Fast approximation algorithms for $p$-centres in large $\delta$-hyperbolic graphs

TL;DR

This paper introduces a quasilinear time algorithm for the $p$-center problem in large $elta$-hyperbolic graphs, achieving an additive error proportional to the hyperbolic constant, significantly improving efficiency over prior methods.

Contribution

The authors develop a fast, practical approximation algorithm for the $p$-center problem in large $elta$-hyperbolic graphs with provable error bounds, surpassing previous algorithms in efficiency.

Findings

Algorithm runs in $O(p(elta+1)(n+m)\u2217 $ time.

Achieves additive error of at most $3elta$ for $p \u2265 3$.

Outperforms prior algorithms with higher complexity and lower accuracy.

Abstract

We provide a quasilinear time algorithm for the $p$-center problem with an additive error less than or equal to 3 times the input graph's hyperbolic constant. Specifically, for the graph $G=(V,E)$ with $n$ vertices, $m$ edges and hyperbolic constant $\delta$, we construct an algorithm for $p$-centers in time $O(p(\delta+1)(n+m)\log(n))$ with radius not exceeding $r_p + \delta$ when $p \leq 2$ and $r_p + 3\delta$ when $p \geq 3$, where $r_p$ are the optimal radii. Prior work identified $p$-centers with accuracy $r_p+\delta$ but with time complexity $O((n^3\log n + n^2m)\log(diam(G)))$ which is impractical for large graphs.