Separator Theorem for Minor-Free Graphs in Linear Time

TL;DR

The paper resolves a long-standing open problem by giving a linear-time algorithm that, for any fixed h, either finds a balanced separator of size O(poly(h) sqrt(n)) in K_h-minor-free graphs or outputs a K_h minor. The approach hinges on a novel combination of vertex-weighted BFS, a vertex-weighted KPR decomposition, and a stochastic-connector framework that ties separator size to minor existence, with the capacity to certify a minor model when a small separator cannot be found. The authors provide precise bounds (separator size O(h^{13} sqrt(n)) and running time O(h^{13} n)) and show how to obtain linear-time results for genus-bounded graphs as a corollary, while also offering a reduction to sparse graphs to maintain efficiency. This work advances the practical generalization of Lipton-Tarjan's planar separator theory to broader minor-free graph classes and introduces techniques, such as the stochastic connector, that may have independent interest for sparsest-cut related problems.

Abstract

The planar separator theorem by Lipton and Tarjan [FOCS '77, SIAM Journal on Applied Mathematics '79] states that any planar graph with $n$ vertices has a balanced separator of size $O(\sqrt{n})$ that can be found in linear time. This landmark result kicked off decades of research on designing linear or nearly linear-time algorithms on planar graphs. In an attempt to generalize Lipton-Tarjan's theorem to nonplanar graphs, Alon, Seymour, and Thomas [STOC '90, Journal of the AMS '90] showed that any minor-free graph admits a balanced separator of size $O(\sqrt{n})$ that can be found in $O(n^{3/2})$ time. The superlinear running time in their separator theorem is a key bottleneck for generalizing algorithmic results from planar to minor-free graphs. Despite extensive research for more than two decades, finding a balanced separator of size $O(\sqrt{n})$ in (linear) $O(n)$ time for minor-free graphs remains a major open problem. Known algorithms either give a separator of size much larger than $O(\sqrt{n})$ or have superlinear running time, or both. In this paper, we answer the open problem affirmatively. Our algorithm is very simple: it runs a vertex-weighted variant of breadth-first search (BFS) a constant number of times on the input graph. Our key technical contribution is a weighting scheme on the vertices to guide the search for a balanced separator, offering a new connection between the size of a balanced separator and the existence of a clique-minor model. We believe that our weighting scheme may be of independent interest.

Figures (2)

• Figure 1: An algorithm for finding a $(1-\frac{1}{200h^2})$-balanced separator with size $O(\operatorname{poly}(h)\sqrt{n})$ of a $K_h$-minor-free graph $G$. $T_{i}(v)$ denotes the vertex set of the subtree of $T_{i}$ rooted at $v$.
• Figure 2: An algorithm that finds a $(1-\frac{1}{200h^2})$-balanced separator with size $O(\operatorname{poly}(h)\sqrt{n})$ of $G$ if exists, or outputs a $K_h$-minor model otherwise.

Theorems & Definitions (36)

• Theorem 1: Alon, Seymour, and Thomas AST90ConfAST90
• Theorem 2
• Lemma 1: Vertex-Weighted KPR
• Definition 1: Stochastic Connector
• Lemma 2: Corresponds to Lemma 4.3 in KL24
• Lemma 3
• proof
• Theorem 2
• Lemma 4
• Lemma 5