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

TL;DR

This paper tackles the problem of efficiently producing small, α-balanced vertex separators in broad graph classes by introducing and analyzing a simple polynomial-time method called reweighted spectral partitioning (RSP).RSP is a rounding procedure for a semidefinite program γ^{(n)}(G), then dimension-reduces to a 1D embedding and uses a sweep-based rounding to extract a separator, with optional partition-oracle support to improve guarantees.A refined Cheeger-style inequality is proved, linking vertex expansion ψ(G), γ^{(n)}(G), γ^{(1)}(G) and padded-decomposability α(G); this yields near-best separator bounds across planar, genus-g, and $K_h$-minor-free graph classes, plus novel bounds for ball-intersection and nearest-neighbor geometries.Algorithmically, the paper provides existential, padded-partition, and no-oracle guarantees, achieving $O( ext{√}n)$ planar separators and sublinear-in-n bounds in other hereditary classes, while offering a geometric–spectral framework that unifies several prior results.The work also contributes independent results, including improved Fiedler-type bounds for genus graphs, new circle-packing-based arguments, and a planar/separator-theorem-style spectral proof, highlighting the practical potential of a simple, graph-only input algorithm.

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, $O(\min\{(\log g)^2,\logΔ\}\cdot\sqrt{gn})$-sized separators for genus-$g$ graphs of maximum degree $Δ$, and $O(\min\{\log h,\sqrt{\logΔ}\}(h\log h\log\log h)\sqrt{n})$-sized separators for $K_h$-minor-free graphs of maximum degree $Δ$. 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.

Figures (12)

• Figure 1: Examples of vertex separators produced by reweighted spectral partitioning: A $100$-vertex grid graph (left), a $50$-vertex random planar graph (centre), and $50$-vertex $2$-dimensional $5$-nearest-neighbour graph (right). The separator vertices are the large red ones.
• Figure 2: Two examples sets of balls, and the corresponding geometric intersection graphs. The left example has ply $1$, while the right example has ply $2$.
• Figure 3: Illustration of a geodesic ball (sphere cap) on the unit sphere $S^2$ embedded in $\mathbb{R}^3$, along with its corresponding centre $f(v)$. The distance $s(v)$ is the longest Euclidean distance from $f(v)$ to any point in the ball.
• Figure 4: Cross-section of a unit ball, and the different associated quantities.
• Figure 5: Non-cellular embeddings: planar graph (left) and $K_5$ in the flat torus (right). A non-cellular face in each is traced.

Theorems & Definitions (84)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4: Refined Cheeger-Style Inequality for Vertex Expansion
• Theorem 1.5: Bounds on $\gamma^{(d)}$
• Corollary 1.6: Existential Separator Bounds
• Corollary 1.7: Separator Bounds via Reweighted Spectral Partitioning with Padded Partitions
• Corollary 1.8: Separator Bounds via Reweighted Spectral Partitioning with No Partition Oracle
• Theorem 2.1: Cheeger inequality for edge expansion alon1985lambda1mohar1989isoperimetricjerrum1988conductance
• Corollary 2.2