A Schur Complement Cheeger Inequality
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Cheeger's inequality shows that any undirected graph G with minimum nonzero normalized Laplacian eigenvalue λ_G has a cut with conductance at most O(√(λ_G)). Qualitatively, Cheeger's inequality says that if the relaxation time of a graph is high, there is a cut that certifies this. However, there is a gap in this relationship, as cuts can have conductance as low as Θ(λ_G). To better approximate the relaxation time of a graph, we consider a more general object. 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 Ω(λ_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(λ_G).
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