Gap Amplification for Small-Set Expansion via Random Walks

In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness ε and soundness 12 is at least as difficult as Small-Set Expansion with completeness ε and soundness f (ε), for any function f (ε) which grows faster than √ ε. We achieve this amplification via random walks – the output graph corresponds to taking random walks on the original graph. An interesting feature of our reduction is that unlike gap amplification via parallel repetition, the size of the instances (number of vertices) produced by the reduction remains the same. ∗University of California, Berkeley. Research supported byNSFCareerAward andAlfred Sloan P. Fellowship. Email: [email protected] †University of California, Berkeley. This material is based upon work supported by a Berkeley Chancellor’s Fellowship and the National Science Foundation Graduate Research Fellowship Program under Grant No. DGE 1106400. Email: [email protected]