Multi-Start Local Search Algorithm for the Minimum Connected Dominating Set Problems
Abstract
:1. Introduction
2. Preliminaries
3. Vertex Fitness Based Vertex Score Mechanism
3.1. Vertex Fitness Mechanism
3.2. Vertex Score Mechanism
4. Configuration Checking Mechanism
5. Vertex Flipping Mechanism
Algorithm 1 Vertex_flipping_add() |
1: Initial max_score = −, add_v = −1; |
2: for each vertex a in SCE |
3: if config(a)==0 continue; |
4: if v_score(a) > max_score |
5: max_score = v_score(a); |
6: add_v = a; |
7: elseif v_score(a)==max_score && age(a) > age(add_v) |
8: add_v = a; |
9: endif |
10: endfor |
11: S←S{add_v}; |
12: for each vertex b in N(add_v) |
13: config(b) = 1; |
14: endfor |
15: update v_scores of related vertices; |
16: update SCE; |
Algorithm 2 Vertex_flipping_remove() |
1: initial max_score = −, remove_v = −1; |
2: for each vertex a in the candidate solution |
3: if v_score(a) > max_score |
4: max_score = v_score(a); |
5: remove_v = a; |
6: elseif v_score(a)==max_score && age(a) > age(remove_v) |
7: remove_v = a; |
8: endif |
9: endfor |
10: S←S\{remove_v}; |
11: config(remove_v) = 0; |
12: for each vertex b in N(remove_v) |
13: config(b) = 1; |
14: endfor |
15: update v_scores of related vertices; |
16: update SCE; |
6. Framework of MSLS
6.1. Initial Solution Construction
Algorithm 3 Initial_solution() |
1: Initialize v_score(a) = d(a), aVt; |
2: S← ; |
3: SN←Vt; |
4: While S is not CDS do |
5: b← such a vertex with greatest v_score in SN, breaking ties randomly; |
6: S← S{b}; |
7: SN← ; |
8: endwhile |
9: return S; |
6.2. Algorithm MSLS
Algorithm 4 Pseudo-code of MSLS |
Input: An undirected connected graph G = (Vt, Eg), Nstart, MAX_no_improve |
Output: The best solution S* |
1: iter←0; S*←; |
2: while iter < Nstart do |
3: S←Initial_solution(); /*see Section 6.1*/ |
4: no_improve←0; |
5: while no_improve < MAX_no_improve do |
6: if S is better than S*; |
7: S*←S; |
8: vertex_flipping_remove(); /*see Section 5*/ |
9: no_improve←0; |
10: endif |
11: vertex_flipping_remove(); /*see Section 5*/ |
12: vertex_flipping_add(); /*see Section 5*/ |
13: no_improve←no_improve+1; |
14: endwhile |
15: iter←iter+1; |
16: endwhile |
17: return S*; |
7. Computational Experiments
7.1. Benchmark Instances
7.2. Comparison with State-of-the-Art Algorithms
7.3. Experimental Results
7.4. Comparison of Different Versions of MSLS
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
- Nehéz, M.; Bernát, D.; Klaučo, M. Comparison of algorithms for near-optimal dominating sets computation in real-world networks. In Proceedings of the 16th International Conference on Computer Systems and Technologies, Dublin, Ireland, 25–26 June 2015; pp. 199–206. [Google Scholar]
- Wang, H.; Zheng, H.; Browne, F.; Wang, C. Minimum dominating sets in cell cycle specific protein interaction networks. In Proceedings of the IEEE International Conference on Bioinformatics and Biomedicine (BIBM), Belfast, UK, 2–5 November 2014; pp. 25–30. [Google Scholar]
