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Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding Under ETH

Authors Rishav Gupta , Bingkai Lin , Xin Zheng

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Maximum Likelihood Decoding
  • Parameterized Complexity
  • Hardness of Approximation
  • Exponential Time Hypothesis

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Abstract

We present a simple deterministic reduction which, assuming the Exponential Time Hypothesis (ETH), yields tight lower bounds for approximating the parameterized Maximum Likelihood Decoding problem (MLD) and the parameterized Nearest Codeword Problem (NCP) within some fixed constant factor. Our starting point is the ETH-based exponential-time hardness of (c, s)-Gap MAXLIN established in [Nir Bitansky et al., 2024]. We transform a (c, s)-Gap MAXLIN instance into an instance of γ-Gap k-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-ETH [Pasin Manurangsi, 2020], which is a much stronger assumption than ETH. Under ETH, the strongest known lower bound was n^{Ω(k/poly log k)} due to [Mitali Bafna et al., 2025]. Unlike previous approaches that rely on reductions from the hardness of approximating  2-CSP, our reduction provides a more direct and conceptually simpler route to achieving the optimal lower bounds.

Cite As Get BibTex

Rishav Gupta, Bingkai Lin, and Xin Zheng. Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding Under ETH. In 41st Computational Complexity Conference (CCC 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 383, pp. 1:1-1:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CCC.2026.1

Author Details

Rishav Gupta
  • National University of Singapore, Singapore
Bingkai Lin
  • State Key Laboratory for Novel Software Technology, Nanjing University, China
Xin Zheng
  • State Key Laboratory for Novel Software Technology, Nanjing University, China

Funding

Supported by the "111 Cente" (No. B26023) and NRF grant NRF-NRFI09-0005.

Acknowledgements

The authors would like to thank the anonymous reviewers for their comments and suggestions which help to improve the paper significantly. Special thanks to Reviewer B for simplifying the proof for Lemma 29.

References

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