A fast and simple randomized parallel algorithm for the maximal independent set problem
Journal of Algorithms
A simple parallel algorithm for the maximal independent set problem
SIAM Journal on Computing
Approximation algorithms for scheduling unrelated parallel machines
Mathematical Programming: Series A and B
Ramsey theory (2nd ed.)
Further algorithmic aspects of the local lemma
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Coloring non-uniform hypergraphs: a new algorithmic approach to the general Lovász local lemma
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Nonrepetitive colorings of graphs
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Pattern avoidance: themes and variations
Theoretical Computer Science - Combinatorics on words
ACM SIGecom Exchanges
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
An approximation algorithm for max-min fair allocation of indivisible goods
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
On allocations that maximize fairness
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Improved algorithmic versions of the Lovász Local Lemma
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Acyclic edge colorings of graphs
Journal of Graph Theory
Santa Claus Meets Hypergraph Matchings
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
The complexity of nonrepetitive coloring
Discrete Applied Mathematics
A constructive proof of the Lovász local lemma
Proceedings of the forty-first annual ACM symposium on Theory of computing
MaxMin allocation via degree lower-bounded arborescences
Proceedings of the forty-first annual ACM symposium on Theory of computing
A constructive proof of the general lovász local lemma
Journal of the ACM (JACM)
On Allocating Goods to Maximize Fairness
FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
Solving MAX-r-SAT above a tight lower bound
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Deterministic algorithms for the Lovász Local Lemma
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Approximation Algorithms for the Edge-Disjoint Paths Problem via Raecke Decompositions
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
An algorithmic approach to the lovász local lemma. I
Random Structures & Algorithms
A parallel algorithmic version of the local lemma
Random Structures & Algorithms
Bounding Ramsey numbers through large deviation inequalities
Random Structures & Algorithms
Random Structures & Algorithms
IPCO'13 Proceedings of the 16th international conference on Integer Programming and Combinatorial Optimization
Constraint satisfaction, packet routing, and the lovasz local lemma
Proceedings of the forty-fifth annual ACM symposium on Theory of computing
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The Lovász Local Lemma (LLL) is a powerful tool that gives sufficient conditions for avoiding all of a given set of “bad” events, with positive probability. A series of results have provided algorithms to efficiently construct structures whose existence is non-constructively guaranteed by the LLL, culminating in the recent breakthrough of Moser and Tardos [2010] for the full asymmetric LLL. We show that the output distribution of the Moser-Tardos algorithm well-approximates the conditional LLL-distribution, the distribution obtained by conditioning on all bad events being avoided. We show how a known bound on the probabilities of events in this distribution can be used for further probabilistic analysis and give new constructive and nonconstructive results. We also show that when a LLL application provides a small amount of slack, the number of resamplings of the Moser-Tardos algorithm is nearly linear in the number of underlying independent variables (not events!), and can thus be used to give efficient constructions in cases where the underlying proof applies the LLL to super-polynomially many events. Even in cases where finding a bad event that holds is computationally hard, we show that applying the algorithm to avoid a polynomial-sized “core” subset of bad events leads to a desired outcome with high probability. This is shown via a simple union bound over the probabilities of non-core events in the conditional LLL-distribution, and automatically leads to simple and efficient Monte-Carlo (and in most cases RNC) algorithms. We demonstrate this idea on several applications. We give the first constant-factor approximation algorithm for the Santa Claus problem by making a LLL-based proof of Feige constructive. We provide Monte Carlo algorithms for acyclic edge coloring, nonrepetitive graph colorings, and Ramsey-type graphs. In all these applications, the algorithm falls directly out of the non-constructive LLL-based proof. Our algorithms are very simple, often provide better bounds than previous algorithms, and are in several cases the first efficient algorithms known. As a second type of application we show that the properties of the conditional LLL-distribution can be used in cases beyond the critical dependency threshold of the LLL: avoiding all bad events is impossible in these cases. As the first (even nonconstructive) result of this kind, we show that by sampling a selected smaller core from the LLL-distribution, we can avoid a fraction of bad events that is higher than the expectation. MAX k-SAT is an illustrative example of this.