Expander Decomposition for Non-Uniform Vertex Measures

A $(\phi,\epsilon)$-expander-decomposition of a graph $G$ (with $n$ vertices and $m$ edges) is a partition of $V$ into clusters $V_1,\ldots,V_k$ with conductance $\Phi(G[V_i]) \ge \phi$, such that there are $O(\epsilon m)$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. [Agassy, Dorman, and Kaplan, ICALP 2023] (ADK) gave a randomized $\tilde{O}(m)$ time algorithm for computing a $(\phi, \phi\log^2 {n})$-expander decomposition. In this paper we generalize this result for a broader notion of expansion. Let $\mu \in \mathbb{R}_{\ge 0 }^n$ be a vertex measure. A standard generalization of conductance of a cut $(S,\overline{S})$ is its $\mu$-expansion $\Phi^{\mu}_G(S,\overline{S}) = |E(S,\overline{S})|/\min \{\mu(S),\mu(\overline{S})\}$, where $\mu(S) = \sum_{v\in S} \mu(v)$. We present a randomized $\tilde{O}(m)$ time algorithm for computing a $(\phi, \phi \log^2 {n}\cdot\frac{\mu(V)}{m})$-expander decomposition with respect to $\mu$-expansion. A substantial portion of the exposition is adapted from ADK, and this work serves as a convenient reference for generalized expander decomposition.