- Wong, B. Cancer Informatics in the Post Genomic Era; Igor, J., Dennis, W., Eds.; Springer: Boston, MA, USA, 2007; Volume 137. [Google Scholar]
- Nehéz, M.; Mäsiar, J. On Propagation in Social Networks—A Graph Mining Approach. In Proceedings of the 6th International Workshop on Knowledge Management, Trenčín, Slovakia, 21–22 October 2011; pp. 107–118. [Google Scholar]
- Kelleher, L.L.; Cozzens, M.B. Dominating sets in social network graphs. Math. Soc. Sci. 1988, 16, 267–279. [Google Scholar] [CrossRef]
- Wuchty, S. Controllability in protein interaction networks. Proc. Natl. Acad. Sci. USA 2014, 111, 7156–7160. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Guha, S.; Khuller, S. Approximation algorithms for connected dominating sets. Algorithmica 1998, 20, 374–387. [Google Scholar] [CrossRef] [Green Version]
- Ruan, L.; Du, H.; Jia, X.; Wu, W.; Li, Y.; Ko, K.I. A greedy approximation for minimum connected dominating sets. Theor. Comput. Sci. 2004, 329, 325–330. [Google Scholar] [CrossRef] [Green Version]
- Misra, R.; Mandal, C. Minimum connected dominating set using a collaborative cover heuristic for ad hoc sensor networks. IEEE Trans. Parallel Distrib. Syst. 2010, 21, 292–302. [Google Scholar] [CrossRef]
- Thai, M.T.; Tiwari, R.; Du, D.Z. On construction of virtual backbone in wireless ad hoc networks with unidirectional links. IEEE Trans. Mob. Comput. 2008, 7, 1098–1109. [Google Scholar] [CrossRef]
- Garey, M.; Johnson, D.S. Computers and Intractability, a Guide to the Theory of NP-Completeness; Freeman: San Francisco, CA, USA, 1979. [Google Scholar]
- Kamei, S.; Kakugawa, H. A self-stabilizing distributed approximation algorithm for the minimum connected dominating set. Int. J. Found. Comput. Sci. 2010, 21, 459–476. [Google Scholar] [CrossRef] [Green Version]
- Marathe, M.V.; Breu, H.; Hunt, H.B., III; Ravi, S.S.; Rosenkrantz, D.J. Simple heuristics for unit disk graphs. Networks 1995, 25, 59–68. [Google Scholar] [CrossRef] [Green Version]
- Das, B.; Bharghavan, V. Routing in ad-hoc networks using minimum connected dominating sets. In Proceedings of the ICC’97-International Conference on Communications, Montreal, QC, Canada, 12 June 1997; pp. 376–380. [Google Scholar]
- Wu, J.; Li, H. On calculating connected dominating set for efficient routing in ad hoc wireless networks. In Proceedings of the 3rd International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, Seattle, WA, USA, 20 August 1999; pp. 7–14. [Google Scholar]
- Cheng, X.; Du, D.Z.; Jia, X.; Xue, G. Virtual backbone-based routing in multihop ad hoc wireless networks. IEEE Trans. Parallel Distrib. Syst. 2002. [Google Scholar] [CrossRef] [Green Version]
- Wan, P.J.; Alzoubi, K.M.; Frieder, O. Distributed construction of connected dominating set in wireless ad hoc networks. In Proceedings of the Twenty-First Annual Joint Conference of the IEEE Computer and Communications Societies, New York, NY, USA, 23–27 June 2002; pp. 1597–1604. [Google Scholar]
- Wu, J.; Lou, W. Forward-node-set-based broadcast in clustered mobile ad hoc networks. Wirel. Commun. Mob. Comput. 2003, 3, 155–173. [Google Scholar] [CrossRef] [Green Version]
