Separator Theorem for Minor-Free Graphs in Linear Time

2025-12-01

AI Summary

For the balanced separator problem on $H$-minor-free graphs, this paper presents the first linear-time $O(n)$ algorithm that constructs a balanced separator of size $O(sqrt{n})$, breaking a three-decade-old dual bottleneck in both time complexity and separator size. Methodologically, we introduce a novel framework based on vertex-weighted breadth-first search (BFS): through a constant number of carefully designed weighted BFS traversals—guided by a weight assignment scheme tied to clique-minor structure—we efficiently identify high-quality separator vertices. Compared to prior state-of-the-art algorithms—which either require $Omega(n^{3/2})$ time or yield separators of size $omega(sqrt{n})$—our approach achieves essential improvements in both runtime efficiency and separator quality. This result provides a foundational tool for divide-and-conquer algorithms, graph embeddings, and partitioning tasks on minor-free graphs.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Search
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Generalizing planar separator theorem to minor-free graphs
• Finding balanced O(sqrt(n)) separators in linear time
• Overcoming superlinear time bottleneck for minor-free graphs

Innovation

• Vertex-weighted BFS for balanced separator search
• Weighting scheme linking separator size to clique-minor model
• Linear-time algorithm for minor-free graphs using constant BFS runs