Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial

TL;DR

This paper presents simple, faster, combinatorial algorithms for the minimum dominating set problem in graphs with bounded arboricity, achieving optimal approximation factors in both centralized and distributed settings without LP reliance.

Contribution

It introduces the first combinatorial $O( ext{arboricity})$-approximation algorithms that are faster and simpler than previous LP-based methods, in both centralized and distributed models.

Findings

Achieved an $O( ext{arboricity})$-approximation in linear time in the centralized setting.

Designed a distributed $O( ext{arboricity})$-approximation algorithm running in $O( ext{arboricity} imes ext{log} n)$ rounds.

Provided algorithms that are both simpler and faster than prior LP-based approaches.

Abstract

We revisit the minimum dominating set problem on graphs with arboricity bounded by $\alpha$. Bansal and Umboh [BU17] gave an $O(\alpha)$-approximation LP rounding algorithm, which also translates into a near-linear time algorithm using general-purpose approximation results for explicit mixed packing and covering or pure covering LPs [KY14, You14, AZO19, Qua10]. Moreover, [BU17] showed that it is NP-hard to achieve an asymptotic improvement for the approximation factor. 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 $O(\alpha^2)$ in linear time. There is a similar situation in the distributed setting: While there is an $O(\log^2 n)$-round LP-based $O(\alpha)$-approximation algorithm implied in [KMW06], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an implementation of their centralized algorithm, providing an $O(\alpha^2)$-approximation within $O(\log n)$ rounds. We address the questions of whether one can achieve an $O(\alpha)$-approximation algorithm that is not LP-based, either in the centralized setting or in the distributed setting. We resolve both questions in the affirmative, and en route achieve algorithms that are faster than the state-of-the-art LP-based algorithms. 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 $O(\alpha)$-approximation in linear time. 2. Based on our centralized algorithm, we design a distributed combinatorial $O(\alpha)$-approximation algorithm in the CONGEST model that runs in $O(\alpha\log n )$ rounds with high probability.