Fast Approximation Algorithms for p-Centers in Large $$\delta $$ δ -Hyperbolic Graphs

Katherine Edwards, W. Sean Kennedy, Iraj Saniee

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 $$G=(V,E)$$ G = ( V , E ) with n vertices, m edges and hyperbolic constant $$\delta $$ δ , we construct an algorithm for p-centers in time $$O(p(\delta +1)(n+m)\log (n))$$ O ( p ( δ + 1 ) ( n + m ) log ( n ) ) with radius not exceeding $$r_p + \delta $$ r p + δ when $$p \le 2$$ p ≤ 2 and $$r_p + 3\delta $$ r p + 3 δ when $$p \ge 3$$ p ≥ 3 , where $$r_p$$ r p are the optimal radii. Prior work identified p-centers with accuracy $$r_p+\delta $$ r p + δ but with time complexity $$O((n^3\log n + n^2m)\log ({{\mathrm{diam}}}(G)))$$ O ( ( n 3 log n + n 2 m ) log ( diam ( G ) ) ) which is impractical for large graphs.