Expander Decomposition with Almost Optimal Overhead
Nikhil Bansal, Arun Jambulapati, Thatchaphol Saranurak
TL;DR
The paper resolves a long-standing gap in expander decomposition by providing the first polynomial-time algorithm that achieves near-optimal flow-expander overhead, matching the existential lower bound up to a subpolynomial factor. It introduces a novel combination of concurrent-flow programming and a spreading-metric framework, along with a consensus-scale clustering mechanism and a heavy-cluster handling case, to decompose a graph into φ-flow-expanders while removing only a small fraction of edges. The main technical advance lies in telescoping the cost via invariant node-mass $A$, enabling a logarithmic boundary term to be balanced against a recursive cost that scales with the preserved mass, resulting in overhead $\gamma=O(\log n\exp(\sqrt{\log\log n}))$. The method extends to capacitated graphs and other variants, offering a near-optimal, practically applicable tool for routing, sparsification, and related graph algorithms, and opens questions about extending the approach to tree, vertex, and directed expander decompositions.
Abstract
We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $φ$, our algorithm removes at most a $φ\log^{1+o(1)}n$ fraction of edges so that every remaining connected component is a $φ$-\emph{flow}-expander (a stronger guarantee than being a $φ$-\emph{cut}-expander). This achieves overhead $\log^{1+o(1)}n$, nearly matching the $Ω(\log n)$ graph-theoretic lower bound that already holds for cut-expander decompositions, up to a $\log^{o(1)}n$ factor. Prior polynomial-time algorithms required removing $O(φ\log^{1.5}n)$ and $O(φ\log^{2}n)$ fractions of edges to guarantee $φ$-cut-expander and $φ$-flow-expander components, respectively.