Rounding via Low Dimensional Embeddings

A regular graph \(G = (V,E)\) is an \((\varepsilon,\gamma)\) small-set expander if for any set of vertices of fractional size at most \(\varepsilon\), at least \(\gamma\) of the edges that are adjacent to it go outside. In this paper, we give a unified approach to several known complexity-theoretic results on small-set expanders. In particular, we show: 1. Max-Cut: we show that if a regular graph \(G = (V,E)\) is an \((\varepsilon,\gamma)\) small-set expander that contains a cut of fractional size at least \(1-\delta\), then one can find in \(G\) a cut of fractional size at least \(1-O\left(\frac{\delta}{\varepsilon\gamma^6}\right)\) in polynomial time. 2. Improved spectral partitioning, Cheeger's inequality and the parallel repetition theorem over small-set expanders. The general form of each one of these results involves square-root loss that comes from certain rounding procedure, and we show how this can be avoided over small set expanders. Our main idea is to project a high dimensional vector solution into a low-dimensional space while roughly maintaining \(\ell_2^2\) distances, and then perform a pre-processing step using low-dimensional geometry and the properties of \(\ell_2^2\) distances over it. This pre-processing leverages the small-set expansion property of the graph to transform a vector valued solution to a different vector valued solution with additional structural properties, which give rise to more efficient integral-solution rounding schemes.