Gap Amplification for Small-Set Expansion via Random Walks

Abstract: 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 $\frac{1}{2}$ 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 $\sqrt{\epsilon}$. We achieve this amplification via random walks -- our gadget is the graph with adjacency matrix corresponding to a random walk 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.