Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH

TL;DR

This paper establishes tight lower bounds for approximating parameterized Maximum Likelihood Decoding and Nearest Codeword problems under ETH, using a novel reduction and combinatorial cover families.

Contribution

It introduces a simple deterministic reduction from Gap-MAXLIN to parameterized decoding problems, achieving optimal lower bounds under ETH with a new combinatorial approach.

Findings

Achieves tight lower bounds for approximation under ETH.

Introduces a novel combinatorial object called a cover family.

Provides both randomized and derandomized constructions of cover families.

Abstract

We present a simple deterministic reduction which, assuming the Exponential Time Hypothesis ($\mathsf{ETH}$), yields tight lower bounds for approximating the parameterized Maximum Likelihood Decoding problem ($\mathsf{MLD}$) and the parameterized Nearest Codeword Problem ($\mathsf{NCP}$) within some fixed constant factor. Our starting point is the ETH-based exponential-time hardness of $(c,s)$-Gap-$\mathsf{MAXLIN}$ established in [BHI+24]. We transform a $(c,s)$-Gap-$\mathsf{MAXLIN}$ instance into an instance of $\gamma$-Gap $k$-$\mathsf{MLD}$ via a novel combinatorial object that we call a cover family. We provide both a randomized construction of the required cover families and a subsequent derandomization. Prior to our work, $n^{\Omega(k)}$ hardness for constant-factor approximation was only shown under the randomized Gap Exponential Time Hypothesis Gap-$\mathsf{ETH}$ [Man20], which is a much stronger assumption than $\mathsf{ETH}$. Under $\mathsf{ETH}$, the strongest known lower bound was $n^{\Omega(k/\operatorname{poly} \log k)}$ due to [BKM25]. Unlike previous approaches that rely on reductions from the hardness of approximating $2$-$\mathsf{CSP}$, our reduction provides a more direct and conceptually simpler route to achieving the optimal lower bounds.