Deterministic algorithms for the Lovász Local Lemma

The Lovász Local Lemma [6] (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 it. It is often used in combination with the probabilistic method for non-constructive existence proofs. A prominent application is to k-CNF formulas, where LLL implies that, if every clause in the formula shares variables with at most d < 2k/e other clauses then such the formula has a satisfying assignment. Recently, a randomized algorithm to efficiently construct a satisfying assignment was given by Moser [17]. Subsequently Moser and Tardos [18] gave a randomized algorithm to construct the structures guaranteed by the LLL in a very general algorithmic framework. We address the main problem left open by Moser and Tardos of derandomizing these algorithms efficiently. Specifically, for a k-CNF formula with m clauses and d < 2k/(l+)/e for some c E (0, 1), we give an algorithm that finds a satisfying assignment in time O(m2(1+1/E)). This improves upon the deterministic algorithms of Moser and of Moser- Tardos with running times mn (k2) and mD(k 1/c) which are superpolynomial for k = w(1) and upon other previous algorithms which work only for d </= 2k/ 16 /e. Our algorithm works efficiently for the asymmetric version of LLL under the algorithmic framework of Moser and Tardos [18] and is also parallelizable, i.e., has polylogarithmic running time using polynomially many processors.

Department dc:contributor.department

Massachusetts Institute of Technology. Dept. of Electrical Engineering and Computer Science.

Grantor dc:publisher

Massachusetts Institute of Technology

Year dc:date.issued

2010

Author dc:creator

• Haeupler, Bernhard

Advisor dc:contributor.advisor

• David R. Karger.

• Electrical Engineering and Computer Science.

Statement dc:rights

• M.I.T. theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission. See provided URL for inquiries about permission.

Licence dc:rights.uri

• http://dspace.mit.edu/handle/1721.1/7582

Language dc:language.iso

eng

Handle dc:identifier.uri

http://hdl.handle.net/1721.1/60165

OAI identifier oai:identifier

oai:dspace.mit.edu:1721.1/60165