Expander Decomposition with Almost Optimal Overhead

TL;DR

This paper introduces the first polynomial-time algorithm for near-optimal flow-expander decomposition, removing minimal edges to produce components with strong expansion guarantees, nearly matching theoretical lower bounds.

Contribution

It provides a novel polynomial-time algorithm achieving almost optimal overhead for flow-expander decomposition, improving upon prior methods with higher edge removal fractions.

Findings

Achieves overhead of log^{1+o(1)}n, close to the log n lower bound.

Removes at most phi\u007flog^{1+o(1)}n fraction of edges.

Produces phi ext{-}flow-expander components with minimal edge removal.

Abstract

We present the first polynomial-time algorithm for computing a near-optimal \emph{flow}-expander decomposition. Given a graph $G$ and a parameter $\phi$, our algorithm removes at most a $\phi\log^{1+o(1)}n$ fraction of edges so that every remaining connected component is a $\phi$-\emph{flow}-expander (a stronger guarantee than being a $\phi$-\emph{cut}-expander). This achieves overhead $\log^{1+o(1)}n$, nearly matching the $\Omega(\log n)$ 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 $O(\phi\log^{1.5}n)$ and $O(\phi\log^{2}n)$ fractions of edges to guarantee $\phi$-cut-expander and $\phi$-flow-expander components, respectively.