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A New Notion of Commutativity for the Algorithmic Lovász Local Lemma

Authors David G. Harris, Fotis Iliopoulos, Vladimir Kolmogorov

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Author Details

David G. Harris
  • University of Maryland, College Park, MD, USA
Fotis Iliopoulos
  • Institute for Advanced Study, Princeton, NJ, USA
Vladimir Kolmogorov
  • Institute of Science and Technology, Klosterneuburg, Austria

Funding

  • Iliopoulos, Fotis: This material is based upon work directly supported by the IAS Fund for Math and indirectly supported by the National Science Foundation Grant No. CCF-1900460. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. This work is also supported by the National Science Foundation Grant No. CCF-1815328.
  • Kolmogorov, Vladimir: Supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160.

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David G. Harris, Fotis Iliopoulos, and Vladimir Kolmogorov. A New Notion of Commutativity for the Algorithmic Lovász Local Lemma. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 31:1-31:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.31

Abstract

The Lovász Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser & Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. 
Besides conditions for convergence, many other natural questions can be asked about algorithms; for instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?". These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Probabilistic algorithms
  • Mathematics of computing → Combinatorics
Keywords
  • Lovász Local Lemma
  • Resampling
  • Moser-Tardos algorithm
  • latin transversal
  • commutativity

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References

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