it is easy to see that the operator Dη(A) is trans- lation invariant by construction. Now using Theorem 3.2, 4 From Integrality gaps to Dictatorship Tests On the other hand, we have k 4.1 Rounding Scheme For functions f, g ∈ L2(H ), k define the rounding procedure Round as follows: 0 0 X X 0 d 0 η,f,g hf , Dη(A)g i = hTρf , aijhui, vji Pd(Tρg )i ij d=0 Round X 0 0 η,f,g = aijhTρf ,Thui,vj i(Tρg )i Input: An m × n matrix A = (aij) with SDP solution ij (d) {u1, u2,..., um}, {v1, v2,..., vn} ⊆ B We can assume that all vectors ui and vj have unit norm. ◦ Compute f,¯ g¯ the multilinear extensions of f, g. By Lemma 2.2 , we have

◦ Generate k × d matrix Φ all of whose entries X ¯0 0 E aijTρf (Φui)Tρg¯ (Φvj)(4.8) are mutually independent normal variables of Φ √ ij standard deviation 1/ d. X (4.9) = a hT f 0,T (T g0)i ◦ Output the assignment ij ρ hui,vj i ρ ij ¯ xi = trunc Tρf(Φui) , From the above equations we have yj = trunc Tρg¯(Φvj) . 0 0 X ¯0 0 (4.10) hf , Dη(A)g i = E aijTρf (Φui)Tρg¯ (Φvj) Φ ij The expected value of the solution returned Roundη,f,g(A) is given by: By the invariance principle (Theorem 2.1), we have ¯0 ¯0 Cη/2 P ¯ (4.11) kTρf − trunc Tρf k 6 τ Roundη,f,g(A) = E aijtrunc Tρf(Φui)trunc Tρg¯(Φvj). Φ ij and 4.2 Relaxed Influence Condition The following 0 ¯0 Cη/2 lemma shows that we could replace the condition (4.12) kTρg¯ − trunc Tρg k 6 τ . MaxComInf(Tρf, Tρg) 6 τ in Definition√ 3.1 by the condition MaxInf Tρf, MaxInf Tρg τ with a small Now we shall apply the simple yet powerful bootstrap- 6 k×d loss in the soundness. The proof is omitted here due to ping trick. Let us define new vectors in L2(G ), space constraints. 0 ¯0 0 0 (ui)Φ = Tρf (Φui)(vj)Φ = Tρg¯ (Φvj)

k Lemma 4.1. Let A be a dictatorship test on L2(H ), and k and let f, g be a pair of functions in L2(H ) with (u00) = trunc T f¯0(Φu )(v00) = trunc T g¯0(Φv ) kfk∞, kgk∞ 6 1 and MaxComInf(Tρf, Tρg) 6 τ for i Φ ρ i j Φ ρ j ρ = 1 − η. Then for every τ 0 > 0, there are 0 00 Cη/2 functions f 0, g0 ∈ L (Hk) with kf 0k , kg0k 1 and Equation (4.11) implies that kui − ui k 6 τ and 2 ∞ ∞ 6 0 00 Cη/2 0 0 0 kv − v k τ . Using the bootstrapping argument MaxInf Tρf , MaxInf Tρg 6 τ such that j j 6 (Lemma 2.1), we ﬁnish the proof

0 0 pτ 0 hTρf , ATρg i > hTρf, ATρgi − 2kAk /τ η . X 00 00 (4.13) Roundη,f 0,g0 (A) = aijhui , vj i ij With this background, we now present the soundness X 0 0 X 0 00 0 X 00 0 00 analysis. = aijhui, vji− aijhui−ui , vji− aijhui , vj−vj i ij ij ij (4.11) 4.3 Proof of Theorem 3.1 X 0 0 Cη/2 Cη/2 > aijhui, vji − 2τ opt(A) − 2τ opt(A) ij Proof. By Lemma 4.1, there exists function 1 √ 0 0 k 0 0 hf, D Agi − 4τ Cη/2opt(A) − 4τ /4opt(A)/ η . f , g ∈ L2(H ) with kf k∞, kg k∞ 6 1 and > η 0 0 √ MaxInf Tρf , MaxInf Tρg 6 τ such that 5 From Dictatorship Tests to Integrality Gaps

0 0 1/4 √ In this section, we outline the key ideas in the proof of hf , Dη(A)g i > hf, Dη(A)gi − 2kD(A)k · τ / η Theorem 3.2. Due to space constraints, the details are 1/4 √ > hf, Dη(A)gi − 4opt(A) · τ / η . deferred to the full version. 