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URN: urn:nbn:de:0030-drops-147244
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A New Notion of Commutativity for the Algorithmic Lovász Local Lemma

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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.

BibTeX - Entry

@InProceedings{harris_et_al:LIPIcs.APPROX/RANDOM.2021.31,
  author =	{Harris, David G. and Iliopoulos, Fotis and Kolmogorov, Vladimir},
  title =	{{A New Notion of Commutativity for the Algorithmic Lov\'{a}sz Local Lemma}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)},
  pages =	{31:1--31:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-207-5},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{207},
  editor =	{Wootters, Mary and Sanit\`{a}, Laura},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2021/14724},
  URN =		{urn:nbn:de:0030-drops-147244},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2021.31},
  annote =	{Keywords: Lov\'{a}sz Local Lemma, Resampling, Moser-Tardos algorithm, latin transversal, commutativity}
}

Keywords: Lovász Local Lemma, Resampling, Moser-Tardos algorithm, latin transversal, commutativity
Seminar: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)
Issue date: 2021
Date of publication: 15.09.2021


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