Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866914547770589184 |
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| author | Gupta, Rishav Lin, Bingkai Zheng, Xin |
| author_facet | Gupta, Rishav Lin, Bingkai Zheng, Xin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_08797 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH Gupta, Rishav Lin, Bingkai Zheng, Xin Computational Complexity 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. |
| title | Tight Lower Bound for Approximating Parametrized Maximum Likelihood Decoding under ETH |
| topic | Computational Complexity |
| url | https://arxiv.org/abs/2605.08797 |