Random $K$-satisfiability ($K$-SAT) is a paradigmatic model system for studying phase transitions in constraint satisfaction problems and for developing empirical algorithms. The statistical properties of the random $K$-SAT solution space have been extensively investigated, but most earlier efforts focused on solutions that are typical. Here we consider maximally flexible solutions which satisfy all the constraints only using the minimum number of variables. Such atypical solutions have high internal entropy because they contain a maximum number of null variables which are completely free to choose their states. Each maximally flexible solution indicates a dense region of the solution space. We estimate the maximum fraction of null variables by the replica-symmetric cavity method, and implement message-passing algorithms to construct maximally flexible solutions for single $K$-SAT instances.

中文翻译：

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随机K可满足性公式的最大灵活解。

随机 $ķ$-可满足性（$ķ$-SAT）是一个范式模型系统，用于研究约束满足问题中的相变并开发经验算法。随机的统计特性$ķ$-SAT解决方案空间已被广泛研究，但最早期的工作集中在典型的解决方案上。在这里，我们考虑最大灵活的解决方案，该解决方案仅使用最少数量的变量即可满足所有约束。这样的非典型解具有很高的内部熵，因为它们包含最大数量的完全可以自由选择其状态的空变量。每个最大灵活的解都表示解空间的密集区域。我们通过复制对称腔方法估计空变量的最大分数，并实现消息传递算法来构造单个变量的最大灵活解$ķ$-SAT实例。