Strong Inapproximability for a Promise Rank Problem

TL;DR

This paper proves the hardness of approximating the rank of matrices within certain bounds over finite fields, impacting complexity theory and inapproximability results.

Contribution

It introduces new hardness results for a promise rank problem using moment matrices and superposition soundness, with novel reduction techniques.

Findings

Proves NP-hardness of approximating matrix rank within specific bounds.

Develops reduction methods with different time complexities for finite fields.

Connects the problem's hardness to PCP-free inapproximability of coding and lattice problems.

Abstract

Given a linear subspace of $n \times n$ matrices over $\mathbb F_{2^r}$ that is promised to contain a matrix of rank $1$, we prove that it is hard to find a matrix of rank $n^{o(1/\log \log n)}$, assuming NP doesn't have sub-exponential algorithms. In addition to being a basic problem, the hardness of this problem, even for the exact version, drove recent PCP-free inapproximability results for minimum distance and shortest vector problems concerning codes and lattices. The proof combines the concept of superposition soundness introduced by Khot and Saket with moment matrices. To produce a rank-gap of $1$ vs. $k$, the reduction runs in time $n^{O(\log k)}$. We also give another moment-matrix-based construction which runs in time $n^{O(k)}$ but works for any finite field $\mathbb F_q$.