Low-Congestion Shortcuts for Graphs Excluding Dense Minors

TL;DR

This paper introduces a simple, constructive method to find low-congestion shortcuts in graphs excluding dense minors, leading to near-optimal distributed algorithms for key problems like MST and shortest paths.

Contribution

It provides a new, elementary proof and distributed construction algorithm for low-congestion shortcuts in graphs with excluded minors, generalizing and strengthening previous results.

Findings

Achieves $O( ilde{ ext{delta}} D)$ congestion and dilation bounds

Provides a distributed algorithm with $ ilde{ ext{O}}( ext{delta} D)$ rounds

Enables near-optimal distributed algorithms for fundamental problems

Abstract

We prove that any $n$-node graph $G$ with diameter $D$ admits shortcuts with congestion $O(\delta D \log n)$ and dilation $O(\delta D)$, where $\delta$ is the maximum edge-density of any minor of $G$. Our proof is simple, elementary, and constructive - featuring a $\tilde{\Theta}(\delta D)$-round distributed construction algorithm. Our results are tight up to $\tilde{O}(1)$ factors and generalize, simplify, unify, and strengthen several prior results. For example, for graphs excluding a fixed minor, i.e., graphs with constant $\delta$, only a $\tilde{O}(D^2)$ bound was known based on a very technical proof that relies on the Robertson-Seymour Graph Structure Theorem. A direct consequence of our result is that many graph families, including any minor-excluded ones, have near-optimal $\tilde{\Theta}(D)$-round distributed algorithms for many fundamental communication primitives and optimization problems including minimum spanning tree, minimum cut, and shortest-path approximations.