A Cheeger Inequality for Small Set Expansion
Combinatorics
2023-04-19 v2 Discrete Mathematics
Abstract
The discrete Cheeger inequality, due to Alon and Milman (J. Comb. Theory Series B 1985), is an indispensable tool for converting the combinatorial condition of graph expansion to an algebraic condition on the eigenvalues of the graph adjacency matrix. We prove a generalization of Cheeger's inequality, giving an algebraic condition equivalent to small set expansion. This algebraic condition is the p-to-q hypercontractivity of the top eigenspace for the graph adjacency matrix. Our result generalizes a theorem of Barak et al (STOC 2012) to the low small set expansion regime, and has a dramatically simpler proof; this answers a question of Barak (2014).
Cite
@article{arxiv.2304.07612,
title = {A Cheeger Inequality for Small Set Expansion},
author = {Akhil Jalan},
journal= {arXiv preprint arXiv:2304.07612},
year = {2023}
}