We introduce and initiate the study of a new model of reductions called the random noise model. In this model, the truth table $T_f$ of the function $f$ is corrupted on a randomly chosen $\delta$-fraction of instances. A randomized algorithm $A$ is a $\left(t, \delta, 1-\varepsilon\right)$-recovery reduction for $f$ if: 1. With probability $1-\varepsilon$ over the choice of $\delta$-fraction corruptions, given access to the corrupted truth table, the algorithm $A$ computes $f(\phi)$ correctly with probability at least $2/3$ on every input $\phi$. 2. The algorithm $A$ runs in time $O(t)$. We believe this model, which is a natural relaxation of average-case complexity, both has practical motivations and is mathematically interesting. Pointing towards this, we show the existence of robust deterministic polynomial-time recovery reductions with the highest tolerable noise level for many of the canonical NP-complete problems - SAT, kSAT, kCSP, CLIQUE and more. Our recovery reductions are optimal for non-adaptive algorithms under complexity-theoretic assumptions. Notably, all our recovery reductions follow as corollaries of one black box algorithm based on group theory and permutation group algorithms. This suggests that recovery reductions in the random noise model are important to the study of the structure of NP-completeness. Furthermore, we establish recovery reductions with optimal parameters for Orthogonal Vectors and Parity $k$-Clique problems. These problems exhibit structural similarities to NP-complete problems, with Orthogonal Vectors admitting a $2^{0.5n}$-time reduction from kSAT on $n$ variables and Parity $k$-Clique, a subexponential-time reduction from 3SAT. This further highlights the relevance of our model to the study of NP-completeness.

翻译：我们引入并初步研究了一种称为随机噪声模型的新归约模型。在该模型中，函数$f$的真值表$T_f$在随机选择的$\delta$比例实例上发生损坏。随机算法$A$是$f$的$\left(t, \delta, 1-\varepsilon\right)$-恢复归约，若满足：1. 在$\delta$比例损坏实例的选择上，以至少$1-\varepsilon$的概率，给定对损坏真值表的访问权限，算法$A$能在每个输入$\phi$上以至少$2/3$的概率正确计算$f(\phi)$；2. 算法$A$的运行时间为$O(t)$。我们认为这一模型作为平均情况复杂度的自然松弛，既具有实际动机，又在数学上富有意义。为此，我们证明了对于许多经典NP完全问题——SAT、kSAT、kCSP、CLIQUE等——存在具有最高可容忍噪声水平的鲁棒确定性多项式时间恢复归约。在复杂性理论假设下，我们的恢复归约对于非自适应算法是最优的。值得注意的是，所有恢复归约均可作为基于群论和置换群算法的单一黑盒算法的推论得出。这表明随机噪声模型中的恢复归约对于研究NP完全性的结构具有重要意义。此外，我们为正交向量问题和奇偶$k$-团问题建立了具有最优参数的恢复归约。这些问题与NP完全问题具有结构相似性：正交向量问题允许从$n$变量kSAT进行$2^{0.5n}$时间归约，而奇偶$k$-团问题允许从3SAT进行亚指数时间归约。这进一步凸显了我们的模型与NP完全性研究的相关性。