Arora, Rao and Vazirani [2] showed that the standard semidefinite programming (SDP) relaxation of the Sparsest Cut problem with the triangle inequality constraints has an integrality gap of O( √ log n). They conjectured that the gap is bounded from above by a constant. In this paper, we disprove this conjecture (referred to as the ARV-Conjecture) by constructing an Ω(log log n) integrality gap instance. Khot and Vishnoi [16] had earlier disproved the non-uniform version of the ARV-Conjecture.A simple "stretching" of the integrality gap instance for the Sparsest Cut problem serves as an Ω(log log n) integrality gap instance for the SDP relaxation of the Minimum Linear Arrangement problem. This SDP relaxation was considered in [6,11], where it was shown that its integrality gap is bounded from above by O( √ log n log log n).
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 L p Subspace Approximation, is a generalization of the classic least squares regression problem. Here, the input consists of a set of points X = {a 1 ,. .. , a m } ⊆ R n and a parameter k (possibly depending on n). The goal is to find a subspace H of R n of dimension k that minimizes the p norm of the Euclidean distances to the points in X. 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, ∞), and for k = n− 1, it is NP-hard to approximate this problem to within a factor of γ p − for constant > 0, where γ p is the pth norm of a standard Gaussian random variable. This matches the γ p approximation algorithm obtained by Deshpande, Tulsiani, and Vishnoi who also showed the same hardness result under the UGC. The second problem we study is the related L p Quadratic Grothendieck Maximization Problem, considered by Kindler, Naor, and Schechtman. Here, the input is a multilinear quadratic form n i, j=1 a ij x i x j and the goal is to maximize the quadratic form over the p unit ball, namely, all x with n i=1 |x i | p 1. The problem is polynomial time solvable for p = 2. We show that for any constant p ∈ (2, ∞), it is NP-hard to approximate the quadratic form to within a factor of γ 2 p − for any > 0. The same hardness factor was shown under the UGC by Kindler et al. We also obtain a γ 2 p-approximation algorithm for the problem using the convex relaxation of the problem defined by Kindler et al. A γ 2 p approximation algorithm has also been independently obtained by Naor and Schechtman. 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 do not explicitly involve the Gaussian distribution.
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