Bypassing UGC from some Optimal Geometric Inapproximability Results

Abstract

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 seems critical in these proofs. In this work we bypass the 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 = {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 sum of the p th powers of the distances to the points. 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 < ∞) it is NP-hard to approximate this problem to within a factor of γ p − for constant > 0, where γ p is the pth moment of a standard Gaussian variable. This matches the factor γ p approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi [12], 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 [29]. Here, the input is a multilinear quadratic form ∑ n i, j=1 a i j 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 polytime solvable for p = 2. We show that for any constant p (2 < p < ∞), it is NP-hard to approximate Val p (A) to within a factor of γ 2 p − for any > 0. The same hardness factor was shown under the UGC in [29]. We also obtain a γ 2 p -approximation algorithm for the problem using the convex relaxation of the problem defined by [29]. A γ 2 p approximation algorithm has also been independently obtained by Naor and Schechtman [33].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.

Approximation resistance from pairwise independent subgroups

“…Previous reductions that bypassed the UG Conjecture for other problems [43,28,25,45] started from Khot's SmoothLabel-Cover [43]. By contrast, our reduction starts from the usual Label-Cover.…”

2 log1-ε n hardness for the closest vector problem with preprocessing

“…An instance of LCPP is called δ-smooth if any two labels i = i of v ∈ V map to different labels of w ∈ W with probability at least 1 − δ over the choice of neighbors w of v. The smoothness property was introduced in [Kho02] and has been used for several hardness of approximation reductions [FGRW09,GRSW10,KS11]. The hardness factor achieved by the the reduction from LCPP to MWSPP is bounded by 1/δ and 1/s where δ is the smoothness parameter and s is the soundness of the LCPP instance.…”

Approximation Algorithms and Semidefinite Programming

“…In 2010, Guruswami et al [GRSW10] obtained tight inapproximability results for two geometric computational problems (which we will not describe here) assuming only P = NP. Previously, the same inapproximability bounds were derived assuming the UGC as well.…”