Reweighted Spectral Partitioning Works: A Simple Algorithm for Vertex Separators in Special Graph Classes

TL;DR

This paper introduces a simple polynomial-time reweighted spectral partitioning algorithm that effectively finds small vertex separators in various special graph classes, improving upon existing bounds.

Contribution

It provides a new spectral algorithm with refined Cheeger inequalities and direct bounds for specific graph classes, enhancing separator size guarantees.

Findings

Achieves small separators for planar graphs, genus-$g$ graphs, and $K_h$-minor-free graphs.

Provides a spectral proof of the planar separator theorem.

Offers improved bounds on the Fiedler value for genus-$g$ graphs.

Abstract

We establish that a simple polynomial-time algorithm that we call reweighted spectral partitioning obtains small 2/3-balanced vertex-separators for a number of graph classes, including $O(\sqrt{n})$-sized separators for planar graphs, $O(\min\{(\log g)^2,\log\Delta\}\cdot\sqrt{gn})$-sized separators for genus-$g$ graphs of maximum degree $\Delta$, and $O(\min\{\log h,\sqrt{\log\Delta}\}(h\log h\log\log h)\sqrt{n})$-sized separators for $K_h$-minor-free graphs of maximum degree $\Delta$. To accomplish this, we first obtain a refined form of a Cheeger-style inequality relating the vertex expansion of a graph and the solution to a semidefinite program defined over the graph. Then, to obtain the guarantees for specific graph classes, we derive direct bounds on the value of the semidefinite program. We also obtain several other results of independent interest, including an improved separator theorem for the intersection graphs of $d$-dimensional balls with bounded ply, a new bound on the Fiedler value of genus-$g$ graphs, and a new "spectral" proof of the planar separator theorem.