Efficient Distributed Decomposition and Routing Algorithms in Minor-Free Networks and Their Applications
Year: 2026
Type: article
Abstract: In the \(\textsf{LOCAL}\) model of distributed computing, low-diameter decomposition is a fundamental tool for algorithm design, as it enables a reduction from general graphs to low-diameter graphs where brute-force information gathering can be performed efficiently. Chang and Su [PODC 2022] showed that any high-conductance network excluding a fixed minor contains a high-degree vertex \(v^{\star}\) , allowing the entire graph topology to be gathered at \(v^{\star}\) efficiently in the \(\textsf{CONGEST}\) model via expander routing . Consequently, in such networks, many problems that admit efficient \(\textsf{LOCAL}\) algorithms via low-diameter decomposition can also be solved efficiently in \(\textsf{CONGEST}\) using expander decomposition . In this work, we present improved decomposition and routing algorithms for networks excluding a fixed minor. We define an \((\epsilon,D,T)\) -decomposition of a graph \(G=(V,E)\) as a partition of \( V \) into clusters of diameter at most \( D \) , with at most \(\epsilon|E|\) inter-cluster edges, such that information gathering within each cluster can be completed in \( T \) rounds in parallel. We show that an \((\epsilon,D,T)\) -decomposition with \(\begin{align*} D=O(\epsilon^{-1})\quad\text{and}\quad T=\min\left\{2^{O\left(\log^{2}\frac{1} {\epsilon}\right)}\cdot O(\log\Delta),\ \operatorname{poly}(\epsilon^{-1},\log \Delta)\right\} \nonumber\end{align*}\) can be computed deterministically in \(\begin{align*} O(\epsilon^{-1}\log^{\ast}n)+\min\left\{2^{O\left(\log^{2}\frac{1}{\epsilon} \right)}\cdot O(\log\Delta),\ \operatorname{poly}(\epsilon^{-1},\log\Delta)\right\}\nonumber \end{align*}\) rounds in the \(\textsf{CONGEST}\) model for networks excluding a fixed minor. Our algorithm has a wide range of applications, including the following results in \(\textsf{CONGEST}\) : — A \((1-\epsilon)\) -approximate maximum independent set in networks excluding a fixed minor can be computed deterministically in \(O(\epsilon^{-1}\log^{\ast}n)+\operatorname{poly}(\epsilon^{-1})\) rounds, nearly matching the \(\Omega(\epsilon^{-1}\log^{\ast}n)\) lower bound of Lenzen and Wattenhofer [DISC 2008]. — Property testing of any additive minor-closed property can be performed deterministically in \(O(\log n)\) rounds for constant \(\epsilon\) , or in \(O(\epsilon^{-1}\log n)+\operatorname{poly}(\epsilon^{-1})\) rounds for constant \(\Delta\) , nearly matching the \(\Omega(\epsilon^{-1}\log n)\) lower bound of Levi et al. [PODC 2018].
Source: ACM Transactions on Algorithms
DOI: 10.1145/3805031
Author: Yi-Jun Chang
Institution: National University of Singapore
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W7148733116