David G. Harris
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 Kn.
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 Kn(s) and the first commutative resampling oracle for Hamiltonian cycles of Kn.
- D. Achlioptas and F. Iliopoulos. 2016. Random walks that find perfect objects and the Lovász Local Lemma. J. ACM 63, 3 (2016), Article # 22. Google Scholar
Digital Library
- D. Achlioptas and F. Iliopoulos. 2016. Focused stochastic local search and the Lovász local lemma. In Proceedings of the 27th ACM-SIAM Symposium on Discrete Algorithms (SODA’16), pp. 20248--2038. Google ScholarDigital Library
- D. Achlioptas, F. Iliopoulos, and A. Sinclair. 2019. Beyond the Lovász Local Lemma: Point to set correlations and their algorithmic applications. In Proceedings of the 60th IEEE Symposium on Foundations of Computer Science (FOCS’19). 725--744.Google Scholar
- M. Albert, A. Frieze, and B. Reed. 1995. Multicoloured Hamilton cycles. Electronic J. Combinat. 2, 1 (1995), R10.Google ScholarCross Ref
- R. Bissacot, R. Fernandez, A. Procacci, and B. Scoppola. 2011. An improvement of the Lovász Local Lemma via cluster expansion. Combinat., Probabil. Comput. 20, 5 (2011), 709--719. Google ScholarDigital Library
- G. Blelloch, J. Fineman, and J. Shun. 2012. Greedy sequential maximal independent set and matching are parallel on average. In Proceedings of the 24th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA’12). 308--317. Google ScholarDigital Library
- S. Brandt, O. Fischer, J. Hirvonen, B. Keller, T. Lempiäinen, J. Rybicki, J. Suomela, and J. Uitto. 2015. A lower bound for the distributed Lovász Local Lemma. In Proceedings of the 48th ACM Symposium on Theory of Computing (STOC’15). 479--488. Google ScholarDigital Library
- K. Chung, S. Pettie, and H. Su. 2017. Distributed algorithms for the Lovász local lemma and graph coloring. Distributed Computing 30, 4 (2017), 261--2680. Google ScholarDigital Library
- S. Cook. 1985. A taxonomy of problems with fast parallel algorithms. Info. Control 64, 1--3 (1985), 2--22. Google ScholarDigital Library
- P. Erdős and J. Spencer. 1990. Lopsided Lovász Local Lemma and Latin transversals. Discrete Appl. Math 30, (2, 3) (1990), 151--154. Google ScholarDigital Library
- M. Fischer and M. Ghaffari. 2017. Sublogarithmic distributed algorithms for Lovász Local lemma, and the complexity hierarchy. In Proceedings of the 31st International Symposium on Distributed Computing (DISC’17). 18.Google Scholar
- M. Fischer and A. Noever. 2019. Tight analysis of randomized greedy MIS. ACM Trans. Algor. 16, 1 (2019), Article #6. Google ScholarDigital Library
- M. Ghaffari, D. Harris, and F. Kuhn. 2018. On derandomizing local distributed algorithms. In Proceedings of the 59th IEEE Symposium on Foundations of Computer Science (FOCS’18). 662--673.Google Scholar
- A. Graf and P. Haxell. 2018. Finding independent transversals efficiently. arXiv:1811.02687.Google Scholar
- A. Graf, D. Harris, and P. Haxell. 2019. Algorithms for weighted independent transversals and strong colouring. arXiv:1907.00033.Google Scholar
- B. Haeupler and D. Harris. 2017. Parallel algorithms and concentration bounds for the Lovász Local Lemma via witness DAGs. ACM Trans. Algor. 13, 4 (2017), Article #53. Google ScholarDigital Library
- B. Haeupler, B. Saha, and A. Srinivasan. 2011. New constructive aspects of the Lovász Local Lemma. J. ACM 58, 6 (2011). Google ScholarDigital Library
- D. Harris. 2016. Lopsidependency in the Moser-Tardos framework: Beyond the Lopsided Lovász Local Lemma. ACM Trans. Algor. 13, 1 (2016), Article #17. Google ScholarDigital Library
- D. Harris. 2020. New bounds for the Moser-Tardos distribution. Random Struct. Algor. 57, 1 (2020), 97--131.Google ScholarCross Ref
- D. Harris. 2019. Deterministic algorithms for the Lovász Local Lemma: Simpler, more general, and more parallel. arXiv:1909.08065.Google Scholar
- D. Harris and A. Srinivasan. 2017. Algorithmic and enumerative aspects of the Moser-Tardos distribution. ACM Trans. Algor. 13, 3 (2017), Article #33. Google ScholarDigital Library
- D. Harris and A. Srinivasan. 2017. A constructive Lovász Local Lemma for permutations. Theory Comput. 13, 17 (2017), 1--41.Google ScholarCross Ref
- D. Harris and A. Srinivasan. 2019. The Moser-Tardos framework with partial resampling. J. ACM 66, 5 (2019), Article #36. Google ScholarDigital Library
- N. Harvey and C. Liaw. 2017. Rainbow Hamilton cycles and lopsidependency. Discrete Math. 340, 6 (2017), 1261--1270.Google ScholarCross Ref
- N. Harvey and J. Vondrák. 2020. An algorithmic proof of the Lopsided Lovász Local Lemma via resampling oracles. SIAM J. Comput. 49, 2 (2020), 394--428.Google ScholarCross Ref
- P. Haxell. 2008. An improved bound for the strong chromatic number. J. Graph Theory 58, 2 (2008), 148--158. Google ScholarDigital Library
- F. Iliopoulos. 2018. Commutative algorithms approximate the LLL distribution. In Proceedings of the Conference on Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM’18), pp. 44:1--44:20.Google Scholar
- K. Kolipaka and M. Szegedy. 2011. Moser and Tardos meet Lovász. In Proceedings of the 43rd ACM Symposium on Theory of Computing (STOC’11). 235--244. Google ScholarDigital Library
- V. Kolmogorov. 2018. Commutativity in the algorithmic Lovász Local Lemma. SIAM J. Comput. 47, 6 (2018), 2029--2056.Google ScholarCross Ref
- L. Lu, A. Mohr, and L. Székely. 2012. Quest for negative dependency graphs. Recent Advances in Harmonic Analysis and Applications. Springer, pp. 243-256.Google Scholar
- L. Lu and L. Székély. 2007. Using Lovász Local Lemma in the space of random injections. Electronic J. Combinat. 13-R63 (2007).Google Scholar
- L. Lu and L. Székély. 2011. A new asymptotic enumeration technique: The Lovász local lemma. arXiv:0905.3983 (2011).Google Scholar
- M. Luby. 1996. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 4 (1996), 1036--1053. Google ScholarDigital Library
- C. McDiarmid. 1995. Hypergraph coloring and the Lovász Local Lemma. J. Discrete Math. 167--168 (1995), 481-486. Google ScholarDigital Library
- R. Moser and G. Tardos. 2010. A constructive proof of the general Lovász Local Lemma. J. ACM 57, 2 (2010), Article #11. Google ScholarDigital Library
- W. Pegden. 2014. An extension of the Moser-Tardos algorithmic Local Lemma. SIAM J. Discrete Math. 28, 2 (2014), 911--917.Google ScholarCross Ref
- J. B. Shearer. 1985. On a problem of Spencer. Combinatorica 5, 3 (1985), 241--245. Google ScholarDigital Library
Index Terms
- Oblivious Resampling Oracles and Parallel Algorithms for the Lopsided Lovász Local Lemma
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