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The Unique Games Conjecture, integrality gap for cut problems and embeddability of negative type metrics into `1
- In Proc. 46th IEEE Symp. on Foundations of Comp. Sci
, 2005
"... In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an n-point negative type metric which requires distortion ..."
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Cited by 180 (13 self)
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In this paper we disprove the following conjecture due to Goemans [17] and Linial [25] (also see [5, 27]): “Every negative type metric embeds into `1 with constant distortion. ” We show that for every δ> 0, and for large enough n, there is an n-point negative type metric which requires distortion at-least (log log n)1/6−δ to embed into `1. Surprisingly, our construction is inspired by the Unique Games Conjecture (UGC) of Khot [20], establishing a previously unsuspected connection between PCPs and the theory of metric embeddings. We first prove that the UGC implies super-constant hardness results for (non-uniform) Sparsest Cut and Minimum Uncut problems. It is already known that the UGC also implies an optimal hardness result for Maximum Cut [21]. Though these hardness results rely on the UGC, we demonstrate, nevertheless, that the corresponding PCP reductions can be used to construct “integrality gap instances ” for the respective problems. Towards this, we first construct an integrality gap instance for a natural SDP relaxation of Unique Games. Then, we “simulate ” the PCP reduction, and “translate ” the integrality gap instance of Unique Games to integrality gap instances for the respective cut problems! This enables us to prove
Correlation clustering with a fixed number of clusters
- Theory of Computing
"... Abstract: We continue the investigation of problems concerning correlation clustering or clustering with qualitative information, which is a clustering formulation that has been studied recently (Bansal, Blum, Chawla (2004), Charikar, Guruswami, Wirth (FOCS’03), Charikar, Wirth (FOCS’04), Alon et al ..."
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Cited by 37 (0 self)
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Abstract: We continue the investigation of problems concerning correlation clustering or clustering with qualitative information, which is a clustering formulation that has been studied recently (Bansal, Blum, Chawla (2004), Charikar, Guruswami, Wirth (FOCS’03), Charikar, Wirth (FOCS’04), Alon et al. (STOC’05)). In this problem, we are given a complete graph on n nodes (which correspond to nodes to be clustered) whose edges are labeled + (for similar pairs of items) and − (for dissimilar pairs of items). Thus our input consists of only qualitative information on similarity and no quantitative distance measure between items. The quality of a clustering is measured in terms of its number of agreements, which is simply the number of edges it correctly classifies, that is the sum of number of − edges whose endpoints it places in different clusters plus the number of + edges both of whose endpoints it places within the same cluster. In this paper, we study the problem of finding clusterings that maximize the number of agreements, and the complementary minimization version where we seek clusterings that minimize the number of disagreements. We focus on the situation when the number of clusters is stipulated to be a small constant k. Our main result is that for every k, there is a polynomial time approximation scheme for both maximizing agreements and minimizing disagreements.
Semidefinite programming heuristics for surface reconstruction ambiguities
- In ECCV
, 2008
"... Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based ..."
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Cited by 18 (1 self)
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Abstract. We consider the problem of reconstructing a smooth surface under constraints that have discrete ambiguities. These problems arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface. 1
Towards computing the grothendieck constant
- In SODA ’09: Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms
, 2009
"... The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unkno ..."
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Cited by 16 (2 self)
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The Grothendieck constant KG is the smallest constant such that for every d ∈ N and every matrix A = (aij), sup u i,v j ∈B (d) X aij〈ui, vj 〉 � KG · ij sup x i,y j ∈[−1,1] X ij aijxiyj, where B (d) is the unit ball in R d. Despite several efforts [15, 23], the value of the constant KG remains unknown. The Grothendieck constant KG is precisely the integrality gap of a natural SDP relaxation for the KM,N-Quadratic Programming problem. The input to this problem is a matrix A = (aij) and the objective is to maximize the quadratic form P ij aijxiyj over xi, yj ∈ [−1, 1]. In this work, we apply techniques from [22] to the KM,N-Quadratic Programming problem. Using some standard but non-trivial modifications, the reduction in [22] yields the following hardness result: Assuming the Unique Games Conjecture [9], it is NP-hard to approximate the KM,N-Quadratic Programming problem to any factor better than the Grothendieck constant KG. By adapting a “bootstrapping ” argument used in a proof of Grothendieck inequality [5], we are able to perform a tighter analysis than [22]. Through this careful analysis, we obtain the following new results: ◦ An approximation algorithm for KM,N-Quadratic Programming that is guaranteed to achieve an approximation ratio arbitrarily close to the Grothendieck constant KG (optimal approximation ratio assuming the Unique Games Conjecture). ◦ We show that the Grothendieck constant KG can be computed within an error η, in time depending only on η. Specifically, for each η, we formulate an explicit finite linear program, whose optimum is η-close to the Grothendieck constant. We also exhibit a simple family of operators on the Gaussian Hilbert space that is guaranteed to contain tight examples for the Grothendieck inequality.