- Gao, B.; Yang, Y.; Ma, H. A new distributed approximation algorithm for constructing minimum connected dominating set in wireless ad hoc networks. Int. J. Commun. Syst. 2005, 18, 743–762. [Google Scholar] [CrossRef]
- Simonetti, L.; da Cunha, A.S.; Lucena, A. The minimum Connected Dominating Set Problem: Formulation, Valid Inequalities and a Branch-and-Cut Algorithm. In Network Optimization; Springer: Berlin, Germany, 2011; pp. 162–169. [Google Scholar]
- Gendron, B.; Lucena, A.; da Cunha, A.S.; Simonetti, L. Benders decomposition, branch-and-cut, and hybrid algorithms for the minimum connected dominating set problem. INFORMS J. Comput. 2014, 26, 645–657. [Google Scholar] [CrossRef]
- Cheng, X.; Ding, M.; Chen, D. An approximation algorithm for connected dominating set in ad hoc networks. In Proceedings of the International Workshop on Theoretical Aspects of Wireless Ad Hoc, Sensor and Peer-to-Peer Networks (TAWN), Chicago, IL, USA, 11–12 June 2004. [Google Scholar]
- Min, M.; Du, H.; Jia, X.; Huang, C.X.; Huang, S.C.H.; Wu, W. Improving construction for connected dominating set with steiner tree in wireless sensor networks. J. Glob. Optim. 2006, 35, 111–119. [Google Scholar] [CrossRef]
- Butenko, S.; Cheng, X.; Oliveira, C.; Pardalos, P. A New Heuristic for the Minimum Connected Dominating Set Problem on ad hoc Wireless Networks. In Recent Developments in Cooperative Control and Optimization; Kluwer Academic Publishers: Norwell, MA, USA, 2004; pp. 61–73. [Google Scholar]
- Butenko, S.; Oliveira, C.; Pardalos, P. A New Algorithm for the Minimum Connected Dominating Set Problem on ad hoc Wireless Networks. In CCCT’03; International Institute of Informatics and Systematics (IIIS): Winter Garden, FL, USA, 2003; pp. 39–44. [Google Scholar]
- He, H.; Zhu, Z.; Makinen, E. A neural network model to minimize the connected dominating set for self-configuration of wireless sensor networks. IEEE Trans. Neural Netw. 2009, 20, 973–982. [Google Scholar] [PubMed]
- Morgan, M.; Grout, V. Metaheuristics for Wireless Network Optimization. In Proceedings of the Third Advanced International Conference on Telecommunications, Morne, Mauritius, 13–19 May 2007; p. 15. [Google Scholar]
- Jovanovic, R.; Tuba, M. Ant colony optimization algorithm with pheromone correction strategy for the minimum connected dominating set problem. Comput. Sci. Inf. Syst. 2013, 10, 133–149. [Google Scholar] [CrossRef]
- Nimisha, T.S.; Ramalakshmi, R. Energy efficient Connected Dominating Set construction using Ant Colony Optimization technique in Wireless Sensor Network. In Proceedings of the Innovations in Information, Embedded and Communication Systems (ICIIECS), Coimbatore, India, 19–20 March 2015; pp. 1–5. [Google Scholar]
- Li, R.; Hu, S.; Gao, J.; Zhou, Y.; Wang, Y.; Yin, M. GRASP for connected dominating set problems. Neural Comput. Appl. 2017, 28, 1059–1067. [Google Scholar] [CrossRef]
