最小距离问题和最短向量问题的单侧错误参数化归约
One-Sided-Error Parameterized Reductions for the Minimum Distance and Shortest Vector Problems
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中文总结 AI 辅助
该研究对MDP和SVP的双侧错误归约部分去随机化，提出单侧错误归约，结合OR函数构造与电路下界假设，得到MDP和SVP近似的确定性W[1]-困难性及欧几里得SVP近似的确定性NP-困难性。
中文摘要 AI 辅助
为最小距离问题（MDP）和最短向量问题（SVP）获取确定性归约是出了名的困难。在双侧错误随机归约下，Bennett、Cheraghchi、Guruswami和Ribeiro（STOC 2023）证明了这些问题的近似参数化困难性。我们对他们的归约进行部分去随机化，提出单侧错误随机归约：在FPT多对一单侧错误随机归约下，MDP难以在任意常数因子内近似，属于W[1]-困难问题；对于每一个p≥1，ℓ_p范数下的SVP难以在任意低于2^(1/p)的常数因子内近似，属于W[1]-困难问题。我们通过证明当目标问题具有OR函数时，单侧错误随机归约可以被条件去随机化，展示了其有用性。在标准的困难性与随机性假设（即针对非确定性电路的合理下界假设）下，我们证明了形式化该去随机化的一般性定理，其中OR函数将多个实例合并为一个保留其析取关系的实例。我们为相关的MDP和SVP间隙问题构建了此类OR函数，从而获得确定性W[1]-困难性：对于每个固定有限域上的MDP，其在任意常数因子内的近似属于W[1]-困难问题；对于每个固定整数p，ℓ_p范数下的SVP在任意低于2^(1/p)的因子内的近似属于W[1]-困难问题。将相同框架应用于Micciancio的单侧错误随机归约（ToC 2012），在相同的电路下界假设下，可获得欧几里得SVP在任意常数因子内的近似的确定性多项式时间NP-困难性。
英文摘要
It is notoriously difficult to obtain deterministic reductions for the Minimum Distance Problem (MDP) and the Shortest Vector Problem (SVP). Under two-sided-error randomized reductions, Bennett, Cheraghchi, Guruswami, and Ribeiro (STOC 2023) proved parameterized hardness of approximation for these problems. We partially derandomize their reductions and present one-sided-error randomized reductions: MDP is W[1]-hard to approximate within an arbitrary constant factor under FPT many-one one-sided-error randomized reductions; For every $p \ge 1$, SVP in the $\ell_p$ norm is W[1]-hard to approximate within an arbitrary constant factor below $2^{1/p}$. We demonstrate the usefulness of one-sided-error randomized reductions by showing that they can be conditionally derandomized when the target problem has an OR function. Under a standard hardness-vs-randomness assumption, namely a plausible lower-bound assumption against nondeterministic circuits, we prove a general theorem formalizing this derandomization. Here, an OR function combines several instances into one instance that preserves their disjunction. We construct such OR functions for the relevant MDP and SVP gap problems, and thereby obtain deterministic W[1]-hardness for approximating MDP over every fixed finite field within every constant factor, and for approximating SVP in $\ell_p$ norms for every fixed integer $p$ within every factor below $2^{1/p}$. Applying the same framework to Micciancio's one-sided-error randomized reduction (ToC 2012) yields, under the same circuit lower-bound assumption, deterministic polynomial-time NP-hardness of approximating Euclidean SVP within every constant factor.
