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Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player

Daniel Agassy, Dani Dorfman, Haim Kaplan

TL;DR

This work advances expander decompositions by introducing a non-stop spectral cut-player that extends OSVV-style ideas directly to conductance, avoiding subdivision tricks used in earlier work. The authors integrate this spectral cut-player into SW's cut-matching framework to compute a $(\phi,\phi\log^2 n)$-expander decomposition in $\tilde{O}(m/\phi)$ time, bringing inter-cluster edges to within a $O(\log n)$ factor of the optimum. The approach handles shrinking game graphs and uses a potential-based analysis to guarantee that the remaining piece becomes a near-expander, while embedding arguments bound congestion and preserve near-linear running time. The result is a near-optimal trade-off between conductance guarantees and inter-cluster edge counts, with potential applications to sparsification and fast graph algorithms such as solvers and sparsifiers.

Abstract

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 at most $εm$ inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized $\tilde{O}(m/φ)$ time algorithm for computing a $(φ, φ\log^2 {n})$-expander decomposition. This improves upon the $(φ, φ\log^3 {n})$-expander decomposition also obtained in $\tilde{O}(m/φ)$ time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal. One crucial component of SW's algorithm is non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW.

Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player

TL;DR

This work advances expander decompositions by introducing a non-stop spectral cut-player that extends OSVV-style ideas directly to conductance, avoiding subdivision tricks used in earlier work. The authors integrate this spectral cut-player into SW's cut-matching framework to compute a -expander decomposition in time, bringing inter-cluster edges to within a factor of the optimum. The approach handles shrinking game graphs and uses a potential-based analysis to guarantee that the remaining piece becomes a near-expander, while embedding arguments bound congestion and preserve near-linear running time. The result is a near-optimal trade-off between conductance guarantees and inter-cluster edge counts, with potential applications to sparsification and fast graph algorithms such as solvers and sparsifiers.

Abstract

A -expander-decomposition of a graph (with vertices and edges) is a partition of into clusters with conductance , such that there are at most inter-cluster edges. Such a decomposition plays a crucial role in many graph algorithms. We give a randomized time algorithm for computing a -expander decomposition. This improves upon the -expander decomposition also obtained in time by [Saranurak and Wang, SODA 2019] (SW) and brings the number of inter-cluster edges within logarithmic factor of optimal. One crucial component of SW's algorithm is non-stop version of the cut-matching game of [Khandekar, Rao, Vazirani, JACM 2009] (KRV): The cut player does not stop when it gets from the matching player an unbalanced sparse cut, but continues to play on a trimmed part of the large side. The crux of our improvement is the design of a non-stop version of the cleverer cut player of [Orecchia, Schulman, Vazirani, Vishnoi, STOC 2008] (OSVV). The cut player of OSSV uses a more sophisticated random walk, a subtle potential function, and spectral arguments. Designing and analysing a non-stop version of this game was an explicit open question asked by SW.
Paper Structure (30 sections, 71 theorems, 103 equations, 8 algorithms)

This paper contains 30 sections, 71 theorems, 103 equations, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Approximating conductance via cut-matching
  4. KRV's Cut-Matching Game for Conductance
  5. OSVV's Cut-Matching Game for Conductance
  6. Expander decomposition via spectral Cut-Matching
  7. SW's Cut-Matching for expander decomposition
  8. Our contribution: Spectral cut player for expander decomposition
  9. Cut player
  10. Matching player
  11. Analysis of the spectral cut player
  12. The Algorithm
  13. $F\IfNoValueTF{t}{}{_{t}}\IfNoValueTF{-NoValue-}{}{(-NoValue-)}$ is embeddable in $G$
  14. $A_T$ is a near expander in $F\IfNoValueTF{T}{}{_{T}}\IfNoValueTF{-NoValue-}{}{(-NoValue-)}$
  15. Proof of Theorem \ref{['theorem:our']}
  16. ...and 15 more sections

Key Result

Lemma 2.7

Let $G=(V,E)$ be an undirected graph with $n=|V|$. Denote by $N\in\mathop{\mathbb R}^{n\times n}$ its normalized Laplacian, and denote by $0=\nu_1\le\nu_2\le \ldots \le \nu_n$ the eigenvalues of the normalized Laplacian. Then, $\Phi(G)\ge \frac{\nu_2}{2}$.

Theorems & Definitions (149)

  • Definition 2.1: $d_G\IfNoValueTF{-NoValue-}{}{(-NoValue-)}$,$\mathop{\mathrm{\mathbf{vol}}}\nolimits_G(S)$
  • Definition 2.2: $G\{A\}$
  • Definition 2.3: $d$-Matching
  • Definition 2.4: $d_G\IfNoValueTF{-NoValue-}{}{(-NoValue-)}$-stochastic
  • Definition 2.5: Laplacian, Normalized Laplacian
  • Definition 2.6: Conductance
  • Lemma 2.7: Cheeger Bound cheeger1970lower
  • Definition 2.8: Expander, Near-Expander
  • Definition 2.9: Embedding
  • Lemma 2.10
  • ...and 139 more