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

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 $p\le 2$ and $r_p+3\delta$ when $p\ge 3$, where $r_p$ are the optimal radii. Prior work identified $p$-centers with accuracy $r_p+\delta$ but with time complexity $O((n^3\log n+n^{2}m)\log (diam(G)))$ which is impractical for large graphs.

19 pages