聚焦随机算法的一个局部引理
A Local Lemma for Focused Stochastic Algorithms
摘要：
我们开发了一个框架，集中随机局部搜索算法的严格分析。这些算法通过重复选择当前状态中违反的某些约束并移动到解决违反的随机附近状态来搜索状态空间，同时希望不会引入许多新的约束。最近，一类重要的具有可证明性能保证的局部搜索算法出现在Lovász局部引理（LLL）的算法化中，LLL是一种通过在状态空间上引入背景测度来证明满足状态存在的非构造性工具。虽然功能强大，但这类算法的状态转换必须在精确的意义上与背景度量完全兼容。在许多应用程序中，这是一个非常严格的要求，需要在类之外进行操作。在这里，我们介绍了\emph{measure distortion}的概念，并开发了一个框架，用于分析任意集中的随机局部搜索算法，恢复LLL算法的特殊情况下，没有失真。我们的框架作为输入的任意这样的算法和任意的概率测度，并显示了如何使用该测度作为算法进展的衡量标准，即使是独立于该测度设计的算法。
英文摘要：
We develop a framework for the rigorous analysis of focused stochastic local search algorithms. These are algorithms that search a state space by repeatedly selecting some constraint that is violated in the current state and moving to a random nearby state that addresses the violation, while hopefully not introducing many new ones. An important class of focused local search algorithms with provable performance guarantees has recently arisen from algorithmizations of the Lovász Local Lemma (LLL), a non-constructive tool for proving the existence of satisfying states by introducing a background measure on the state space. While powerful, the state transitions of algorithms in this class must be, in a precise sense, perfectly compatible with the background measure. In many applications this is a very restrictive requirement and one needs to step outside the class. Here we introduce the notion of \emph{measure distortion} and develop a framework for analyzing arbitrary focused stochastic local search algorithms, recovering LLL algorithmizations as the special case of no distortion. Our framework takes as input an arbitrary such algorithm and an arbitrary probability measure and shows how to use the measure as a yardstick of algorithmic progress, even for algorithms designed independently of the measure.
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