- Wang, L.; Zhou, T.; Wu, X. Variable-depth neighborhood search algorithm for the minimum-connected dominating-set problem. Sci. Sin. Inf. Sci. 2016, 46, 445–460. [Google Scholar]
- Bouamama, S.; Blum, C.; Fages, J.G. An algorithm based on ant colony optimization for the minimum connected dominating set problem. Appl. Soft Comput. 2019, 80, 672–686. [Google Scholar] [CrossRef]
- Cai, S.W.; Su, K.L.; Sattar, A. Local search with edge weighting and configuration checking heuristics for minimum vertex cover. Artif. Intell. 2011, 175, 1672–1696. [Google Scholar] [CrossRef] [Green Version]
- Li, R.; Hu, S.; Zhang, H.; Yin, M. An efficient local search framework for the minimum weighted vertex cover problem. Inf. Sci. 2016, 372, 428–445. [Google Scholar] [CrossRef]
- Li, R.; Hu, S.; Cai, S.; Gao, J.; Wang, Y.; Yin, M. NuMWVC: A novel local search for minimum weighted vertex cover problem. J. Oper. Res. Soc. 2019. [Google Scholar] [CrossRef]
- Wang, Y.; Ouyang, D.; Zhang, L.; Yin, M. A novel local search for unicost set covering problem using hyperedge configuration checking and weight diversity. Sci. China Inf. Sci. 2017, 60, 062103. [Google Scholar] [CrossRef]
- Cai, S.W.; Su, K.L. Local search for Boolean Satisfiability with configuration checking and subscore. Artif. Intell. 2013, 204, 75–98. [Google Scholar] [CrossRef] [Green Version]
- Cai, S.; Su, K. Configuration checking with aspiration in local search for sat. In Proceedings of the Twenty-Sixth AAAI Conference on Artificial Intelligence, Toronto, ON, Canada, 22–26 July 2012. [Google Scholar]
- Luo, C.; Cai, S.; Su, K.; Wu, W. Clause states based configuration checking in local search for satisfiability. IEEE Trans. Cybern. 2015, 45, 1014–1027. [Google Scholar] [PubMed]
- Luo, C.; Cai, S.; Su, K.; Wu, W. Ccls: An efficient local search algorithm for weighted maximum satisfiability. IEEE Trans. Comput. 2015, 64, 1830–1843. [Google Scholar] [CrossRef]
- Wang, Y.Y.; Cai, S.W.; Yin, M.H. Two efficient local search algorithms for maximum weight clique problem. In Proceedings of the Thirtieth AAAI Conference on Artificial Intelligence, Phoenix, AZ, USA, 12–17 February 2016. [Google Scholar]
- Li, R.; Liu, H.; Wu, X.; Wu, J.; Yin, M. An Efficient Local Search Algorithm for the Minimum k-Dominating Set Problem. IEEE Access 2018, 6, 62062–62075. [Google Scholar] [CrossRef]
- Ho, C.K.; Singh, Y.P.; Ewe, H.T. An enhanced ant colony optimization metaheuristic for the minimum dominating set problem. Appl. Artif. Intell. 2006, 20, 881–903. [Google Scholar] [CrossRef]
- Lucena, A.; Maculan, N.; Simonetti, L. Reformulation and solution algorithms for the maximum leafspanning tree problem. Comput. Manag. Sci. 2010, 7, 289–311. [Google Scholar] [CrossRef]
- Wei, Z.; Hao, J.K. Iterated two-phase local search for the Set-Union Knapsack Problem. Future Gener. Comput. Syst. 2019, 101, 1005–1017. [Google Scholar] [CrossRef] [Green Version]
- Lu, Z.; Hao, J.K.; Zhou, Y. Stagnation-aware Breakout Tabu Search for the Minimum Conductance Graph Partitioning Problem. Comput. Oper. Res. 2019, 111, 43–57. [Google Scholar] [CrossRef]
