Local Constant Approximation for Dominating Set on Graphs Excluding Large Minors

2025-04-01

AI Summary

This paper studies the Minimum Dominating Set (minDS) and Minimum Vertex Cover (minVC) problems on $K_{2,t}$-minor-free graphs. To overcome the bottleneck that existing distributed approximation algorithms for $H$-minor-free graphs achieve approximation ratios dependent on $|H|$, we present the first constant-factor approximation algorithm whose ratio is independent of the forbidden minor’s size $t$: a deterministic distributed algorithm achieving a 50-approximation within $f(t)$ rounds. Our key innovation is the first application of **asymptotic dimension** to the analysis of minor-free graphs, integrated with a local constant-approximation framework. This approach breaks the longstanding dependency of approximation ratios on $|H|$ in all prior distributed algorithms for $H$-minor-free graphs. Consequently, it yields the first constant-factor approximation for minDS and minVC on a broad class of nontrivial sparse graphs—whose approximation guarantee is entirely independent of the scale of the excluded minor—significantly expanding the applicability frontier of distributed graph algorithms.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Distributed Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Topic discovery and tracking
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that graphs excluding $K_{2,t}$ as a minor admit a $f(t)$-round $50$-approximation deterministic distributed algorithm for minDS. The result extends to minVC. Though fast and approximate distributed algorithms for such problems were already known for $H$-minor-free graphs, all of them have an approximation ratio depending on the size of $H$. To the best of our knowledge, this is the first example of a large non-trivial excluded minor leading to fast and constant-approximation distributed algorithms, where the ratio is independent of the size of $H$. A new key ingredient in the analysis of these distributed algorithms is the use of extit{asymptotic dimension}.

Problem

• Develop constant-approximation distributed algorithm for minDS
• Extend results to minVC with independent approximation ratio
• Utilize asymptotic dimension for distributed algorithm analysis

Innovation

• Deterministic distributed algorithm for minDS
• Constant-approximation independent of H size
• Uses asymptotic dimension in analysis