We give an efficient algorithm to strongly refute \emph{semi-random} instances of all Boolean constraint satisfaction problems. The number of constraints required by our algorithm matches (up to polylogarithmic factors) the best-known bounds for efficient refutation of fully random instances. Our main technical contribution is an algorithm to strongly refute semi-random instances of the Boolean $k$-XOR problem on $n$ variables that have $\widetilde{O}(n^{k/2})$ constraints. (In a semi-random $k$-XOR instance, the equations can be arbitrary and only the right-hand sides are random.) One of our key insights is to identify a simple combinatorial property of random XOR instances that makes spectral refutation work. Our approach involves taking an instance that does not satisfy this property (i.e., is \emph{not} pseudorandom) and reducing it to a partitioned collection of $2$-XOR instances. We analyze these subinstances using a carefully chosen quadratic form as a proxy, which in turn is bounded via a combination of spectral methods and semidefinite programming. The analysis of our spectral bounds relies only on an off-the-shelf matrix Bernstein inequality. Even for the purely random case, this leads to a shorter proof compared to the ones in the literature that rely on problem-specific trace-moment computations.

中文翻译：

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强烈反驳所有半随机布尔 CSP

我们给出了一个有效的算法来强烈反驳所有布尔约束满足问题的 \emph{semi-random} 实例。我们的算法所需的约束数量匹配（最多为多对数因子）有效驳斥完全随机实例的最著名界限。我们的主要技术贡献是一种算法，可以在具有 $\widetilde{O}(n^{k/2})$ 约束的 $n$ 变量上强烈反驳布尔 $k$-XOR 问题的半随机实例。（在半随机 $k$-XOR 实例中，方程可以是任意的，只有右侧是随机的。）我们的一个关键见解是确定随机 XOR 实例的一个简单组合属性，它使谱反驳起作用. 我们的方法涉及采用不满足此属性的实例（即，是 \emph{not} 伪随机）并将其减少为 $2$-XOR 实例的分区集合。我们使用精心选择的二次形式作为代理来分析这些子实例，而后者又通过谱方法和半定规划的组合来限定。我们的谱边界分析仅依赖于现成的矩阵 Bernstein 不等式。即使对于纯随机的情况，与依赖于特定问题的跟踪矩计算的文献中的证明相比，这也导致了更短的证明。