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 at most $\epsilon m$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. [ADK23] 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 $(S,\bar{S})$ is its $\mu$-expansion $\Phi^{\mu}_G(S,\bar{S}) = |E(S,\bar{S})|/\min \mu(S)),\mu(\bar{S})\}$, where $\mu(S) = \sum_{v\in S} \mu(v)$. We present a randomized $\tilde{O}(m)$ time algorithm for computing a $(\phi, \phi \log^2 {n}\left(\frac{\mu(V)}{m}\right))$-expander decomposition with respect to $\mu$-expansion.