Improved Lower Bounds for Approximating Parameterized Nearest Codeword and Related Problems under ETH

TL;DR

This work addresses the fine-grained complexity of approximating parameterized linear-code problems (notably Nearest Codeword and Maximum Likelihood Decoding) under ETH. It introduces a direct gap-creating self-reduction for $k$-MLD$_p$ that yields a Gap-$k'$-MLD$_p$ instance with $k'=O(k^2\log k)$ and a $(3/2-\varepsilon)$-gap, using a threshold-graph/code-based construction and a merging technique to control dimension. Leveraging gap amplification results, the authors derive ETH/randomized ETH lower bounds for a family of problems ($k$-NCP, $k$-MDP, $k$-CVP, $k$-SVP) across finite fields and norms, improving on prior $f(k)n^{Ω(\text{polylog }k)}$ barriers. The results move the landscape closer to Gap-ETH-type hardness for constant-factor approximations, while highlighting remaining gaps and suggesting avenues to close the $n^{o(k)}$-time gap under weaker assumptions than Gap-ETH.

Abstract

In this paper we present a new gap-creating randomized self-reduction for parameterized Maximum Likelihood Decoding problem over $\mathbb{F}_p$ ($k$-MLD$_p$). The reduction takes a $k$-MLD$_p$ instance with $k\cdot n$ vectors as input, runs in time $f(k)n^{O(1)}$ for some computable function $f$, outputs a $(3/2-\varepsilon)$-Gap-$k'$-MLD$_p$ instance for any $\varepsilon>0$, where $k'=O(k^2\log k)$. Using this reduction, we show that assuming the randomized Exponential Time Hypothesis (ETH), no algorithms can approximate $k$-MLD$_p$ (and therefore its dual problem $k$-NCP$_p$) within factor $(3/2-\varepsilon)$ in $f(k)\cdot n^{o(\sqrt{k/\log k})}$ time for any $\varepsilon>0$. We then use reduction by Bhattacharyya, Ghoshal, Karthik and Manurangsi (ICALP 2018) to amplify the $(3/2-\varepsilon)$-gap to any constant. As a result, we show that assuming ETH, no algorithms can approximate $k$-NCP$_p$ and $k$-MDP$_p$ within $γ$-factor in $f(k)n^{o(k^{\varepsilon_γ})}$ time for some constant $\varepsilon_γ>0$. Combining with the gap-preserving reduction by Bennett, Cheraghchi, Guruswami and Ribeiro (STOC 2023), we also obtain similar lower bounds for $k$-MDP$_p$, $k$-CVP$_p$ and $k$-SVP$_p$. These results improve upon the previous $f(k)n^{Ω(\mathsf{poly} \log k)}$ lower bounds for these problems under ETH using reductions by Bhattacharyya et al. (J.ACM 2021) and Bennett et al. (STOC 2023).

Figures (3)

• Figure 1: A simplified pictorial illustration of our main construction. For detailed illustration see Figure \ref{['figure for gap creating']}.
• Figure 2: Illustration for the vectors of Lemma \ref{['lemma: gap creating (warm up)']} in the completeness setting. We can choose each $\vec{b}_{j,\vec{\sigma}_j}$ as $\vec{\sigma}_j=(C(\vec{v}_1)[j], \cdots, C(\vec{v}_k)[j])$.
• Figure 3: A pictorial illustration for the construction in Theorem \ref{['theorem: gap amplification for mld in BGKM18']}.

Theorems & Definitions (44)

• Theorem 1: informal; See Theorem \ref{['theorem: gap creating (main)']} for a formal statement
• Corollary 2
• Corollary 3
• Corollary 4
• Definition 5: Error-correcting Codes
• Definition 6: Linear Codes
• Definition 7: 3-SAT
• Theorem 10: Chernoff Bound
• Definition 11: $\varepsilon$-Collision Number
• Lemma 12: KN21, See also Theorem 10 in LRSW23