Abstract
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.
- Alon, N. 1991. A parallel algorithmic version of the Local Lemma. Rand. Struct. Algor. 2, 367--378. Google ScholarDigital Library
- Alon, N. 1994. Explicit Ramsey graphs and orthonormal labelings. Electron. J. Combin. 1, 12--8.Google Scholar
- Alon, N. and Krivelevich, M. 1997. Constructive bounds for a Ramsey-type problem. Graphs Combinat. 13, 217--225.Google ScholarDigital Library
- Alon, N. and Grytczuk, J. 2008. Breaking the rhythm on graphs. Disc. Math. 308, 8, 1375--1380.Google ScholarDigital Library
- Alon, N. and Pudlak, P. 1999. Constructive lower bounds for off-diagonal Ramsey numbers. Israel J. Math. 122, 243--251.Google ScholarCross Ref
- Alon, N. and Spencer, J. H. 2008. The Probabilistic Method 3rd Ed. Wiley.Google Scholar
- Alon, N., Babai, L., and Itai, A. 1986. A fast and simple randomized parallel algorithm for the maximal independent set problem. J. Algor. 7, 567--583. Google ScholarDigital Library
- Alon, N., Sudakov, B., and Zaks, A. 2001. Acyclic edge colorings of graphs. J. Graph Theory 37, 3, 157--167. Google ScholarDigital Library
- Alon, N., Grytczuk, J., Haluszczak, M., and Riordan, O. 2002. Nonrepetitive colorings of graphs. Rand. Struct. Algor. 21, 3--4, 336--346. Google ScholarDigital Library
- Alon, N., McDiarmid, C., and Reed, B. 2007. Acyclic coloring of graphs. Rand. Struct. Algor. 2, 3, 277--288. Google ScholarDigital Library
- Alon, N., Gutin, G., Kim, E. J., Szeider, S., and Yeo, A. 2010. Solving MAX-r-SAT above a tight lower bound. In Proceedings of the 21th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’10). ACM. Google ScholarDigital Library
- Andrews, M. 2010. Approximation algorithms for the edge-disjoint paths problem via Raecke decompositions. In Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science (FOCS’10). 277--286. Google ScholarDigital Library
- Asadpour, A. and Saberi, A. 2007. An approximation algorithm for max-min fair allocation of indivisible goods. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC’07). 114--121. Google ScholarDigital Library
- Asadpour, A., Feige, U., and Saberi, A. 2008. Santa claus meets hypergraph matchings. In Proceedings of the 11th International Workshop, APPROX’08, and on 12th International Workshop, RANDOM’08 on Approximation, Randomization and Combinatorial Optimization. 10--20. Google ScholarDigital Library
- Bansal, N. and Sviridenko, M. 2006. The Santa Claus problem. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing (STOC’06). 31--40. Google ScholarDigital Library
- Bateni, M., Charikar, M., and Guruswami, V. 2009. Maxmin allocation via degree lower-bounded arborescences. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC’09). 543--552. Google ScholarDigital Library
- Beck, J. 1991. An algorithmic approach to the Lovász Local Lemma. Rand. Struct. Algor. 2, 4, 343--365. Google ScholarDigital Library
- Bezáková, I. and Dani, V. 2005. Allocating indivisible goods. SIGecom Exch. 5, 3, 11--18. Google ScholarDigital Library
- Bresar, B., Grytczuk, J., Klavzar, S., Niwczyk, S., and Peterin, I. 2007. Nonrepetitive colorings of trees. Disc. Math. 307, 2, 163--172.Google ScholarDigital Library
- Chakrabarty, D., Chuzhoy, J., and Khanna, S. 2009. On allocating goods to maximize fairness. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science (FOCS’09). Google ScholarDigital Library
- Chandrasekaran, K., Goyal, N., and Haeupler, B. 2010. Deterministic algorithms for the Lovász Local Lemma. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’10). 992--1004. Google ScholarDigital Library
- Currie, J. D. 2005. Pattern avoidance: Themes and variations. Theor. Comput. Sci. 339, 1, 7--18. Google ScholarDigital Library
- Czumaj, A. and Scheideler, C. 2000. Coloring non-uniform hypergraphs: A new algorithmic approach to the general Lovász local lemma. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’00). 30--39. Google ScholarDigital Library
- Erdős, P. 1947. Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53, 292--294.Google ScholarCross Ref
- Erdős, P. and Lovász, L. 1975. Problems and results on 3-chromatic hypergraphs and some related questions. In Infinite and Finite Sets, Colloq. Math. Soc., J. Bolyai Series, vol. 11, North-Holland, 609--627.Google Scholar
- Feige, U. 2008a. On allocations that maximize fairness. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’08). 287--293. Google ScholarDigital Library
