Moser and tardos meet Lovász
Moser and tardos meet Lovász
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DOI：
10.1145/1993636.1993669
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发表时间：
2011-06
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影响因子：
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通讯作者：
K. Kolipaka;M. Szegedy
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文献类型：
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作者：
K. Kolipaka;M. Szegedy
贝克的早期作品[3]在参数上具有重大折衷的Lovász本地引理（LL）。对于固定依赖图G的最大程度，获得的确切标准是在[12]中给出的确切标准。 lovász局部引理的概率分配的概率分配也表明：Moser-Tardos算法的顺序和平行性都可以通过更严格的分析来有效还证明，每当p∈Lo（g）/（1+ε）时，顺序和并行版本的预期运行时间最多为n/ε和o（1/εlog n/ε） <1。这是n的数量G中的节点。通过在我们的分析中添加几行，我们可以再现Shearer的结果（对于一般LLL）。 Moser-Tardos算法的存在可以在存在的情况下有用。和Sokal [10]。这扩展了Haeupler，Saha和Srinivasan的最新工作之一[6]。 ）变量版本有时比通用版本大。
Beck's early work [3] gave an efficient version of the Lovász Local Lemma(LLL) with significant compromise in the parameters. Following several improvements [1,7,4,13], Moser [8], and Moser and Tardos [9] obtained asymptotically optimal results in terms of the maximal degree. For a fixed dependency graph G the exact criterion under which LLL applies is given by Shearer in [12]. For a dependency structure G, let LO(G) be the set of those probability assignments to the nodes of G for which the Lovász Local Lemma holds. We show that: Both the sequential and parallel ersions of the Moser-Tardos algorithm are efficient up to the Shearer's bound, by giving a tighter analysis. We also prove that, whenever p ∈ LO(G)/(1+ε), the expected running times of the sequential and parallel versions are at most n/ε and O(1/ε log n/ε), the later when ε < 1. Here n is the number of nodes in G. By adding a few lines to our analysis we can reprove Shearer's result (for the general LLL). Our alternative proof for the Shearer's bound not only highlights the connection between the variable and general versions of LLL, but also illustrates that variants of the Moser-Tardos algorithm can be useful in existence proofs. We obtain new formulas for phase transitions in the hardcore lattice gas model, non-trivially equivalent to the ones studied by Scott and Sokal [10]. We prove that if p ∈ LO(G)/(1+ε), the running time of the Moser-Tardos algorithm is polynomial not only in the number of events, but also in the number of variables. This extends one of the results from the more recent work of Haeupler, Saha, and Srinivasan [6]. Our new formulas immediately give a majorizing lemma that connects LLL bounds on different graphs. We show that the LLL bound for the (special case of the) variable version is sometimes larger than for the general version. This is the first known separation between the variable and the general versions of LLL.