Simple Combinatorial Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs
share
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.
READ FULL TEXT
