[论文解读] Variable Version Lovász Local Lemma: Beyond Shearer's Bound

Kun He, Li Liang+3 more|arXiv (Cornell University)|Sep 15, 2017

Advanced Graph Theory Research参考文献 35被引用 3

一句话总结

本文建立了变量版本洛瓦兹局部引理（variable-LLL）的首个必要且充分条件，解决了长期悬而未决的开放问题。它精确界定了两类事件-变量图——环状图与树状图——的边界，并证明：当且仅当基础图包含长度至少为4的诱导环时，变量-LLL与抽象-LLL边界之间存在间隙。

ABSTRACT

A tight criterion under which the abstract version Lovász Local Lemma (abstract-LLL) holds was given by Shearer decades ago. However, little is known about that of the variable version LLL (variable-LLL) where events are generated by independent random variables, though this model of events is applicable to almost all applications of LLL. We introduce a necessary and sufficient criterion for variable-LLL, in terms of the probabilities of the events and the event-variable graph specifying the dependency among the events. Based on this new criterion, we obtain boundaries for two families of event-variable graphs, namely, cyclic and treelike bigraphs. These are the first two non-trivial cases where the variable-LLL boundary is fully determined. As a byproduct, we also provide a universal constructive method to find a set of events whose union has the maximum probability, given the probability vector and the event-variable graph. Though it is #P-hard in general to determine variable-LLL boundaries, we can to some extent decide whether a gap exists between a variable-LLL boundary and the corresponding abstract-LLL boundary. In particular, we show that the gap existence can be decided without solving Shearer's conditions or checking our variable-LLL criterion. Equipped with this powerful theorem, we show that there is no gap if the base graph of the event-variable graph is a tree, while gap appears if the base graph has an induced cycle of length at least 4. The problem is almost completely solved except when the base graph has only 3-cliques, in which case we also get partial solutions. A set of reduction rules are established that facilitate to infer gap existence of an event-variable graph from known ones. As an application, various event-variable graphs, in particular combinatorial ones, are shown to be gapful/gapless.

研究动机与目标

• 通过建立变量-LLL有效性的必要且充分条件，弥合对变量-LLL理解上的空白。
• 确定非平凡事件-变量图族（包括环状图与树状图）的变量-LLL边界的精确界限。
• 开发一种通用的构造性方法，以在给定概率向量与依赖结构下，最大化事件的并集概率。
• 在不求解谢雷条件的情况下，刻画变量-LLL与抽象-LLL边界之间是否存在间隙的条件。
• 将该框架应用于组合事件-变量图，判断其是否具有间隙或无间隙。

提出的方法

• 基于事件概率与事件-变量图结构，提出一个新准则，通过涉及独立集的不等式组表达。
• 引入一种约化框架，利用结构特性（如诱导环）从已知的基础图类型推断间隙的存在性。
• 构建一个通用算法，以计算在给定事件-变量图与概率向量下，事件并集可能达到的最大概率。
• 将计算最大并集概率的问题约化为计数一个Holant实例的满足赋值数，该问题为#P-难。
• 利用二分查找与约化方法，证明确定最大概率向量（INT）的问题为#P-难。
• 利用概率空间中块划分与3SAT实例之间的联系，证明相关难题结果。

实验结果

研究问题

• RQ1 在给定事件-变量图与事件概率的情况下，变量-LLL成立的确切必要且充分条件是什么？
• RQ2 哪些事件-变量图族的变量-LLL边界可以被完全刻画？
• RQ3 在什么情况下，变量-LLL边界与抽象-LLL边界（谢雷界）之间存在间隙？
• RQ4 是否可以在不求解谢雷条件或验证新准则的情况下，判断间隙是否存在？
• RQ5 在给定事件-变量图与概率向量下，确定事件并集最大概率的问题是否在计算上是困难的？

主要发现

• 本文基于事件概率与事件-变量图结构，建立了变量-LLL的必要且充分条件，解决了长期悬而未决的开放问题。
• 对于两类非平凡图族——环状图与树状图——变量-LLL边界被完全确定。
• 当且仅当事件-变量图的基础图包含长度至少为4的诱导环时，变量-LLL与抽象-LLL边界之间存在间隙。
• 若基础图为树，则变量-LLL与抽象-LLL边界之间不存在间隙。
• 在给定事件-变量图与概率向量下，计算事件并集最大概率的问题为#P-难。
• 本文提供了一种通用的构造性方法，可在给定概率向量与事件-变量图下，找到一组事件，使其并集概率达到最大可能值。