research-article

Oblivious Resampling Oracles and Parallel Algorithms for the Lopsided Lovász Local Lemma

Author:

David G. HarrisAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 17, Issue 1

Article No.: 1, Pages 1 - 32

https://doi.org/10.1145/3392035

Published: 31 December 2020 Publication History

Abstract

The Lovász Local Lemma (LLL) shows that, for a collection of “bad” events B in a probability space that are not too likely and not too interdependent, there is a positive probability that no events in B occur. Moser and Tardos (2010) gave sequential and parallel algorithms that transformed most applications of the variable-assignment LLL into efficient algorithms. A framework of Harvey and Vondrák (2015) based on “resampling oracles” extended this to sequential algorithms for other probability spaces satisfying a generalization of the LLL known as the Lopsided Lovász Local Lemma (LLLL).

We describe a new structural property that holds for most known resampling oracles, which we call “obliviousness.” Essentially, it means that the interaction between two bad-events B, B^′ depends only on the randomness used to resample B and not the precise state within B itself.

This property has two major consequences. First, combined with a framework of Kolmogorov (2016), it leads to a unified parallel LLLL algorithm, which is faster than previous, problem-specific algorithms of Harris (2016) for the variable-assignment LLLL and of Harris and Srinivasan (2014) for permutations. This gives the first RNC algorithms for rainbow perfect matchings and rainbow Hamiltonian cycles of K_n.

Second, this property allows us to build LLLL probability spaces from simpler “atomic” events. This gives the first resampling oracle for rainbow perfect matchings on the complete s-uniform hypergraph K_n^(s) and the first commutative resampling oracle for Hamiltonian cycles of K_n.

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Cited By

Wang CYin Y(2024)A Sampling Lovász Local Lemma for Large Domain Sizes2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00019(129-150)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00019
Harris D(2023)Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallelRandom Structures & Algorithms10.1002/rsa.2115263:3(716-752)Online publication date: 15-Apr-2023
https://doi.org/10.1002/rsa.21152
Wanless IWood D(2022)A General Framework for Hypergraph ColoringSIAM Journal on Discrete Mathematics10.1137/21M142101536:3(1663-1677)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/21M1421015
Show More Cited By

Index Terms

Oblivious Resampling Oracles and Parallel Algorithms for the Lopsided Lovász Local Lemma
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
  2. Probability and statistics
    1. Probabilistic algorithms
2. Theory of computation
  1. Design and analysis of algorithms
    1. Parallel algorithms

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Published In

cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 17, Issue 1

January 2021

335 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3446616

Editor:
Edith Cohen
Google Research, USA and Tel Aviv University, Israel

Issue’s Table of Contents

Copyright © 2020 Public Domain.

This paper is authored by an employee(s) of the United States Government and is in the public domain. Non-exclusive copying or redistribution is allowed, provided that the article citation is given and the authors and agency are clearly identified as its source.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 31 December 2020

Online AM: 07 May 2020

Accepted: 01 March 2020

Revised: 01 December 2019

Received: 01 October 2018

Published in TALG Volume 17, Issue 1

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Cited By

Wang CYin Y(2024)A Sampling Lovász Local Lemma for Large Domain Sizes2024 IEEE 65th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS61266.2024.00019(129-150)Online publication date: 27-Oct-2024
https://doi.org/10.1109/FOCS61266.2024.00019
Harris D(2023)Deterministic algorithms for the Lovász local lemma: Simpler, more general, and more parallelRandom Structures & Algorithms10.1002/rsa.2115263:3(716-752)Online publication date: 15-Apr-2023
https://doi.org/10.1002/rsa.21152
Wanless IWood D(2022)A General Framework for Hypergraph ColoringSIAM Journal on Discrete Mathematics10.1137/21M142101536:3(1663-1677)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1137/21M1421015
Feng WHe KYin YKhuller SVassilevska Williams V(2021)Sampling constraint satisfaction solutions in the local lemma regimeProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451101(1565-1578)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451101

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