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 epsilon and soundness 1/2 is at least as difficult as Small-Set Expansion with completeness epsilon and soundness f(epsilon), for any function f(epsilon) which grows faster than (epsilon)^(1/2). 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.