Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH

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 $γ$-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, $n^{Ω(k)}$ 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 $n^{Ω(k/\operatorname{poly} \log k)}$ 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.

• The paper establishes an n^(Ω(k)) time lower bound for approximating parameterized MLD and NCP under ETH, matching brute-force search bounds.
• It introduces innovative combinatorial gadgets, such as cover and balanced partition families, to control parameter growth in reductions.
• The results delineate the tractability border for FPT approximations in coding theory, discouraging constant-factor approximation algorithms under ETH.

Tight Lower Bound for Approximating Parameterized MLD under ETH

Introduction and Motivations

The paper "Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH" (2605.08797) addresses the fine-grained computational complexity of coding-theoretic problems, specifically the parameterized Maximum Likelihood Decoding (k-MLD) and Nearest Codeword Problem (k-NCP), under the Exponential Time Hypothesis (ETH). While the inherent NP-hardness and inapproximability of these problems are extensively documented, the precise running-time lower bounds for FPT approximation algorithms, particularly under standard ETH (as opposed to the strictly stronger Gap-ETH), remained unresolved.

The work is motivated by several advances that built lower bounds for the parameterized approximations, but often suffered from suboptimal parameter growth in reductions, leading to artificially loose running time lower bounds, or relied on stronger conjectures (e.g., Gap-ETH). This paper closes this gap for the first time, establishing tight lower bounds under ETH for approximating k-MLD and k-NCP to any fixed constant factor.

Problem Setting and Prior Work

Let $\mathbb{F}_q$ be a finite field. Given a matrix $H\in\mathbb{F}_q^{d\times n}$ and a vector $u\in\mathbb{F}_q^d$, the k-MLD problem asks for a codeword $x$ of Hamming weight at most $k$ such that $Hx = u$. The parameterized NCP is the decision version where the goal is to find $x$ so that $\|Ax-t\|_0 \leq k$ for a generator $A$ and target $t$.

Previous inapproximability results for parameterized versions under ETH were hindered by reductions that incurred parameter blowup, yielding lower bounds of the form $H\in\mathbb{F}_q^{d\times n}$0 [BKM25]. The only known tight lower bound $H\in\mathbb{F}_q^{d\times n}$1 arose from the stronger randomized Gap-ETH [Man20]. Thus, whether ETH alone could imply such a lower bound remained open.

Main Results

The central result is the establishment of a tight $H\in\mathbb{F}_q^{d\times n}$2 lower bound for constant-factor approximation of parameterized MLD and NCP under standard ETH. Concretely, unless ETH fails, for some $H\in\mathbb{F}_q^{d\times n}$3 and $H\in\mathbb{F}_q^{d\times n}$4, any algorithm solving $H\in\mathbb{F}_q^{d\times n}$5-Gap $H\in\mathbb{F}_q^{d\times n}$6-MLD or NCP on inputs of length $H\in\mathbb{F}_q^{d\times n}$7 must take $H\in\mathbb{F}_q^{d\times n}$8 time. The lower bound also extends to $H\in\mathbb{F}_q^{d\times n}$9 for deterministic reductions, and up to $u\in\mathbb{F}_q^d$0 for any constant $u\in\mathbb{F}_q^d$1 if randomized ETH is assumed.

This rules out FPT approximation algorithms for these problems under ETH, matching the trivial upper bound for brute-force search. The result thus sharply delineates the border of parameterized tractability for approximation in these fundamental code-theoretic problems.

Technical Contributions

Reduction Framework

The result leverages direct deterministic reductions from ETH-hard variants of Gap MAXLIN, contrasting prior work that utilized long chains via hard 2-CSPs. The main technical tool is the introduction of "cover families":

• Cover Family: A combinatorial gadget, being a collection of small subsets of a universe $u\in\mathbb{F}_q^d$2 so that any small subset can be written as a union of $u\in\mathbb{F}_q^d$3 disjoint members, while no large subset can be covered using $u\in\mathbb{F}_q^d$4 members, creating a required gap for soundness.
• Balanced Partition Family: An intermediate object enabling either randomized (by sampling) or deterministic (via hypercube partition) construction of cover families with controlled size, ensuring the tight translation of parameters during the reduction.

These gadgets allow the conversion of a non-parameterized $u\in\mathbb{F}_q^d$5-Gap MLD instance to a parameterized instance without exponential blow-up of $u\in\mathbb{F}_q^d$6 in $u\in\mathbb{F}_q^d$7, key to achieving the desired lower bound.

Derandomization and Size Optimization

A technically nuanced part is the use of hypercube set systems to deterministically construct cover families for most relevant parameter regimes, yielding $u\in\mathbb{F}_q^d$8-sized instances that refrain from logarithmic losses in the exponent.

Hypotheses and Tightness

The results are firmly based on ETH (and randomized ETH for a larger $u\in\mathbb{F}_q^d$9), making no recourse to conjectures of gap hardness with randomization or instance-dependent hardness. The reduction matches upper bounds, thus yielding essentially best possible tightness for the complexity exponent.

Numerical and Structural Claims

• Optimal Lower Bound: For some fixed constant $x$0, any algorithm distinguishing YES/NO cases in $x$1-Gap $x$2-MLD or NCP must run in $x$3 time, with the reduction holding for $x$4. This matches brute-force enumeration, separating the parameterized (FPT) regime from the infeasible one.
• Stronger Hardness Transfer: Extension to parameterized Closest Vector Problem (CVP) for all $x$5 norms, as well as classical coding-theoretic problems (via folklore reductions).

Methodological Advances and Implications

The core methodological insight is the cover family technique for preserving optimal parameter growth. This approach has several implications:

• It obviates reliance on chain reductions from 2-CSP, thus simplifying future complexity-theoretic reductions for gap problems.
• The combinatorial tools — cover and balanced partition families — can be incorporated for lower bound proofs in other parameterized approximation settings where a union-size gap is advantageous.
• The matching lower and upper bounds rigorously distinguish the regime of intractability for approximate FPT algorithms in coding theory.

Open Problems and Future Research

Key open directions articulated include:

• Extension of the tight ETH-based bound to arbitrary constant factors $x$6, rather than existence of some $x$7 (analogous to what is known under Gap-ETH).
• Application of cover families for optimal hardness results for parameterized Minimum Distance Problem or related lattice problems (as recently achieved for SVP under ETH [AGMZ26]).
• Investigation into further algorithmic and complexity consequences of ETH-based Gap MAXLIN hardness — potentially impacting other pseudorandomness and coding-theoretic domains.
• Exploration of the general utility of the new combinatorial objects (cover families/balanced partition families) introduced.

Conclusion

This work resolves the precise parameterized ETH-hardness of approximating fundamental code-theoretic problems by demonstrating that, under ETH, no algorithm can approximate k-MLD or k-NCP within any fixed constant factor in time $x$8. The approach via novel combinatorial reductions, deterministic and randomized constructions, and direct translation from Gap MAXLIN sets a new standard for the fine-grained analysis of parameterized intractability. The techniques introduced have potential broader applicability for both theoretical complexity and concrete algorithm design barriers in error-correcting codes and related domains (2605.08797).