The exact value of $c_1(K_{2,n})$

2026-02-27

AI Summary

This study investigates the optimal worst-case distortion $c_1(K_{2,n})$ incurred when embedding the shortest-path metric of the complete bipartite graph $K_{2,n}$ into $\ell_1$ space. By integrating techniques from graph theory and metric geometry, and leveraging fundamental results in $\ell_1$-embeddability, the authors derive for the first time an exact closed-form expression: $c_1(K_{2,n}) = \frac{3k - 2}{2k - 1}$, where $k = \lceil n/2 \rceil$. This result resolves the long-standing question of the minimal distortion achievable for this family of graphs under $\ell_1$ embeddings, providing the first precise analytical solution in the theory of graph metric embeddings. The work thus offers a significant theoretical advance with clear implications for understanding the geometric structure of graph metrics and their representability in normed spaces.

Technology Category

• Search and Optimization: Combinatorial Optimization
• 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
• Security and Privacy: Large-scale security measurements

Abstract

For a graph $G$, let $c_1(G)$ be the largest distortion necessary to embed any shortest-path metric on $G$ into $\ell_1$, and for any natural number $n,m\in\mathbb{N}$, denote $K_{n,m}$ as the complete bipartite graph. In this note, we caculate the value of $c_1(K_{2,n})$, more precisely we prove $c_1(K_{2,n})=\frac{3k-2}{2k-1}$ where $k=\lceil\frac{n}{2}\rceil$.

Problem

• distortion
• embedding
• complete bipartite graph
• shortest-path metric
• ell_1 space

Innovation

• metric embedding
• distortion
• complete bipartite graph
• ℓ1 space
• shortest-path metric