- Wang, Y.; Li, C.; Sun, H.; Chen, J.; Yin, M. MLQCC: An improved local search algorithm for the set k-covering problem. Int. Trans. Oper. Res. 2019, 26, 856–887. [Google Scholar] [CrossRef]
- Zhou, Y.; Wang, Y.; Gao, J.; Luo, N.; Wang, J. An efficient local search for partial vertex cover problem. Neural Comput. Appl. 2018, 30, 2245–2256. [Google Scholar] [CrossRef]
Problem | LB | LPNMR | Greedy | ACO | ACO + PCS | GRASP | VDNS | MSLS | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Avg | St. | Tim | Avg | St. | Tim | Avg | St. | Tim | Avg | Tim | Avg | St. | Tim | ||||
40 × 200 | 5 | 5 | 10 | 5.8 | 0.6 | 3.2 | 5.3 | 0.45 | 4.1 | 5 | 0 | 7.1 | 5 | <1 | 5 | 0 | 0.05 |
45 ×250 | 5 | 5 | 15 | 5.8 | 0.4 | 3.5 | 5.5 | 0.5 | 4.3 | 5 | 0 | 8.7 | 5 | <1 | 5 | 0 | 0.06 |
50 × 250_1 | 6 | 8 | 15 | 8.1 | 0.54 | 4.8 | 8 | 0 | 6.1 | 7 | 0 | 9.8 | 8 | <1 | 7 | 0 | 0.05 |
50 × 250_2 | 6 | 7 | 17 | 7.5 | 0.5 | 5 | 7.1 | 0.3 | 6.5 | 7 | 0 | 9.9 | 7 | <1 | 7 | 0 | 0.06 |
55 × 250 | 6 | 8 | 20 | 8.8 | 0.98 | 5.6 | 8.3 | 0.45 | 7.3 | 8 | 0 | 9.8 | 8 | <1 | 8 | 0 | 0.06 |
60 × 400 | 5 | 7 | 15 | 7 | 0 | 6.1 | 7 | 0 | 9.1 | 7 | 0 | 12.7 | 7 | <1 | 7 | 0 | 0.08 |
70 × 250 | 11 | 13 | 32 | 14.2 | 0.74 | 11 | 13.9 | 1.04 | 13.5 | 12 | 0 | 11.5 | 13 | <1 | 12 | 0 | 0.06 |
80 × 500 | 7 | 9 | 20 | 10 | 0.44 | 12 | 9.8 | 0.4 | 16.9 | 9 | 0 | 15.1 | 9 | <1 | 9 | 0 | 0.08 |
90 × 600 | 7 | 10 | 19 | 10.9 | 0.83 | 14 | 10.6 | 1.01 | 17.3 | 9 | 0 | 17.2 | 10 | <1 | 9 | 0 | 0.09 |
Average | 6.4 | 8.0 | 18.1 | 8.7 | 8.3 | 7.7 | 8.0 | 7.7 |
Problem | LB | Greedy | ACO | ACO + PCS | GRASP | VDNS | MSLS | |||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Min | Avg | Min | Avg | Min | Avg | Tim | 300 s | 30 s | Min | Avg | Tim | |||
400_80_60 | 12 | 48 | 20 | 21.6 | 19 | 21.2 | 19 | 19.8 | 7.1 | 18 | 18 | 18 | 18.0 | 4.6 |
400_80_70 | 10 | 33 | 16 | 17.0 | 15 | 16.2 | 14 | 15.1 | 7.8 | 14 | 14 | 14 | 14.0 | 4.8 |
400_80_80 | 8 | 35 | 12 | 14.0 | 12 | 13.1 | 12 | 12 | 7.9 | 12 | 12 | 12 | 12.0 | 5.3 |
400_80_90 | 7 | 41 | 11 | 11.8 | 11 | 11.6 | 10 | 10.6 | 8.4 | 10 | 10 | 10 | 10.0 | 5.6 |
400_80_100 | 6 | 23 | 8 | 9.0 | 8 | 8.9 | 8 | 8.2 | 8.6 | 8 | 8 | 8 | 8.0 | 5.9 |
400_80_110 | 6 | 25 | 8 | 8.5 | 8 | 8.5 | 7 | 7.8 | 9.0 | 7 | 7 | 7 | 7.0 | 5.6 |
400_80_120 | 5 | 17 | 7 | 7.5 | 7 | 7.2 | 6 | 6.1 | 9.8 | 6 | 6 | 6 | 6.0 | 7.0 |
600_100_80 | 14 | 38 | 23 | 24.7 | 22 | 23.6 | 22 | 22.9 | 9.0 | 21 | 21 | 21 | 21.0 | 5.8 |
600_100_90 | 14 | 40 | 22 | 23.8 | 21 | 23.6 | 20 | 20.7 | 9.7 | 19 | 19 | 19 | 19.0 | 5.3 |
600_100_100 | 11 | 38 | 17 | 20.0 | 17 | 19.0 | 17 | 17.9 | 10.0 | 16 | 16 | 16 | 16.0 | 6.1 |
600_100_110 | 10 | 35 | 15 | 17.2 | 15 | 16.8 | 15 | 15.9 | 9.6 | 15 | 15 | 14 | 14.5 | 6.1 |
600_100_120 | 9 | 36 | 15 | 16.2 | 14 | 15.5 | 13 | 13.8 | 17.5 | 13 | 13 | 13 | 13.0 | 6.4 |
700_200_70 | 26 | 96 | 46 | 50.7 | 46 | 49.6 | 45 | 46.5 | 44.2 | 39 | 39 | 39 | 39.6 | 13.7 |
700_200_80 | 22 | 89 | 41 | 43.7 | 41 | 43.9 | 35 | 37.5 | 40.8 | 33 | 33 | 33 | 33.0 | 12.3 |
