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.
@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/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.31}, 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} }
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