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...
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Zusammenfassung: | 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. |
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DOI: | 10.48550/arxiv.2211.09729 |