# NuCDS: An Efficient Local Search Algorithm for Minimum Connected Dominating Set

IJCAI, pp. 1503-1510, 2020.

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Keywords:

minimum dominating setconnected dominating setmassive graphminimum dominating set problemwireless networkMore(12+)

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Abstract:

The minimum connected dominating set (MCDS) problem is an important extension of the minimum dominating set problem, with wide applications, especially in wireless networks. Despite its practical importance, there are few works on solving MCDS for massive graphs, mainly due to the complexity of maintaining connectivity. In this paper, we ...More

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Introduction

- The minimum dominating set (MDS) problem is to find a dominating set with the minimum number of vertices in the given graph.
- An important generalization of MDS is the minimum connected dominating set (MCDS) problem, whose goal is to find a minimum size dominating set that forms a connected subgraph in the given graph.
- An important application of MCDS is generating a virtual backbone in wireless networks such as mobile ad hoc networks [Al-Karaki and Kamal, 2008], wireless sensors networks [Misra and Mandal, 2009] and vehicular ad hoc networks [Chinnasamy et al, 2019].
- Several exact algorithms [Fomin et al, 2008; Simonetti et

Highlights

- Given an undirected connected graph G = (V, E), a set D ⊆ V is called a dominating set if each vertex in V either belongs to D or is adjacent to at least one vertex from D
- In order to solve the performance bottleneck problem caused by the connectivity constraint of minimum connected dominating set, we introduce a hybrid dynamic connectivity maintenance method (HDC for short)
- But we report the average run time when all algorithms obtain the same minimal and average values in Figure 2, which shows the effectiveness of NuCDS
- We proposed two new algorithmic components, namely the hybrid dynamic connectivity maintenance heuristic and the safety-based vertex selection heuristic, for minimum connected dominating set
- We conducted extensive benchmarks to evaluate the performance of NuCDS and the experimental results showed that our algorithm significantly outperforms its competitors on almost all the instances of any size
- Three modified versions of NuCDS are proposed to verify the effectiveness of HDC and saf ety, especially on massive graphs

Results

- Most instances of classical benchmarks are so easy that all algorithms obtain the same solution quality very quickly
- The authors ignore these instances, but the authors report the average run time when all algorithms obtain the same minimal and average values in Figure 2, which shows the effectiveness of NuCDS.
- NuCDS can solve all these 118 instances within the time limit, while MSLS, ACO-efficient, ACO-RVNS and RNS-TS can only solve 45, 48, 18, and 26 instances, respectively.
- The excellent results of NuCDS on massive graphs can mainly be attributed to the power of the HDC heuristic

Conclusion

- The authors proposed two new algorithmic components, namely the hybrid dynamic connectivity maintenance heuristic and the safety-based vertex selection heuristic, for MCDS.
- Both components were used to develop an efficient local search algorithm named NuCDS.
- As shown in Table 4, three modified versions of NuCDS are proposed to verify the effectiveness of HDC and saf ety, especially on massive graphs.
- The authors compared NuCDS with NuCDS1 to show the effectiveness of saf ety, and with

Summary

## Introduction:

The minimum dominating set (MDS) problem is to find a dominating set with the minimum number of vertices in the given graph.- An important generalization of MDS is the minimum connected dominating set (MCDS) problem, whose goal is to find a minimum size dominating set that forms a connected subgraph in the given graph.
- An important application of MCDS is generating a virtual backbone in wireless networks such as mobile ad hoc networks [Al-Karaki and Kamal, 2008], wireless sensors networks [Misra and Mandal, 2009] and vehicular ad hoc networks [Chinnasamy et al, 2019].
- Several exact algorithms [Fomin et al, 2008; Simonetti et
## Results:

Most instances of classical benchmarks are so easy that all algorithms obtain the same solution quality very quickly- The authors ignore these instances, but the authors report the average run time when all algorithms obtain the same minimal and average values in Figure 2, which shows the effectiveness of NuCDS.
- NuCDS can solve all these 118 instances within the time limit, while MSLS, ACO-efficient, ACO-RVNS and RNS-TS can only solve 45, 48, 18, and 26 instances, respectively.
- The excellent results of NuCDS on massive graphs can mainly be attributed to the power of the HDC heuristic
## Conclusion:

The authors proposed two new algorithmic components, namely the hybrid dynamic connectivity maintenance heuristic and the safety-based vertex selection heuristic, for MCDS.- Both components were used to develop an efficient local search algorithm named NuCDS.
- As shown in Table 4, three modified versions of NuCDS are proposed to verify the effectiveness of HDC and saf ety, especially on massive graphs.
- The authors compared NuCDS with NuCDS1 to show the effectiveness of saf ety, and with

- Table1: Results of NuCDS, MSLS, ACO-efficient, ACO-RVNS and RNS-TS on classical benchmarks
- Table2: Results on SNAP and DIMACS10 benchmarks. To save space, we denote ACO-RVNS and RNS-TS as ACO and RNS
- Table3: Results on the NDR benchmarks. To save space, we denote ACO-efficient and ACO-RVNS as ACOe and ACO, respectively. Moreover, since min and avg of all competitors are no better than that of NuCDS on all instances, we only report min of competitors
- Table4: Three modified versions of NuCDS, where ”+” indicates that the version uses the corresponding strategy while ”-” means not
- Table5: Comparing NuCDS with three modified versions on massive graphs. #Better and #Worse respectively represent the number of instances where NuCDS achieves better and worse result

Related work

- Because of its NP-hardness, much of the research effort in the past decade concerned with solving MCDS has focused on heuristics with the aim of obtaining a good solution within a reasonable time. Two algorithms called MCDS/SA and MCDS/TS based on simulated annealing and tabu search were proposed [Morgan and Grout, 2007]. Hedar and Ismail [2012] designed a simulated annealing algorithm with stochastic local search for MCDS. Later, Jovanovic and Tuba [2013] designed an ant colony optimization algorithm with a so-called pheromone correction strategy. A greedy random adaptive search procedure that incorporated a local search procedure based on a greedy function and tabu search was described in [Li et al, 2017]. Wu et al [2017] used a restricted swap-based neighborhood to improve the tabu search procedure, resulting in the RNS-TS algorithm. Two metaheuristics based on genetic algorithms and simulated annealing were designed to solve MCDS [Hedar et al, 2019]. Li et al [2019] presented a multi-start local search algorithm called MSLS based on three mechanisms including a vertex score, configuration checking, and vertex flipping. Finally, a meta-heuristic algorithm called ACO-RVNS [Bouamama et al, 2019] was proposed, based on ant colony optimization and reduced variable neighborhood search. Experiments show that, for classic graphs with fewer than 5000 vertices, RNS-TS, MSLS, and ACO-RVNS obtain similar state-of-theart performance.

Funding

- This work was supported by Beijing Academy of Artificial Intelligence (BAAI), Youth Innovation Promotion Association, Chinese Academy of Sciences [No 2017150], and NSFC Grant 61806050

Reference

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