We present an efficient algorithm to solve semirandom planted instances of any Boolean constraint satisfaction problem (CSP). The semirandom model is a hybrid between worst-case and average-case input models, where the input is generated by (1) choosing an arbitrary planted assignment $x^*$, (2) choosing an arbitrary clause structure, and (3) choosing literal negations for each clause from an arbitrary distribution "shifted by $x^*$" so that $x^*$ satisfies each constraint. For an $n$ variable semirandom planted instance of a $k$-arity CSP, our algorithm runs in polynomial time and outputs an assignment that satisfies all but a $o(1)$-fraction of constraints, provided that the instance has at least $\tilde{O}(n^{k/2})$ constraints. This matches, up to $polylog(n)$ factors, the clause threshold for algorithms that solve fully random planted CSPs [FPV15], as well as algorithms that refute random and semirandom CSPs [AOW15, AGK21]. Our result shows that despite having worst-case clause structure, the randomness in the literal patterns makes semirandom planted CSPs significantly easier than worst-case, where analogous results require $O(n^k)$ constraints [AKK95, FLP16]. Perhaps surprisingly, our algorithm follows a significantly different conceptual framework when compared to the recent resolution of semirandom CSP refutation. This turns out to be inherent and, at a technical level, can be attributed to the need for relative spectral approximation of certain random matrices - reminiscent of the classical spectral sparsification - which ensures that an SDP can certify the uniqueness of the planted assignment. In contrast, in the refutation setting, it suffices to obtain a weaker guarantee of absolute upper bounds on the spectral norm of related matrices.

翻译：我们提出了一种高效算法，用于解决任意布尔约束满足问题（CSP）的半随机植入实例。半随机模型是输入模型中最坏情况与平均情况之间的混合模型，其生成方式为：(1) 选择任意植入赋值 $x^*$，(2) 选择任意子句结构，(3) 对每个子句的文字否定形式，从以 $x^*$ 为偏移的任意分布中选取，使得 $x^*$ 满足每个约束。对于包含 $n$ 个变量的 $k$ 元半随机植入 CSP 实例，若其实例至少包含 $\tilde{O}(n^{k/2})$ 个约束，则我们的算法可在多项式时间内输出一个满足除 $o(1)$ 比例外所有约束的赋值。该结果在 $polylog(n)$ 因子范围内匹配了解决完全随机植入 CSP 的算法 [FPV15] 以及反驳随机与半随机 CSP 的算法 [AOW15, AGK21] 的子句阈值。本研究表明，尽管子句结构可能为最坏情况，文字模式中的随机性仍使半随机植入 CSP 显著易于最坏情况——相比之下，最坏情况需 $O(n^k)$ 个约束才能获得类似结果 [AKK95, FLP16]。令人意外的是，我们的算法在概念框架上与近期对半随机 CSP 反驳问题的解决存在显著差异。这种差异本质上是固有的，技术层面可归因于对特定随机矩阵相对谱逼近的需求——类似于经典谱稀疏化——这确保半定规划（SDP）能够验证植入赋值的唯一性。相比之下，在反驳场景中，仅需获得相关矩阵谱范数的绝对上界这一较弱保证即可。