Gap Amplification in PCPs Using Lazy Random Walks

Abstract

We show an alternative implementation of the gap amplification step in Dinur’s [4] recent proof of the PCP theorem. We construct a product G ^t of a constraint graph G, so that if every assignment in G leaves an ε-fraction of the edges unsatisfied, then in G ^t every assignment leaves an Ω(tε)-fraction of the edges unsatisfied, that is, it amplifies the gap by a factor Ω(t). The corresponding result in [4] showed that one could amplify the gap by a factor $\Omega(\sqrt{t})$. More than this small quantitative improvement, the main contribution of this work is in the analysis. Our construction uses random walks on expander graphs with exponentially distributed length. By this we ensure that some random variables arising in the proof are automatically independent, and avoid some technical difficulties.