随机$K$ -可满足性公式的最大灵活解
Maximally flexible solutions of a random $K$-satisfiability formula
摘要：
随机$K$ -可满足性（$K$ -SAT）是研究约束满足问题中的相变和开发经验算法的典范模型系统。随机$K$ -SAT解空间的统计特性已被广泛研究，但大多数早期的努力集中在解决方案是典型的。在这里，我们考虑的最大灵活的解决方案，满足所有的约束，只使用最少的变量。这种非典型解具有高的内部熵，因为它们包含最大数量的空变量，这些空变量完全可以自由选择它们的状态。每个最大柔性解表示解空间的一个稠密区域。我们估计的最大比例的空变量的复制对称腔方法，并实现消息传递算法，以构建最大限度地灵活的解决方案，为单个$K$ -SAT实例。
英文摘要：
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.
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