5.1 sdp(G(B)) > Completeness(B) (1 − 5η). To In the second step, we apply the invariance prin- k prove this claim, we need to construct an SDP solu- ciple to construct functions on L2(H ) with the same tion to sdp(G(B)) that achieves nearly the same value properties as f 0, g0. More precisely, we show as Completeness(B). Formally, we need to construct functions f, g whose domain is Gt and outputs are unit Lemma 5.3. For any η > 0, there exists D, τ > vectors. Since we want to achieve a value close to 0 such that the following holds for every operator PtD tD Completeness(B) = λ , the functions f, g should be lin- B = d=0 λdPd on L2(H ): Given two func- 1 t ear or near-linear. Along the lines of [11, 23], we choose tions f, g ∈ L2(G ) with kfk∞, kgk∞ 6 1 and max Inf (U f), Inf (U g) τ, there exists func- the following function f(x) = g(x) = x/kxk which al- i i ρ j ρ 6 0 0 tD 0 0 ways outputs unit vectors, and very close to the linear tions f , g ∈ L2(H ) satisfying kf k∞, kg k∞ 6 1, √ 0 0 function x 7→ x/ t as t increases. Formally, we show the maxi Infi(Tρf ), maxj Infj(Tρg ) 6 τ, and following lemma: 0 0 hTρf ,BTρg i > hf, G(B)gi − ηkBk . Lemma 5.1. 1 ! In particular, the choices D 2 log η/16 and τ 4 > 1−η 6 log t 2 sdp(G(B)) Completeness(B) ρ4 − 2 O(2−35D log D) suffice. > t The invariance principle of [20] only applies to multilin- 1 5 For t > /η , the value of the SDP solution is at least ear polynomials, while the functions f 0, g0 need not be Completeness(B)(1 − 5η). multilinear. To overcome this hurdle, we treat a multivariate Hermite expansion as a multilinear polynomial 5.2 opt(G(B)) 6 Soundnessη,τ (B) (1 + η) + over the ensemble consisting of Hermite polynomials. ηCompleteness(B). For the sake of contradiction, Unfortunately, this step of the proof is complicated with let us suppose opt(G(B)) > Soundnessη,τ (B)(1 + η) + careful truncation arguments and choice of ensembles to ηCompleteness(B). Let the optimum solution be given t apply invariance principle. by two functions f, g ∈ L2(G ). By assumption, we have In conclusion, by applying Lemma 5.3, we obtain kfk∞, kgk∞ 6 1 and 00 00 t0D functions f and g in L2(H ) that have the following hf, G(B)gi > Soundnessη,τ (B)(1+η)+ηCompleteness(B) . properties: To get a contradiction, we will construct low influence 1. k 00 00 functions in L2(H ) that have a objective value greater kf k∞, kg k∞ 6 1 than Soundnessη,τ (B) on the dictatorship test B. This construction is obtained in two steps: 2. 00 00 In the first step, we obtain functions f 0, g0 over a max Infi(Tρf ), max Infj(Tρg ) τ. i j 6 larger dimensional space with the same objective value but are also guaranteed to have no influential coordinates. Further the functions f 00, g00 satisfy, This is achieved by defining f 0, g0 as follows for large 00 00 0 0 enough R, hTρf ,BTρg i > hf , G(B)g i − ηkBk = hf, G(B)gi − ηkBk R 2R Rt = Soundnessη,τ (B)(1 + η) + ηCompleteness(B) − ηkBk . 0 1 X 1 X 1 X f (x) = f √ xi, √ xi,..., √ xi R R R Recall that kBk = λ = Completeness(B). By the i=1 i=R+1 i=(R−1)t+1 1 0 00 00 t0D R 2R Rt choice of k > t D, the functions f , g ∈ L2(H ) ⊂ k 00 00 0 1 X 1 X 1 X L2(H ). Thus we have two functions f , g with no g (x) = g √ xi, √ xi,..., √ xi . R R R influential variables, but yielding a value higher than the i=1 i=R+1 i=(R−1)t+1 Soundnessη,τ (B). A contradiction. For a sufficiently large choice of R (say R = d1/ητe), the 0 0 t0 functions f , g ∈ L2(G ) have no influential coordinates. Acknowledgments We thank Ryan O’Donnell for Formally, we show the following lemma: bringing [14] to our attention. t Lemma 5.2. Given two functions f, g ∈ L2(G ) 0 0 References with kfk∞, kgk∞ 6 1 , there exists f , g ∈ t·d1/(1−ρ)τe 0 0 L2(G ) with kfk∞, kgk∞ 6 1 and 0 0 maxi Infi(Uρf ), maxj Infj(Uρg ) 6 τ and [1] N. Alon, K. Makarychev, Y. Makarychev, and A. Naor. Quadratic forms on graphs. In STOC, pages 486–493, 0 0 hf , G(B)g i = hf, G(B)gi. 2005. [2] N. Alon and A. Naor. Approximating the cut-norm via hardness threshold of the `p Grothendieck problem. In Grothendieck’s inequality. SIAM J. Comput, 35(4):787– SODA, pages 64–73, 2008. 