[Paper Review] Variable Version Lovász Local Lemma: Beyond Shearer's Bound

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

Advanced Graph Theory Research35 references3 citations

TL;DR

This paper establishes the first necessary and sufficient criterion for the variable version of the Lovász Local Lemma (variable-LLL), resolving a long-standing open problem. It provides exact boundaries for two classes of event-variable graphs—cyclic and treelike bigraphs—and proves that a gap exists between variable-LLL and abstract-LLL boundaries if and only if the base graph contains an induced cycle of length at least 4.

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.

Motivation & Objective

• To close the gap in understanding the variable-LLL by establishing a necessary and sufficient condition for its validity.
• To determine the exact boundaries of variable-LLL for non-trivial families of event-variable graphs, including cyclic and treelike bigraphs.
• To develop a universal constructive method for maximizing the union probability of events under given probability vectors and dependency structures.
• To characterize when a gap exists between variable-LLL and abstract-LLL boundaries without solving Shearer’s conditions.
• To apply the framework to combinatorial event-variable graphs and determine whether they are gapful or gapless.

Proposed method

• Proposes a new criterion for variable-LLL based on event probabilities and the event-variable graph’s structure, expressed via a system of inequalities involving independent sets.
• Introduces a reduction framework to infer gap existence from known base graph types, using structural properties like induced cycles.
• Constructs a universal algorithm to compute the maximum possible union probability of events under a given event-variable graph and probability vector.
• Reduces the problem of computing the maximum union probability to counting satisfying assignments of a Holant instance, which is #P-hard.
• Uses binary search and reductions to show that the problem of determining the maximum probability vector (INT) is #P-hard.
• Leverages the connection between block partitions in the probability space and 3SAT instances to prove hardness results.

Experimental results

Research questions

• RQ1 What is the exact necessary and sufficient condition for the variable-LLL to hold, given an event-variable graph and event probabilities?
• RQ2 For which families of event-variable graphs can the variable-LLL boundary be fully characterized?
• RQ3 When does a gap exist between the variable-LLL boundary and the abstract-LLL boundary (Shearer’s bound)?
• RQ4 Can gap existence be decided without solving Shearer’s conditions or verifying the new criterion?
• RQ5 Is the problem of determining the maximum union probability of events under a given event-variable graph and probability vector computationally hard?

Key findings

• The paper establishes a necessary and sufficient criterion for variable-LLL in terms of event probabilities and the event-variable graph, resolving a long-standing open problem.
• The variable-LLL boundary is fully determined for two non-trivial families: cyclic and treelike bigraphs.
• A gap between variable-LLL and abstract-LLL boundaries exists if and only if the base graph of the event-variable graph contains an induced cycle of length at least 4.
• If the base graph is a tree, no gap exists between variable-LLL and abstract-LLL boundaries.
• The problem of computing the maximum union probability of events under a given event-variable graph and probability vector is #P-hard.
• The paper provides a universal constructive method to find a set of events whose union achieves the maximum possible probability, given the probability vector and event-variable graph.