Proceedings of the ninth annual ACM-SIAM symposium on Discrete algorithms
On the power of unique 2-prover 1-round games
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Projective clustering in high dimensions using core-sets
Proceedings of the eighteenth annual symposium on Computational geometry
Hardness Results for Coloring 3 -Colorable 3 -Uniform Hypergraphs
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
Maximizing Quadratic Programs: Extending Grothendieck's Inequality
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Proceedings of the thirty-seventh annual ACM symposium on Theory of computing
On Non-Approximability for Quadratic Programs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
Approximating the Cut-Norm via Grothendieck's Inequality
SIAM Journal on Computing
A 3-Query Non-Adaptive PCP with Perfect Completeness
CCC '06 Proceedings of the 21st Annual IEEE Conference on Computational Complexity
Sampling-based dimension reduction for subspace approximation
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
Efficient subspace approximation algorithms
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Approximating the Radii of Point Sets
SIAM Journal on Computing
Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?
SIAM Journal on Computing
Vertex cover might be hard to approximate to within 2-ε
Journal of Computer and System Sciences
Hardness of Reconstructing Multivariate Polynomials over Finite Fields
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Unique games on expanding constraint graphs are easy: extended abstract
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Optimal algorithms and inapproximability results for every CSP?
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
On hardness of learning intersection of two halfspaces
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Gaussian Bounds for Noise Correlation of Functions and Tight Analysis of Long Codes
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Agnostic Learning of Monomials by Halfspaces Is Hard
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
The UGC Hardness Threshold of the Lp Grothendieck Problem
Mathematics of Operations Research
Coresets and sketches for high dimensional subspace approximation problems
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Subexponential Algorithms for Unique Games and Related Problems
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
Beating the Random Ordering Is Hard: Every Ordering CSP Is Approximation Resistant
SIAM Journal on Computing
Rounding Semidefinite Programming Hierarchies via Global Correlation
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Algorithms and hardness for subspace approximation
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Proceedings of the 4th conference on Innovations in Theoretical Computer Science
Approximation resistance from pairwise independent subgroups
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Approximation resistance on satisfiable instances for predicates with few accepting inputs
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
Column Subset Selection Problem is UG-hard
Journal of Computer and System Sciences
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The Unique Games conjecture (UGC) has emerged in recent years as the starting point for several optimal inapproximability results. While for none of these results a reverse reduction to Unique Games is known, the assumption of bijective projections in the Label Cover instance nevertheless seems critical in these proofs. In this work we bypass the need for UGC assumption in inapproximability results for two geometric problems, obtaining a tight NP-hardness result in each case. The first problem, known as the Lp Subspace Approximation, is a generalization of the classic least squares regression problem. Here, the input consists of a set of points S = {a1,..., am} ⊆ Rn and a parameter k (possibly depending on n). The goal is to find a subspace H of Rn of dimension k that minimizes the lp norm of the Euclidean distances to the points in S. For p = 2, k = n − 1, this reduces to the least squares regression problem, while for p = ∞, k = 0 it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed p (2 p k = n − 1, it is NP-hard to approximate this problem to within a factor of γp − ε for constant ε 0, where γp is the pth norm of a standard Gaussian random variable. This matches the γp approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi [9] who also showed the same hardness result under the Unique Games Conjecture. The second problem we study is the related Lp Quadratic Grothendieck Maximization Problem, considered by Kindler, Naor and Schechtman [24]. Here, the input is a multilinear quadratic form [EQUATION] and the goal is to maximize the quadratic form over the lp unit ball, namely all x with Σni=1 |xi|p = 1. The problem is polynomial time solvable for p = 2. We show that for any constant p (2 p 2p − ε for any ε 0. The same hardness factor was shown under the UGC in [24]. We also obtain a γ2p-approximation algorithm for the problem using the convex relaxation of the problem defined by [24]. A γ2p approximation algorithm has also been independently obtained by Naor and Schechtman [27]. These are the first approximation thresholds, proven under P ≠ NP, that involve the Gaussian random variable in a fundamental way. Note that the problem statements themselves have no mention of Gaussians.