Dynamic algorithms for k-center on graphs

TL;DR

This work delivers the first efficient dynamic algorithms for the graph-based $k$-center problem under edge updates. Itleverages a reduction to k-bounded maximal independent sets on a family of threshold graphs, and combines this with dynamic shortest-path tools to maintain near-optimal centers. The paper presents fully dynamic $(2+\epsilon)$-approximation algorithms (deterministic for uniform weights and randomized for general weights) with near-linear in $k$ and polylog factors in $n$ update times, alongside incremental and decremental variants achieving $(4+\epsilon)$ and $(2+\epsilon)$-approximations, respectively. The approach hinges on maintaining small dominating sets and ruling sets on compact subgraphs, plus efficient partially dynamic SSSP subroutines, enabling practical dynamic clustering on graphs and enabling applications such as dynamic diameter approximation via maintained distances.

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 $(2+ε)$-approximation algorithm and a randomized incremental $(4+ε)$-approximation algorithm, both with amortized update time $kn^{o(1)}$ for weighted graphs. Moreover, we show a reduction that leads to a fully dynamic $(2+ε)$-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 $(1+ε)$-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 $(2+ε)$-approximation of the graph diameter and the fastest known algorithms for such a diameter approximation also rely on maintaining approximate single-source distances.

Theorems & Definitions (58)

• theorem 1.1
• theorem 1.2
• theorem 1.3
• Definition 3.1: $k$-center on graph
• theorem 3.2: Decremental $(1+\epsilon)$-SSSP, bernstein2021deterministic
• theorem 3.3: Incremental $(1+\epsilon)$-SSSP, HKN2014hopsetschechik2018LackiN22
• Definition 4.1: $k$-bounded $(\alpha, \beta)$-ruling set problem
• Definition 4.2: $r$-threshold graph
• lemma 4.2
• lemma 4.3