Moser and Tardos (J. ACM (JACM) 57 (2), 11 2010 ) gave an algorithmic proof of the lopsided Lovász local lemma (LLL) in the variable framework, where each of the undesirable events is assumed to depend on a subset of a collection of independent random variables. For the proof, they define a notion of a lopsided dependency between the events suitable for this framework. In this work, we strengthen this notion, defining a novel directed notion of dependency and prove the LLL for the corresponding graph. We show that this graph can be strictly sparser (thus the sufficient condition for the LLL weaker) compared with graphs that correspond to other extant lopsided versions of dependency. Thus, in a sense, we address the problem “find other simple local conditions for the constraints (in the variable framework) that advantageously translate to some abstract lopsided condition” posed by Szegedy ( 2013 ). We also give an example where our notion of dependency graph gives better results than the classical Shearer lemma. Finally, we prove Shearer’s lemma for the dependency graph we define. For the proofs, we perform a direct probabilistic analysis that yields an exponentially small upper bound for the probability of the algorithm that searches for the desired assignment to the variables not to return a correct answer within n steps. In contrast, the method of proof that became known as the entropic method, gives an estimate of only the expectation of the number of steps until the algorithm returns a correct answer, unless the probabilities are tinkered with.

中文翻译：

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指导 Lovász 局部引理和希勒引理

Moser 和 Tardos (J. ACM (JACM) 57 (2), 11 2010 ) 给出了变量框架中不平衡 Lovász 局部引理 (LLL) 的算法证明，其中假设每个不良事件依赖于独立随机变量的集合。为了证明，他们定义了适用于该框架的事件之间不平衡依赖的概念。在这项工作中，我们加强了这个概念，定义了一个新的有向依赖概念并证明了相应图的 LLL。我们表明，与对应于其他现存不平衡依赖版本的图相比，该图可以严格稀疏（因此是 LLL 较弱的充分条件）。因此，从某种意义上说，我们解决了 Szegedy (2013) 提出的问题“为约束（在变量框架中）找到其他简单的局部条件，这些条件有利地转化为一些抽象的不平衡条件”。我们还举了一个例子，其中我们的依赖图概念比经典的 Shearer 引理给出了更好的结果。最后，我们证明了我们定义的依赖图的希勒引理。对于证明，我们执行直接概率分析，该分析为算法的概率产生一个指数级小的上限，该算法搜索变量的期望分配以在 n 步内不返回正确答案。相比之下，后来被称为熵方法的证明方法仅给出算法返回正确答案之前的步数期望值的估计，