Monday, July 07, 2014

Favorite Theorems: Compressing the Local Lemma

Not only did Moser give a near optimal construction, but he did so in my favorite STOC talk of the decade.


The Lovász local lemma informally states that if you have a large set of events with limited dependence and they individually have a reasonable chance of happening, then there is a positive probability that they all happen. Moser focused on the special case of Boolean formula satisfiability: If we have a k-CNF formula φ where each clause shares a variable with at most r other clauses with r<2k/8 then φ is satisfiable. Note the result does not depend on the number of variables or clauses.

Moser gave this simple algorithm and a proof using entropy that I recognized as a Kolmogorov complexity argument and so excited me that I immediately wrote up the Kolmogorov proof as a blog post.

Moser's paper kickstarted a whole research agenda, with tight bounds using even simpler algorithms (though more complex proofs) and various generalizations. A whole tutorial day of STOC 2014 focused on the local lemma describing research that directly or indirectly came from Moser's most beautiful construction.

3 comments:

  1. Will there be videos from this tutorial day?

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  2. Can someone summarize the tutorial talks in blog? Really, we need more informative blog posts from Computational Complexity, please.

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  3. Given the significance of Moser's results on the LLL, it indeed seems to be a big a loss for our field that his LinkedIn profile suggests that he is no longer involved in basic research.

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