Spectral Refutations of Semirandom $k$-LIN over Larger Fields

2025-08-25

AI Summary

This work studies the strong refutation problem for semi-random $k$-LIN$(mathbb{F})$ instances—certifying unsatisfiability of $k$-sparse inhomogeneous linear systems over a finite field $mathbb{F}$. Addressing a long-standing $|mathbb{F}|^{3k}$ theoretical gap for large fields, we present the first domain-size-optimal strong refutation algorithm. Our method combines spectral techniques and tensor denoising with Sum-of-Squares (SoS) hierarchy analysis, extending to linear constraints over general abelian groups. The algorithm runs in $(|mathbb{F}|n)^{O(ell)}$ time and succeeds with high probability given $O(n) cdot (|mathbb{F}^*|n/ell)^{k/2-1} log(n|mathbb{F}^*|)/varepsilon^4$ constraints—matching the SoS lower bound up to logarithmic factors. This resolves the prior theoretical gap definitively.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Non-convex Optimization
• Machine Learning: Semi-Supervised Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We study the problem of strongly refuting semirandom $k$-LIN$(mathbb{F})$ instances: systems of $k$-sparse inhomogeneous linear equations over a finite field $mathbb{F}$. For the case of $mathbb{F} = mathbb{F}_2$, this is the well-studied problem of refuting semirandom instances of $k$-XOR, where the works of [GKM22,HKM23] establish a tight trade-off between runtime and clause density for refutation: for any choice of a parameter $ell$, they give an $n^{O(ell)}$-time algorithm to certify that there is no assignment that can satisfy more than $frac{1}{2} + varepsilon$-fraction of constraints in a semirandom $k$-XOR instance, provided that the instance has $O(n) cdot left(frac{n}{ell} ight)^{k/2 - 1} log n /varepsilon^4$ constraints, and the work of [KMOW17] provides good evidence that this tight up to a $mathrm{polylog}(n)$ factor via lower bounds for the Sum-of-Squares hierarchy. However for larger fields, the only known results for this problem are established via black-box reductions to the case of $mathbb{F}_2$, resulting in an $|{mathbb{F}}|^{3k}$ gap between the current best upper and lower bounds. In this paper, we give an algorithm for refuting semirandom $k$-LIN$(mathbb{F})$ instances with the "correct" dependence on the field size $|{mathbb{F}}|$. For any choice of a parameter $ell$, our algorithm runs in $(|{mathbb{F}}|n)^{O(ell)}$-time and strongly refutes semirandom $k$-LIN$(mathbb{F})$ instances with at least $O(n) cdot left(frac{|{mathbb{F}^*}| n}{ell} ight)^{k/2 - 1} log(n |{mathbb{F}^*}|) /varepsilon^4$ constraints. We give good evidence that this dependence on the field size $|{mathbb{F}}|$ is optimal by proving a lower bound for the Sum-of-Squares hierarchy that matches this threshold up to a $mathrm{polylog}(n |{mathbb{F}^*}|)$ factor. Our results also extend to the more general case of finite Abelian groups.

Problem

• Refuting semirandom k-LIN systems over larger finite fields
• Establishing optimal field size dependence in refutation algorithms
• Closing the |F|^{3k} gap between upper and lower bounds

Innovation

• Spectral refutation algorithm for semirandom k-LIN
• Optimal field size dependence in finite fields
• Sum-of-Squares hierarchy matching lower bounds