Distributed Approximation Algorithms for Minimum Dominating Set in Locally Nice Graphs

2025-07-07

AI Summary

This paper addresses the distributed Minimum Dominating Set (MDS) problem on graphs embeddable on surfaces of Euler genus $g$. We propose a novel deterministic LOCAL-model algorithm that does not require prior knowledge of the graph’s embedding. By integrating planar-graph MDS techniques with asymptotic dimension analysis, our approach efficiently handles graphs of bounded genus—and more generally, graphs with bounded asymptotic dimension. Our main contributions are: (i) an improved approximation ratio of $34+varepsilon$, surpassing prior bounds of $24g+O(1)$ and $91+varepsilon$; (ii) round complexity $f(g)$, independent of input size; and (iii) the first embedding-oblivious algorithm achieving both this approximation guarantee and sublogarithmic round complexity for bounded-genus and bounded-asymptotic-dimension graph families. This unifies high accuracy and efficiency without relying on explicit surface embeddings.

Technology Category

• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

Abstract

We give a new, short proof that graphs embeddable in a given Euler genus-$g$ surface admit a simple $f(g)$-round $α$-approximation distributed algorithm for Minimum Dominating Set (MDS), where the approximation ratio $αle 906$. Using tricks from Heydt et al. [European Journal of Combinatorics (2025)], we in fact derive that $αle 34 +varepsilon$, therefore improving upon the current state of the art of $24g+O(1)$ due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of $91+varepsilon$ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces. All our distributed algorithms work in the deterministic LOCAL model. They do not require any preliminary embedding of the graph and only rely on two things: a LOCAL algorithm for MDS on planar graphs with ``uniform'' approximation guarantees and the knowledge that graphs embeddable in bounded Euler genus surfaces have asymptotic dimension $2$. More generally, our algorithms work in any graph class of bounded asymptotic dimension where ``most vertices'' are locally in a graph class that admits a LOCAL algorithm for MDS with uniform approximation guarantees.

Problem

• Improving approximation ratio for Minimum Dominating Set in bounded genus graphs
• Developing deterministic LOCAL algorithms without preliminary embedding
• Extending results to graph classes with bounded asymptotic dimension

Innovation

• Uses LOCAL model for distributed MDS algorithms
• Relies on planar graph approximation guarantees
• Leverages bounded asymptotic dimension properties