A Schur Complement Cheeger Inequality
Cheeger's inequality shows that any undirected graph G with minimum normalized Laplacian eigenvalue lambda_G has a cut with conductance at most O(sqrt{lambda_G}). Qualitatively, Cheeger's inequality says that if the mixing time of a graph is high, there is a cut that certifies this. However, this relationship is not tight, as some graphs (like cycles) do not have cuts with conductance o(sqrt{lambda_G}).
To better approximate the mixing time of a graph, we consider a more general object. Specifically, instead of bounding the mixing time with cuts, we bound it with cuts in graphs obtained by Schur complementing out vertices from the graph G. Combinatorially, these Schur complements describe random walks in G restricted to a subset of its vertices. As a result, all Schur complement cuts have conductance at least Omega(lambda_G). We show that unlike with cuts, this inequality is tight up to a constant factor. Specifically, there is a Schur complement cut with conductance at most O(lambda_G).
electrical networks
Cheeger's inequality
mixing time
conductance
Schur complements
Mathematics of computing~Spectra of graphs
65:1-65:15
Regular Paper
Aaron
Schild
Aaron Schild
University of California, Berkeley, CA, USA
Supported by NSF grant CCF-1816861.
10.4230/LIPIcs.ITCS.2019.65
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Aaron Schild
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