Dynamic algorithms for k-center on graphs

• URL: http://arxiv.org/abs/2307.15557v2
• Date: Mon, 8 Jan 2024 22:56:20 GMT
• Title: Dynamic algorithms for k-center on graphs
• Authors: Emilio Cruciani, Sebastian Forster, Gramoz Goranci, Yasamin Nazari, Antonis Skarlatos
• Abstract summary: We give the first efficient algorithms for the $k$-center problem on dynamic graphs undergoing edge updates. We show a reduction that leads to a fully dynamic $(2+epsilon)$-approximation algorithm for the $k$-center problem.
• Score: 3.568439282784197
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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