A new notion of commutativity for the algorithmic Lovász Local Lemma

Harris DG, Iliopoulos F, Kolmogorov V. 2021. A new notion of commutativity for the algorithmic Lovász Local Lemma. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX/RANDOM: Approximation Algorithms for Combinatorial Optimization Problems/ Randomization and Computation, LIPIcs, vol. 207, 31.

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Author
Harris, David G.; Iliopoulos, Fotis; Kolmogorov, VladimirIST Austria
Department
Series Title
LIPIcs
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 and 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 existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. 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.
Publishing Year
Date Published
2021-09-15
Proceedings Title
Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques
Acknowledgement
Fotis Iliopoulos: 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. Vladimir Kolmogorov: Supported by the European Research Council under the European Unions Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no 616160.
Volume
207
Article Number
31
Conference
APPROX/RANDOM: Approximation Algorithms for Combinatorial Optimization Problems/ Randomization and Computation
Conference Location
Virtual
Conference Date
2021-08-16 – 2021-08-18
ISSN
IST-REx-ID

Cite this

Harris DG, Iliopoulos F, Kolmogorov V. A new notion of commutativity for the algorithmic Lovász Local Lemma. In: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. Vol 207. Schloss Dagstuhl - Leibniz Zentrum für Informatik; 2021. doi:10.4230/LIPIcs.APPROX/RANDOM.2021.31
Harris, D. G., Iliopoulos, F., & Kolmogorov, V. (2021). A new notion of commutativity for the algorithmic Lovász Local Lemma. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (Vol. 207). Virtual: Schloss Dagstuhl - Leibniz Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.31
Harris, David G., 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, Vol. 207. Schloss Dagstuhl - Leibniz Zentrum für Informatik, 2021. https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2021.31.
D. G. Harris, F. Iliopoulos, and V. Kolmogorov, “A new notion of commutativity for the algorithmic Lovász Local Lemma,” in Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, Virtual, 2021, vol. 207.
Harris DG, Iliopoulos F, Kolmogorov V. 2021. A new notion of commutativity for the algorithmic Lovász Local Lemma. Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX/RANDOM: Approximation Algorithms for Combinatorial Optimization Problems/ Randomization and Computation, LIPIcs, vol. 207, 31.
Harris, David G., et al. “A New Notion of Commutativity for the Algorithmic Lovász Local Lemma.” Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, vol. 207, 31, Schloss Dagstuhl - Leibniz Zentrum für Informatik, 2021, doi:10.4230/LIPIcs.APPROX/RANDOM.2021.31.
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