Near Optimal Hardness of Approximating $k$-CSP

TL;DR

This paper establishes near-optimal NP-hardness bounds for approximating $k$-CSP problems with large alphabet sizes, nearly matching trivial algorithms and improving previous hardness results.

Contribution

It extends previous hardness results for $k$-CSP to larger alphabets, introducing a new counting lemma for hyperedges in Grassmann graphs.

Findings

Proves NP-hardness of distinguishing high and low satisfiability in $k$-CSPs with large alphabet.

Improves previous hardness bounds from $O(k/R^{k-2})$ to near $1/R^{k-1}$.

Introduces a new counting lemma for hyperedges in pseudo-random Grassmann graphs.

Abstract

We show that for every $k\in\mathbb{N}$ and $\varepsilon>0$, for large enough alphabet $R$, given a $k$-CSP with alphabet size $R$, it is NP-hard to distinguish between the case that there is an assignment satisfying at least $1-\varepsilon$ fraction of the constraints, and the case no assignment satisfies more than $1/R^{k-1-\varepsilon}$ of the constraints. This result improves upon prior work of [Chan, Journal of the ACM 2016], who showed the same result with weaker soundness of $O(k/R^{k-2})$, and nearly matches the trivial approximation algorithm that finds an assignment satisfying at least $1/R^{k-1}$ fraction of the constraints. Our proof follows the approach of a recent work by the authors, wherein the above result is proved for $k=2$. Our main new ingredient is a counting lemma for hyperedges between pseudo-random sets in the Grassmann graphs, which may be of independent interest.