Simple Combinatorial Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs
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We revisit the minimum dominating set problem on graphs with arboricity bounded by α. In the (standard) centralized setting, Bansal and Umboh [BU17] gave an O(α)-approximation LP rounding algorithm. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement. On the other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer [LW10], and Jones et al. [JLR+13], achieve an approximation factor of O(α^2) in linear time. There is a similar situation in the distributed setting: While there are polylog n-round LP-based O(α)-approximation algorithms [KMW06, DKM19], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an O(α^2)-approximation within O(log n) rounds with high probability. We address the question of whether one can achieve a simple, elementary O(α)-approximation algorithm not based on any LP-based methods, either in the centralized setting or in the distributed setting. We resolve these questions in the affirmative. More specifically, our contribution is two-fold: 1. In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an O(α)-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial O(α)-approximation algorithm in the 𝖢𝖮𝖭𝖦𝖤𝖲𝖳 model that runs in O(αlog n ) rounds with high probability. Our round complexity outperforms the best LP-based distributed algorithm for a wide range of parameters.
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