[Paper Review] Fast approximation algorithms for $p$-centres in large $\delta$-hyperbolic graphs

Katherine Edwards, William S. Kennedy+1 more|arXiv (Cornell University)|Apr 25, 2016

Markov Chains and Monte Carlo Methods8 references3 citations

TL;DR

This paper presents a quasilinear-time approximation algorithm for the p-center problem in large δ-hyperbolic graphs, achieving an additive error of at most 3δ from the optimal radius. By leveraging distance-approximating trees and a novel packing-based approach, the algorithm runs in O(p(δ + 1)(n + m) log n) time, enabling practical deployment on massive graphs with hundreds of thousands to millions of nodes.

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.

Motivation & Objective

• To develop a scalable p-center approximation algorithm for large-scale graphs where traditional methods are computationally infeasible.
• To reduce the time complexity of p-center approximation from cubic to quasilinear while maintaining a bounded additive error.
• To leverage the structural properties of δ-hyperbolic graphs—specifically their tree-like metric—to design efficient approximation algorithms.
• To provide a practical alternative to existing methods that are too slow for real-world large networks with small hyperbolic constants.

Proposed method

• Uses a distance-approximating tree T with O(δ log n) distortion to embed the original graph into a tree structure.
• Applies a greedy 2-approximation algorithm to find an initial (p+1)-packing in the approximating tree T.
• Employs a recursive refinement process using geodesic segments and ball carving to iteratively improve the p-center set.
• Maintains a packing of p+1 points with dispersion at least dp+1(G) − O(δ log n) to ensure proximity to optimal solution.
• Uses the dual problem of p-packings to guide the construction of p-centers via iterative refinement of diagonal pairs.
• Combines theoretical guarantees from hyperbolic geometry with algorithmic techniques from metric tree approximation to achieve quasilinear complexity.

Experimental results

Research questions

• RQ1 Can the p-center problem be solved in quasilinear time for large δ-hyperbolic graphs while maintaining a bounded additive error?
• RQ2 What is the minimal additive error achievable in p-center approximation for δ-hyperbolic graphs using subcubic algorithms?
• RQ3 How does the performance of the proposed algorithm compare to prior methods in terms of both accuracy and runtime on real-world networks?
• RQ4 Can distance-approximating trees be used effectively to reduce the complexity of p-center approximation in hyperbolic graphs?
• RQ5 Is the proposed algorithm practically viable for large-scale graphs with p in the range 10–10^4 and n up to 10^9?

Key findings

• The proposed algorithm runs in O(p(δ + 1)(n + m) log n) time, achieving a radius of at most rp + 3δ for p ≥ 3 and rp + δ for p ≤ 2.
• For p ≤ 2, the algorithm achieves an additive error of δ, matching prior best results but with significantly improved time complexity.
• For p ≥ 3, the algorithm achieves a 3δ additive error, which is a trade-off enabling quasilinear runtime.
• Empirical results show that the estimated p-radius from the proposed algorithm is always within 1 unit of the Chepoi-Estellon method’s estimate, despite a larger theoretical error bound.
• On the web-stanford graph (255,265 nodes), the algorithm completed in under two seconds, while the Chepoi-Estellon method was infeasible due to high runtime.
• The tree-approximation heuristic, though theoretically bounded by O(δ log n) error, performed well in practice and offers a fast alternative with quasilinear time complexity.