Bypassing UGC from some optimal geometric inapproximability results

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 L_p Subspace Approximation, is a generalization of the classic least squares regression problem. Here, the input consists of a set of points S = {a₁,..., a_m} ⊆ 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 l_p 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 < ∞), and for k = n − 1, it is NP-hard to approximate this problem to within a factor of γ_p − ε for constant ε > 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 [9] who also showed the same hardness result under the Unique Games Conjecture.

The second problem we study is the related L_p 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 l_p unit ball, namely all x with Σⁿ_i=1 |x_i|^p = 1. The problem is polynomial time solvable for p = 2. We show that for any constant p (2 < p < ∞), it is NP-hard to approximate the quadratic form to within a factor of γ²_p − ε for any ε > 0. The same hardness factor was shown under the UGC in [24]. We also obtain a γ²_p-approximation algorithm for the problem using the convex relaxation of the problem defined by [24]. A γ²_p 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.