Simulating quantum correlations with finite communication
- In Proceedings of 48th Annual IEEE Symposium on Foundations of Computer Science
, 2007
"... Assume Alice and Bob share some bipartite d-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two ..."
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Cited by 13 (1 self)
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Assume Alice and Bob share some bipartite d-dimensional quantum state. A well-known result in quantum mechanics says that by performing two-outcome measurements, Alice and Bob can produce correlations that cannot be obtained locally, i.e., with shared randomness alone. We show that by using only two bits of communication, Alice and Bob can classically simulate any such correlations. All previous protocols for exact simulation required the communication to grow to infinity with the dimension d. Our protocol and analysis are based on a power series method, resembling Krivine’s bound on Grothendieck’s constant, and on the computation of volumes of spherical tetrahedra. 1
Grothendieck Inequalities for Semidefinite Programs with Rank Constraint
, 2010
"... Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank-1 constraint and its easy ..."
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Cited by 12 (3 self)
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Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: A difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give one application in statistical mechanics: Approximating ground states in the n-vector model.
Approximate Kernel Clustering
"... In the kernel clustering problem we are given a large n * n positive semi-definite matrix A = (ai j)with Pn i, j=1 aij = 0 and a small k * k positive semi-definite matrix B = (bi j). The goal is to find a partitionS 1,..., S k of {1,... n} which maximizes the quantity kX i, j=1 0BBBBBBB @ X(p,q)2Si* ..."
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Cited by 11 (5 self)
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In the kernel clustering problem we are given a large n * n positive semi-definite matrix A = (ai j)with Pn i, j=1 aij = 0 and a small k * k positive semi-definite matrix B = (bi j). The goal is to find a partitionS 1,..., S k of {1,... n} which maximizes the quantity kX i, j=1 0BBBBBBB @ X(p,q)2Si*S j a pq1CCCCCCCA bi j. We study the computational complexity of this generic clustering problem which originates in the theoryof machine learning. We design a constant factor polynomial time approximation algorithm for this problem, answering a question posed by Song, Smola, Gretton and Borgwardt. In some cases we manageto compute the sharp approximation threshold for this problem assuming the Unique Games Conjecture (UGC). In particular, when B is the 3 * 3 identity matrix the UGC hardness threshold of this problem isexactly 16 ss27. We present and study a geometric conjecture of independent interest which we show would imply that the UGC threshold when B is the k * k identity matrix is 8ss9 i1- 1k j for every k> = 3.
Linear equations modulo 2 and the L1 diameter of convex bodies
"... We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {ai jk} n ..."
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Cited by 10 (1 self)
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We design a randomized polynomial time algorithm which, given a 3-tensor of real numbers A = {ai jk} n
Algorithms and Hardness for Subspace Approximation
"... The subspace approximation problem Subspace(k, p) asks for a k dimensional linear subspace that fits a given set of m points in Rn optimally. The error for fitting is a generalization of the least squares fit and uses the ℓp norm of the distances (ℓ2 distances) of the points from the subspace, e.g., ..."
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Cited by 9 (0 self)
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The subspace approximation problem Subspace(k, p) asks for a k dimensional linear subspace that fits a given set of m points in Rn optimally. The error for fitting is a generalization of the least squares fit and uses the ℓp norm of the distances (ℓ2 distances) of the points from the subspace, e.g., p = ∞ means minimizing the ℓ2 distance of the farthest point from the subspace. Previous work on subspace approximation considers either the case of small or constant k and p [27, 11, 14] or the case of p = ∞
