Online Graph Embedding in Star Graphs

2026-05-16

AI Summary

This study addresses the problem of online embedding of dynamic guest graphs onto a star-shaped host graph, with the objective of minimizing distortion in pairwise distances. The work presents the first optimal deterministic algorithm, achieving a competitive ratio of 1.5, and introduces a breakthrough randomized algorithm with an improved competitive ratio of $11/9 \approx 1.222$. Both algorithms are proven to be theoretically optimal through matching lower bounds. By integrating techniques from online algorithm design, competitive analysis, and probabilistic methods, this research not only attains the best-known competitive ratios for this setting but also establishes a foundational framework for tackling online embedding problems on more complex host graph topologies.

Abstract

Graph embedding is a fundamental problem of mapping nodes of a guest graph into a host graph while minimizing the distance distortion, with broad applications, including virtual network embeddings into physical topologies, VLSI design, or community detection in social networks. However, in many real-world applications the guest graph changes over time and the embedding can adapt to these changes (e.g. virtual machine migration in network embeddings). Static embeddings are inherently inefficient in comparison to adaptive embeddings, but it remains an unresolved algorithmic challenge to design efficient embedding algorithms that adapt to the demand on-the-fly, i.e., that are online. In this paper, we derive optimal deterministic and randomized online algorithms for the online graph embedding problem in star host graphs. This is an essential building block on the way to design algorithms for more complex host graphs, representing a single node and its neighborhood. We start by presenting a $1.5$-competitive deterministic algorithm and showing that no deterministic algorithm can perform better. Our main contribution is a randomized algorithm that achieves a significantly better competitive ratio of $11/9 \approx 1.222$. Both the deterministic and the randomized algorithms are optimal, which we prove by deriving tight lower bounds for the competitiveness of any algorithm.

Problem

• online graph embedding
• star graphs
• competitive ratio
• dynamic graph
• distance distortion

Innovation

• online graph embedding
• star graphs
• competitive ratio
• randomized algorithm
• deterministic algorithm