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Towards Computing the Grothendieck Constant Towards Computing the Grothendieck Constant Prasad Raghavendra∗ David Steurery 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 2 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 2 f1; −1g such that (d) ui;vj 2B ij xi;yj 2[−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 2 [−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 Szemer´edipartitions 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 2 f1; −1g : 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 kAk1!1 of the matrix A, which is defined as examples for the Grothendieck inequality. kAxk1 kAk1!1 := max : n x2R kxk1 ∗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 yPrinceton 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 p 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 P1 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 2 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 jAf(x)j 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.
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