Minor Excluded Network Families Admit Fast Distributed Algorithms

TL;DR

This paper develops fast distributed algorithms for network optimization problems on graphs excluding a fixed minor, achieving near-optimal round complexity by leveraging the Graph Structure Theorem to construct high-quality shortcuts.

Contribution

It introduces the first use of the Graph Structure Theorem in distributed algorithms, enabling efficient solutions for excluded minor graphs that surpass previous lower bounds.

Findings

Achieves  O(D^2) rounds for key problems on excluded minor graphs.

Shows existence of high-quality shortcuts in these graph families.

Utilizes the Graph Structure Theorem for algorithmic development.

Abstract

Distributed network optimization algorithms, such as minimum spanning tree, minimum cut, and shortest path, are an active research area in distributed computing. This paper presents a fast distributed algorithm for such problems in the CONGEST model, on networks that exclude a fixed minor. On general graphs, many optimization problems, including the ones mentioned above, require $\tilde\Omega(\sqrt n)$ rounds of communication in the CONGEST model, even if the network graph has a much smaller diameter. Naturally, the next step in algorithm design is to design efficient algorithms which bypass this lower bound on a restricted class of graphs. Currently, the only known method of doing so uses the low-congestion shortcut framework of Ghaffari and Haeupler [SODA'16]. Building off of their work, this paper proves that excluded minor graphs admit high-quality shortcuts, leading to an $\tilde O(D^2)$ round algorithm for the aforementioned problems, where $D$ is the diameter of the network graph. To work with excluded minor graph families, we utilize the Graph Structure Theorem of Robertson and Seymour. To the best of our knowledge, this is the first time the Graph Structure Theorem has been used for an algorithmic result in the distributed setting. Even though the proof is involved, merely showing the existence of good shortcuts is sufficient to obtain simple, efficient distributed algorithms. In particular, the shortcut framework can efficiently construct near-optimal shortcuts and then use them to solve the optimization problems. This, combined with the very general family of excluded minor graphs, which includes most other important graph classes, makes this result of significant interest.