A constructive proof of the general Lovasz Local Lemma (0903.0544v3)

Abstract: The Lovasz Local Lemma [EL75] is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In his breakthrough paper [Bec91], Beck demonstrated that a constructive variant can be given under certain more restrictive conditions. Simplifications of his procedure and relaxations of its restrictions were subsequently exhibited in several publications [Alo91, MR98, CS00, Mos06, Sri08, Mos08]. In [Mos09], a constructive proof was presented that works under negligible restrictions, formulated in terms of the Bounded Occurrence Satisfiability problem. In the present paper, we reformulate and improve upon these findings so as to directly apply to almost all known applications of the general Local Lemma.

• The paper presents a constructive algorithm that transforms the Lovász Local Lemma from a non-constructive existence statement to a practical method by efficiently resampling variables.
• It details both sequential and parallel versions, achieving logarithmically reduced expected runtimes and robust bounds on resampling steps.
• The authors propose derandomization strategies to extend these results into deterministic algorithms for complex combinatorial problems.

A Constructive Proof of the General Lovász Local Lemma

This paper explores an algorithmic approach to the Lovász Local Lemma (LLL), a fundamental result in probability theory that provides conditions under which events occur with positive probability. Traditionally, LLL proofs have been non-constructive, which limits their direct applicability to algorithm design. This work advances the understanding of LLL by presenting an efficient, constructive algorithm for finding an assignment that avoids all specified undesired events, addressing limitations in previous constructive interpretations by achieving this with minimal restrictions.

Context and Contributions

The Lovász Local Lemma has been a cornerstone in addressing problems involving rare events and has applications in various fields such as combinatorics and computer science. Beck's initial constructive approach required restrictive conditions, significantly limiting its applicability. Subsequent attempts relaxed these conditions, yet a gap remained between existential guarantees and algorithmic construction.

In this paper, Robin A. Moser and Gábor Tardos present an algorithmic framework that narrows this gap, addressing nearly all known applications of the general LLL. The key innovation lies in using resampling techniques to efficiently converge to a solution meeting the LLL's conditions.

Algorithmic Framework

Their algorithm initializes with a random evaluation of variables in a probability space. It iteratively tests whether any undesirable event occurs. If such an event is found, it randomly resamples the variables involved, ensuring this resampling does not affect other parts of the system. This process repeats until all constraints of the LLL are satisfied.

The authors provide both a sequential and a parallel version of the algorithm. The parallel version shows particular strength, offering significant improvements in the expected runtime when conditions slightly stronger than those proposed in the non-constructive version are met.

Theoretical Insights

The paper proves several important results:

• Expected Resampling Steps: For each event $A$, the expected number of resampling steps is at most $x(A)/(1-x(A))$. This bound highlights the efficiency of the algorithm concerning the degrees of dependencies within events.
• Parallelization: The parallel version of the algorithm not only preserves the correctness of the sequential variant but also optimizes computational resources by reducing the expected number of steps logarithmically.
• Derandomization: Extensions to deterministic algorithms are proposed, demonstrating efficient performance under constraints, such as bounded graph degree.

Implications and Future Directions

The presented algorithm marks a significant step in turning the Lovász Local Lemma into a practical tool for algorithm design. By effectively incorporating probabilistic techniques with constructive methods, the authors pave the way for advancements in fields such as randomized algorithms, satisfiability problems, and combinatorial structure analysis.

Future research might explore:

• Extending the framework to handle even more complex dependencies or larger classes of problems.
• Developing new computational tools to further reduce the complexity of finding satisfying assignments.
• Investigating potential applications in other domains where LLL could provide algorithmic benefits.

The constructive approach to LLL by Moser and Tardos enriches the theoretical landscape and enhances practical methodologies in dealing with dependent random events.