Near Optimal Hardness of Approximating $k$-CSP

TL;DR

This work establishes near-optimal hardness of approximating $k$-CSPs by combining an outer GAP3Lin-based PCP with a novel inner Grassmann-consistency PCP. The core technical advance is a counting lemma for hyperedges between pseudo-random Grassmann subsets, enabling a $(k+1)$-query Grassmann test with near-optimal soundness via global hypercontractivity in the bilinear scheme. The composition yields strong inapproximability results that translate into improved hardness for $k$-Dimensional Matching and related problems, matching the trivial $1/R^{k-1}$ baseline up to vanishing factors while keeping completeness close to 1. These results sharpen the alphabet-soundness tradeoff for constant-arity CSPs and contribute new tools in the analysis of subspace-based PCPs and Grassmann-graph testing.

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.

Theorems & Definitions (65)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 1.8
• Lemma 1.9: Informal version of \ref{['lm: pseudorandom edges']}
• Definition 2.1