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 $S = \{a_1, \dots, a_m\} \subseteq \R^n$ and a parameter $k$ (possibly depending on $n$). The goal is to find a $k$-dimensional subspace $H$ of $\R^n$ that minimizes the $\ell_p$-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 $p = \infty$, $k = 0$ it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any fixed $p$($2 0$, where $\gamma_p$ is the $p$th moment of a standard Gaussian random variable. This matches the $\gamma_p$ approximation algorithm obtained by Deshpande, Tulsiani and Vishnoi 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. Here, the input is a multilinear quadratic form $\sum_{i,j = 1}^n a_{ij} x_i x_j$ and the goal is to maximize the quadratic form over the $\ell_p$ unit ball, namely all $x$ with $\sum_{i=1}^n |x_i|^p = 1$. The problem is polynomial time solvable for $p=2$. We show that for any constant $p$($2 0$. The same hardness factor was earlier shown under the UGC. We also obtain a $\gamma_p^2$-approximation algorithm for the problem using a convex relaxation of the problem. A $\gamma_p^2$ approximation algorithm has also been independently obtained by Naor and Schechtman.
These are the {\em first} approximation thresholds, proven under ${\rm P} \neq {\rm NP}$, that involve the Gaussian random variable in a fundamental way. Note that the problem statements themselves have no mention of Gaussians.
Added appendix on Smooth Label Cover, updated references and some minor edits.
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 $S = \{a_1, \dots, a_m\} \subseteq \R^n$ 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 $p = \infty, k = 0$ it reduces to the problem of finding a ball of minimum radius enclosing all the points. We show that for any finite $p > 2$ it is NP-hard to approximate this problem to within a factor
of $\gamma_p - \epsilon$ for constant $\epsilon > 0$, where $\gamma_p$ is the $p$'th moment of a standard Gaussian. This matches the $\gamma_p$ 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 $\sum_{i,j = 1}^n a_{ij} x_i x_j$ and the goal is to maximize the quadratic form over the $\ell_p$ unit ball, namely all $x$ with $\sum_{i=1}^n |x_i|^p = 1$. The problem is polytime solvable for $p=2$. We show that for any finite constant $p >2$, it is NP-hard to approximate ${\rm Val}_p(A)$ to within a factor of $\gamma_p^2 - \epsilon$ for any $\epsilon > 0$. The same hardness factor was earlier shown under the UGC. We also obtain a $\gamma_p^2$-approximation algorithm for the problem using a convex relaxation of the problem. A $\gamma_p^2$ approximation algorithm has also been independently obtained by Naor and Schechtman.
These are the {\em first} approximation thresholds, proven under ${\rm P} \neq {\rm NP}$, that involve the Gaussian random variable in a fundamental way. Note that the problem statements themselves have no mention of Gaussians.
