Dynamic algorithms for k-center on graphs

TL;DR

This paper introduces the first efficient dynamic graph algorithms for the NP-hard k-center problem, achieving near-optimal approximation ratios with sublinear update times, advancing the understanding of dynamic clustering.

Contribution

It provides the first deterministic and randomized algorithms for dynamic k-center on weighted graphs with near-linear amortized update times, and a reduction to fully dynamic algorithms.

Findings

Deterministic decremental $(2+psilon)$-approximation algorithm.

Randomized incremental $(4+psilon)$-approximation algorithm.

Reduction to fully dynamic $(2+psilon)$-approximation with near-optimal update time.

Abstract

In this paper we give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. In this problem, the goal is to partition the input into $k$ sets by choosing $k$ centers such that the maximum distance from any data point to its closest center is minimized. It is known that it is NP-hard to get a better than $2$ approximation for this problem. While in many applications the input may naturally be modeled as a graph, all prior works on $k$-center problem in dynamic settings are on point sets in arbitrary metric spaces. In this paper, we give a deterministic decremental $(2+\epsilon)$-approximation algorithm and a randomized incremental $(4+\epsilon)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+\epsilon)$-approximation algorithm for the $k$-center problem, with worst-case update time that is within a factor $k$ of the state-of-the-art fully dynamic $(1+\epsilon)$-approximation single-source shortest paths algorithm in graphs. Matching this bound is a natural goalpost because the approximate distances of each vertex to its center can be used to maintain a $(2+\epsilon)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.