Mittagsseminar (in cooperation with J. Lengler, A. Steger, and D. Steurer)

Date and Time: Tuesday, July 17, 2007, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Tobias Christ

On Dinur's Proof of the PCP Theorem - Part I

In this talk we will give a sketch of Inrit Dinur's new (2005) proof of the PCP theorem with a special focus on how expander graphs are used in her construction. Dinur does not prove the PCP theorem itself, but an equivalent version of it, the NP-hardness of approximating a constraint satisfaction problem given by a constraint graph.

Consider the unsat value of a constraint graph, i.e. the minimal fraction of unsatisfied constraints. We perform a combinatorial transformation of the constraint graph to double the unsat value, while increasing the size of the graph only linearly. The core of this construction is a special notion of graph powering for constraint satisfaction problems. For this "amplification of the gap" to work out, it is crucial that the underlying graph is an expander. So before performing the powering, the graph has to be preprocessed. And since the powering increases the alphabet size of the constraint graph, we have to reduce it again making sure not to lose too much of the unsat value. Repeating the whole procedure logarithmically many times finally gives a new constraint graph in polynomial time, whose unsat value either still is 0 if the original problem was satisfiable or it will be larger than some constant if it was unsatisfiable at the beginning.

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