Expander Decomposition for Non-Uniform Vertex Measures
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866917091627499520 |
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| author | Agassy, Daniel Dorfman, Dani Kaplan, Haim |
| author_facet | Agassy, Daniel Dorfman, Dani Kaplan, Haim |
| contents | A $(ϕ,ε)$-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 $Φ(G[V_i]) \ge ϕ$, such that there are $O(εm)$ 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 $(ϕ, ϕ\log^2 {n})$-expander decomposition.
In this paper we generalize this result for a broader notion of expansion. Let $μ\in \mathbb{R}_{\ge 0 }^n$ be a vertex measure. A standard generalization of conductance of a cut $(S,\overline{S})$ is its $μ$-expansion $Φ^μ_G(S,\overline{S}) = |E(S,\overline{S})|/\min \{μ(S),μ(\overline{S})\}$, where $μ(S) = \sum_{v\in S} μ(v)$.
We present a randomized $\tilde{O}(m)$ time algorithm for computing a $(ϕ, ϕ\log^2 {n}\cdot\frac{μ(V)}{m})$-expander decomposition with respect to $μ$-expansion. A substantial portion of the exposition is adapted from ADK, and this work serves as a convenient reference for generalized expander decomposition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_23913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Expander Decomposition for Non-Uniform Vertex Measures Agassy, Daniel Dorfman, Dani Kaplan, Haim Data Structures and Algorithms A $(ϕ,ε)$-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 $Φ(G[V_i]) \ge ϕ$, such that there are $O(εm)$ 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 $(ϕ, ϕ\log^2 {n})$-expander decomposition. In this paper we generalize this result for a broader notion of expansion. Let $μ\in \mathbb{R}_{\ge 0 }^n$ be a vertex measure. A standard generalization of conductance of a cut $(S,\overline{S})$ is its $μ$-expansion $Φ^μ_G(S,\overline{S}) = |E(S,\overline{S})|/\min \{μ(S),μ(\overline{S})\}$, where $μ(S) = \sum_{v\in S} μ(v)$. We present a randomized $\tilde{O}(m)$ time algorithm for computing a $(ϕ, ϕ\log^2 {n}\cdot\frac{μ(V)}{m})$-expander decomposition with respect to $μ$-expansion. A substantial portion of the exposition is adapted from ADK, and this work serves as a convenient reference for generalized expander decomposition. |
| title | Expander Decomposition for Non-Uniform Vertex Measures |
| topic | Data Structures and Algorithms |
| url | https://arxiv.org/abs/2510.23913 |