- Feige, U. 2008b. On estimation algorithms vs approximation algorithms. In Proceedings of the Annual Conference on Foundation of Structure Technology and Theoritical Computer Science (FSTTCS). LIPIcs Series, vol. 2. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 357--363.Google Scholar
- Graham, R. L., Rothschild, B. L., and Spencer, J. H. 1990. Ramsey Theory, 2nd Ed. Wiley. Google ScholarDigital Library
- Grünbaum, B. 1973. Acyclic colorings of planar graphs. Israel J. Math. 14, 390--408.Google ScholarCross Ref
- Grytczuk, J. 2008. Thue type problems for graphs, points, and numbers. Disc. Math. 308, 19, 4419--4429.Google ScholarDigital Library
- Haxell, P. 1995. A condition for matchability in hypergraphs. Graphs Combinat. 11, 3, 245--248.Google ScholarDigital Library
- Krivelevich, M. 1995. Bounding Ramsey numbers through large deviation inequalities. Rand. Struct. Algor. 7, 2, 145--155. Google ScholarCross Ref
- Kündgen, A. and Pelsmajer, M. J. 2008. Nonrepetitive colorings of graphs of bounded tree-width. Disc. Math. 308, 19, 4473--4478.Google ScholarDigital Library
- Leighton, F. T., Maggs, B. M., and Rao, S. B. 1994. Packet routing and jobshop scheduling in O(congestion + dilation) steps. Combinatorica 14, 167--186.Google ScholarCross Ref
- Lenstra, J. K., Shmoys, D. B., and Tardos, E. 1990. Approximation algorithms for scheduling unrelated parallel machines. Math. Program. 46, 259--271. Google ScholarDigital Library
- Luby, M. 1986. A simple parallel algorithm for the maximal independent set problem. SIAM J. Comput. 15, 4, 1036--1053. Google ScholarDigital Library
- Marx, D. and Schaefer, M. 2009. The complexity of nonrepetitive coloring. Disc. Appl. Math. 157, 1, 13--18. Google ScholarDigital Library
- Molloy, M. and Reed, B. 1998. Further algorithmic aspects of the Local Lemma. In Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC’98). ACM, 524--529. Google ScholarDigital Library
- Molloy, M. and Reed, B. 2001. Graph Colouring and the Probabilistic Method. Springer-Verlag.Google Scholar
- Moser, R. A. 2008. Derandomizing the Lovász Local Lemma more effectively. CoRR abs/0807.2120.Google Scholar
- Moser, R. 2009. A constructive proof of the Lovász Local Lemma. In Proceedings of the 41st Annual ACM Symposium on Theory of Computing (STOC’09). 343--350. Google ScholarDigital Library
- Moser, R. and Tardos, G. 2010. A constructive proof of the general Lovász Local Lemma. J. ACM 57, 2, 1--15. Google ScholarDigital Library
- Muthu, R., Narayanan, N., and Subramanian, C. R. 2007. Improved bounds on acyclic edge colouring. Disc. Math. 307, 23, 3063--3069.Google ScholarDigital Library
- Saha, B. and Srinivasan, A. 2010. A new approximation technique for resource-allocation problems. In Proceedings of the 1st Annual Symposium on Innovations in Computer Science (ICS’10). 342--357.Google Scholar
- Schaefer, M. and Umans, C. 2002. Completeness in the polynomial-time hierarchy: A compendium. SIGACT News 33, 3, 32--49.Google ScholarDigital Library
- Srinivasan, A. 2008. Improved algorithmic versions of the Lovász Local Lemma. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA’08). 611--620. Google ScholarDigital Library
- Thue, A. 1906. Über unendliche Zeichenreihen. Norske Vid Selsk. Skr. I. Mat. Nat. Kl. Christiana 7, 1--22.Google Scholar
- Vizing, V. G. 1964. On an estimate of the chromatic class of a p graph (in russian). Metody Diskret. Analiz. 3, 25--30.Google Scholar
Index Terms
- New Constructive Aspects of the Lovász Local Lemma
Recommendations
A constructive proof of the general lovász local lemma
The Lovász Local Lemma discovered by Erdős and Lovász in 1975 is a powerful tool to non-constructively prove the existence of combinatorial objects meeting a prescribed collection of criteria. In 1991, József Beck was the first to demonstrate that a ...
A constructive proof of the Lovász local lemma
STOC '09: Proceedings of the forty-first annual ACM symposium on Theory of computingThe Lovasz Local Lemma [2] is a powerful tool to prove the existence of combinatorial objects meeting a prescribed collection of criteria. The technique can directly be applied to the satisfiability problem, yielding that a k-CNF formula in which each ...
Deterministic algorithms for the Lovász Local Lemma
SODA '10: Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete algorithmsThe Lovász Local Lemma [5] (LLL) is a powerful result in probability theory that states that the probability that none of a set of bad events happens is nonzero if the probability of each event is small compared to the number of events that depend on ...
Comments