### DIMACS Theoretical Computer Science Seminar

Title: Moser and Tardos meet Lovasz

Speaker: Kashyap Kolipaka, Rutgers University

Date: Wednesday, December 1, 2010 11:00-12:00pm

Location: CoRE Bldg, CoRE 431, Rutgers University, Busch Campus, Piscataway, NJ

Abstract:

Beck's early work gave an Efficient Version of the Variable Version of the Lovasz Local Lemma (LLL), but with compromised parameters. This was followed by several improvements (Noga Alon, Artur Czumaj and Christian Scheideler, Michael Molloy and Bruce Reed, Aravind Srinivasan). Most recently Moser and Moser and Tardos obtained asymptotically optimal results (in terms of the maximal degree), employing a remarkable and very natural argument.

As for the original (non-algorithmic, non-variable) version of LLL, Shearer gives the exact criterion when LLL applies. For a dependency structure $G$ let Shearer(G) be the set of those vectors $p=p_{1},\ldots,p_{n}$ of probabilities for which in every setting the LLL applies. We show that whenever $p\in \Shearer(G)/(1+\epsilon)$, the TM algorithm runs in expected time at most $n /\epsilon$. Thus, whenever LLL holds, it can be made efficient, not only asymptotically, and for equal probabilities as in TM, but always.

We prove this sharp statement, which improves upon the MT result, without compromising the elegance of their argument. We uncover new mathematics that highlights the connection between the efficient and non-efficient versions of LLL. The central object is a matrix associated with the independent sets of the dependency graph.

Joint work with Mario Szegedy, Rutgers University.