Fast Approximation Algorithms for p-Centers in Large $$\delta $$ δ -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 \(O(p(\delta +1)(n+m)\log _2(n))\) 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 \(O((n^3\log _2 n + n^2m)\log _2({{\mathrm{diam}}}(G)))\) which is impractical for large graphs.