# Two-goal Local Search and Inference Rules for Minimum Dominating Set

IJCAI, pp. 1467-1473, 2020.

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Keywords:

minimum dominating setBest from Multiple Selectionsmassive graphlocal search algorithmgoal localMore(11+)

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Abstract:

Minimum dominating set (MinDS) is a canonical NP-hard combinatorial optimization problem with applications. For large and hard instances one must resort to heuristic approaches to obtain good solutions within reasonable time. This paper develops an efficient local search algorithm for MinDS, which has two main ideas. The first one is a n...More

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Introduction

- One typical problem is to find a smallest group of influential individuals or a set of initial seeds in a social network, so that all participants can be reached with only one hop from the seeds.
- This problem is equivalent to finding a minimum dominating set (MinDS) for the network.
- The main focus of such algorithms is on theoretical aspects

Highlights

- Given an undirected graph G = (V, E), a dominating set D is a subset of vertices such that each vertex in V \ D has at least one adjacent vertex in D
- A number of works have been done on exact algorithms for minimum dominating set, which mainly focus on improving the upper bound of running time
- We propose a new local search framework for minimum dominating set (Algorithm 1)
- To improve the performance of local search for minimum dominating set on large instances, we propose three inference rules, which can fix a considerable portion of the vertices for massive graphs
- We develop a local search algorithm for minimum dominating set named FastDS (Algorithm 3), which is based on our two-goal framework and employs inference rules
- We proposed a two-goal local search framework for minimum dominating set and proposed three inference rules

Methods

- The authors evaluate FastDS on 7 benchmarks, including 4 standard benchmarks in the literatures and 3 massive benchmarks.

UDG: This is a widely used standard benchmark for MinDS [Hedar and Ismail, 2010; Potluri and Singh, 2011; Potluri and Singh, 2013; Chalupa, 2018]. - T1: This data set consists of 520 instances where each instance has two different weight functions [Jovanovic et al, 2010].
- The authors select these original graphs where the weight of each vertex is set to 1.
- There are 52 families, each of which contains 10 instances with the same size.
- The authors do not report the results on graphs with 50 vertices

Results

**Results on Standard Benchmarks**

Results on UDG, T1 and BHOSLIB benchmarks are reported in Tables 1, 2 and 3 respectively.- The DIMACS instances are so easy that CC2FS, FastMWDS and FastDS find the.
- Cit-HepPh 3078(3078.6) 3192(3209.3) 3087(3088.9) 3078(3078.9).
- Cit-HepTh 2935(2935.7) 2985(2993.4) 2944(2948.6) 2936(2936.1).
- CC2FS is dominated by FastMWDS on all the instances and not reported in Tables 5 and 6.
- FastDS performs best for all the massive benchmarks.
- It obtains the best solutions for 19 SNAP instances, 28 DIMACS10 instances and 63 Repository bn-human-B*1 1189968 (1189990) (1190752) (1190256) (1189877.3).

Conclusion

- The authors proposed a two-goal local search framework for MinDS and proposed three inference rules.
- The resulting algorithm FastDS is robust and efficient on standard benchmarks and massive benchmarks, and significantly outperforms state-ofthe-art algorithms on massive benchmarks.
- The authors would like to study the ideas for other subset problems

Summary

## Introduction:

One typical problem is to find a smallest group of influential individuals or a set of initial seeds in a social network, so that all participants can be reached with only one hop from the seeds.- This problem is equivalent to finding a minimum dominating set (MinDS) for the network.
- The main focus of such algorithms is on theoretical aspects
## Methods:

The authors evaluate FastDS on 7 benchmarks, including 4 standard benchmarks in the literatures and 3 massive benchmarks.

UDG: This is a widely used standard benchmark for MinDS [Hedar and Ismail, 2010; Potluri and Singh, 2011; Potluri and Singh, 2013; Chalupa, 2018].- T1: This data set consists of 520 instances where each instance has two different weight functions [Jovanovic et al, 2010].
- The authors select these original graphs where the weight of each vertex is set to 1.
- There are 52 families, each of which contains 10 instances with the same size.
- The authors do not report the results on graphs with 50 vertices
## Results:

**Results on Standard Benchmarks**

Results on UDG, T1 and BHOSLIB benchmarks are reported in Tables 1, 2 and 3 respectively.- The DIMACS instances are so easy that CC2FS, FastMWDS and FastDS find the.
- Cit-HepPh 3078(3078.6) 3192(3209.3) 3087(3088.9) 3078(3078.9).
- Cit-HepTh 2935(2935.7) 2985(2993.4) 2944(2948.6) 2936(2936.1).
- CC2FS is dominated by FastMWDS on all the instances and not reported in Tables 5 and 6.
- FastDS performs best for all the massive benchmarks.
- It obtains the best solutions for 19 SNAP instances, 28 DIMACS10 instances and 63 Repository bn-human-B*1 1189968 (1189990) (1190752) (1190256) (1189877.3).
## Conclusion:

The authors proposed a two-goal local search framework for MinDS and proposed three inference rules.- The resulting algorithm FastDS is robust and efficient on standard benchmarks and massive benchmarks, and significantly outperforms state-ofthe-art algorithms on massive benchmarks.
- The authors would like to study the ideas for other subset problems

- Table1: Experiment results on UDG benchmark
- Table2: Experiment results on T1 benchmark. To save space, we denote FastMWDS as MWDS
- Table3: Experiment results on BHOSLIB benchmark
- Table4: Averaged run time on standard benchmarks
- Table5: Experiment results on SNAP and DIMACS10 benchmarks same solutions on all the instances quickly, but RLSo and ScBppw are worse. For all standard benchmarks, FastDS has the best performance in terms of solution quality, except slightly worse than CC2FS and FastMWDS on UDG benchmark. Over all standard benchmarks, CC2FS, FastMWDS and FastDS are essentially better than the other algorithms. We compare the averaged run time of these three algorithms on these benchmarks (Table 4), where the run time of each run of an algorithm is the time to reach the final solution
- Table6: Experiment results on Repository benchmark instances. On average, the best solutions found by FastDS have 1320 and 23732 fewer vertices than the strongest combenchmark UDG T1 DIMACS BHOSLIB SNAP DIMACS10 Repository
- Table7: Summary results of all benchmarks. We report for each algorithm the number of instances (or families) where it finds the best Dmin among the algorithms in experiments
- Table8: The fix ratio of the inference rules and construction time with and without rules petitor FastMWDS for SNAP and DIMACS10 benchmarks, and have 14150 fewer vertices than the strongest competitor ScBppw for Repository benchmark

Funding

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

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