803, 2006. [14] H. König. Spherical functions and the Grothendieck [3] S. Arora, E. Berger, E. Hazan, G. Kindler, and M. Safra. inequality. Unpublished manuscript, 2005. Available at On non-approximability for quadratic programs. In http://shrimp.bayou.uni-linz.ac.at/Vienna2005/ FOCS, pages 206–215, 2005. pdf/koenig.pdf. [4] A. Coja-Oghlan, C. Cooper, A. Frieze. An efficiently [15] J. L. Krivine. Constantes de Grothendieck et fonctions regular concept for sparse graphs and matrices. To de type positif sur les sphères. Adv. Math., 31:16–30, appear in SODA, 2009. 1977-1978. [5] B.W. Johnson and J. Lindenstrauss. Basic concepts [16] J. Lindenstrauss and A. Pelczyński.Absolutely sum- in the geometry of Banach spaces. Handbook of the ming operators in Lp spaces and their applications. geometry of Banach spaces, North Holland, Amsterdam, Studia Math, 29:275–326, 1968. 1:1–84, 2001. [17] N. Linial and A. Shraibman. Lower bounds in commu- [6] M. Charikar and A. Wirth. Maximizing quadratic nication complexity based on factorization norms. In programs: Extending Grothendieck’s inequality. In STOC, pages 699–708, 2007. FOCS, pages 54–60, 2004. [18] A. Megretski. Relaxation of quadratic programming [7] A. Frieze and R. Kannan. Quick approximation to in operator theory and systems analysis. Systems, matrices and applications. Combinatorica, 19(2):175– approximation, singular integral operators and related 220, 1999. topics, 2001. [8] A. Grothendieck. Résuméde la théoriemétriquedes [19] E. Mossel. Gaussian bounds for noise correlation of produits tensoriels topologiques. Bol. Soc. Mat, 8:1–79, functions. Submitted. Extended abstract appeared in 1953. FOCS, 2008. [9] S. Khot. On the power of unique 2-prover 1-round [20] E. Mossel, R. O’Donnell, and K. Oleszkiewicz. Noise games. In STOC, pages 767–775, 2002. stability of functions with low influences: invariance [10] S. Khot, G. Kindler, E. Mossel, and R. O’Donnell. and optimality. To appear in Ann. Math., 2008. Optimal inapproximability results for MAX-CUT and [21] A. Nemirovski, C. Roos, and T. Terlaky. On maximiza- other 2-variable CSPs? SIAM J. Comput, 37(1):319– tion of quadratic form over intersection of ellipsoids with 357, 2007. common center. Mathematical Programming, 86(3):463– [11] S. Khot and R. O’Donnell. SDP gaps and UGC- 473, 1999. hardness for MaxCutGain. In FOCS, pages 217–226, [22] P. Raghavendra. Optimal algorithms and inapproxima- 2006. bility results for every CSP? In STOC, pages 245–254, [12] S. Khot and N. K. Vishnoi. The unique games conjec- 2008. ture, integrality gap for cut problems and embeddability [23] J. Reeds. A new lower bound on the real Grothendieck of negative type metrics into `1. In FOCS, pages 53–62, constant. Unpublished Manuscript, 1993. Available at 2005. http://www.dtc.umn.edu/˜reedsj. [13] G. Kindler, A. Naor, and G. Schechtman. The UGC