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Fast Algorithms for Directed Graph Partitioning Using Flows and Reweighted Eigenvalues
We consider a new semidefinite programming relaxation for directed edge expansion, which is obtained by adding triangle inequalities to the reweighted eigenvalue formulation.
Cheeger Inequalities for Directed Graphs and Hypergraphs Using Reweighted Eigenvalues
The first main result is a Cheeger inequality relating the vertex expansion $\vec{\psi}(G)$ of a directed graph $G$ to the vertex-capacitated maximum reweighted second eigenvalue $\vec{\lambda}_2^{v*}$: \[ \vec{\lambda}_2^{v*} \lesssim \vec{\psi}(G) \lesssim \sqrt{\vec{\lambda}_2^{v*} \cdot \log (\Delta/\vec{\lambda}_2^{v*})}.
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