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

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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)


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
  • Lovász Local Lemma
  • Resampling
  • Moser-Tardos algorithm
  • latin transversal
  • commutativity


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