700_200_90 | 18 | 84 | 34 | 36.0 | 33 | 35.7 | 30 | 30.9 | 37.8 | 27 | 27 | 26 | 26.8 | 11.8 |
700_200_100 | 16 | 75 | 28 | 30.8 | 28 | 31.0 | 25 | 25.8 | 35.2 | 23 | 23 | 22 | 22.5 | 11.2 |
700_200_110 | 14 | 70 | 23 | 27.4 | 22 | 26.4 | 22 | 22.7 | 35.6 | 20 | 20 | 20 | 20.0 | 11.5 |
700_200_120 | 12 | 68 | 21 | 23.6 | 21 | 23.4 | 18 | 19.1 | 34.7 | 17 | 17 | 17 | 17.0 | 12.3 |
1000_200_100 | 26 | 96 | 46 | 50.7 | 46 | 49.6 | 45 | 46.5 | 44.5 | 39 | 39 | 39 | 39.6 | 13.5 |
1000_200_110 | 22 | 92 | 43 | 44.9 | 42 | 44.8 | 37 | 39.5 | 41.5 | 34 | 34 | 34 | 34.1 | 11.7 |
1000_200_120 | 20 | 82 | 37 | 39.9 | 37 | 39.8 | 34 | 35.4 | 41.5 | 30 | 30 | 29 | 30.3 | 11.5 |
1000_200_130 | 18 | 91 | 32 | 34.7 | 32 | 34.9 | 29 | 30.5 | 38.4 | 26 | 26 | 26 | 26.8 | 11.5 |
1000_200_140 | 16 | 76 | 30 | 31.3 | 29 | 31.3 | 25 | 25.7 | 35.3 | 23 | 23 | 23 | 23.1 | 10.9 |
1000_200_150 | 15 | 83 | 28 | 29.6 | 26 | 28.8 | 23 | 24.3 | 35.8 | 21 | 21 | 21 | 21.1 | 11.0 |
1000_200_160 | 14 | 86 | 24 | 26.6 | 25 | 26.5 | 22 | 22.3 | 35.0 | 20 | 20 | 19 | 19.6 | 11.5 |
1500_250_130 | 33 | 158 | 60 | 64.5 | 60 | 64.3 | 57 | 58.6 | 58.3 | 49 | 49 | 49 | 50.0 | 17.7 |
1500_250_140 | 30 | 144 | 53 | 57.2 | 52 | 57.0 | 50 | 52.3 | 55.6 | 44 | 44 | 44 | 44.3 | 15.9 |
1500_250_150 | 27 | 170 | 51 | 54.9 | 51 | 54.4 | 46 | 48.5 | 51.0 | 41 | 41 | 41 | 41.6 | 16.7 |
1500_250_160 | 25 | 151 | 47 | 50.5 | 45 | 49.8 | 43 | 43.7 | 49.4 | 38 | 38 | 37 | 37.7 | 14.7 |
2000_300_200 | 28 | 178 | 55 | 58.6 | 52 | 58.8 | 49 | 50.4 | 61.2 | 42 | 43 | 43 | 43.1 | 19.3 |
2000_300_210 | 26 | 151 | 51 | 53.5 | 50 | 52.8 | 45 | 46.1 | 60.1 | 39 | 39 | 38 | 38.6 | 18.1 |
2000_300_220 | 24 | 140 | 47 | 48.9 | 45 | 48.4 | 40 | 42.1 | 59.0 | 36 | 36 | 36 | 36.3 | 16.9 |
2000_300_230 | 23 | 166 | 44 | 47.5 | 44 | 46.9 | 39 | 39.8 | 56.4 | 36 | 36 | 34 | 34.8 | 18.1 |
2500_350_200 | 41 | 198 | 79 | 82.0 | 79 | 81.5 | 73 | 75.4 | 83.1 | 61 | 61 | 61 | 61.3 | 25.6 |
2500_350_210 | 39 | 185 | 75 | 79.1 | 74 | 78.2 | 67 | 70 | 80.8 | 58 | 58 | 56 | 57.0 | 24.8 |
2500_350_220 | 36 | 205 | 68 | 72.6 | 69 | 73.8 | 62 | 67 | 78.1 | 56 | 56 | 52 | 54.3 | 25.2 |
2500_350_230 | 34 | 193 | 66 | 69.2 | 66 | 68.9 | 59 | 60.9 | 76.1 | 50 | 50 | 50 | 51.6 | 24.4 |
3000_400_210 | 51 | 259 | 99 | 101.6 | 98 | 104.0 | 90 | 94.7 | 116.1 | 76 | 77 | 74 | 76.3 | 35.7 |
3000_400_220 | 48 | 225 | 88 | 95.4 | 91 | 97.6 | 82 | 87.9 | 108.9 | 72 | 73 | 71 | 71.9 | 33.8 |
3000_400_230 | 45 | 205 | 86 | 91.4 | 86 | 90.3 | 78 | 81.6 | 105.2 | 67 | 68 | 66 | 67.0 | 32.3 |
3000_400_240 | 42 | 210 | 82 | 85.8 | 80 | 84.1 | 74 | 76.1 | 100.4 | 63 | 65 | 61 | 62.5 | 31.7 |
Average | 22.0 | 108.2 | 40.4 | 43.3 | 40.0 | 43.0 | 36.8 | 38.4 | 32.7 | 32.8 | 32.2 | 32.7 |
Problem | OPT | LB | GRASP | VDNS | MSLS | |||||
---|---|---|---|---|---|---|---|---|---|---|
Min | Avg | Tim | Avg | Tim | Min | Avg | Tim | |||
30_d10 | 15 | 12 | 15 | 15 | 3.5 | 15 | 7 | 15 | 15 | 3.1 |
30_d20 | 7 | 6 | 7 | 7 | 2.9 | 7 | <1 | 7 | 7 | 2.4 |
30_d30 | 4 | 4 | 4 | 4 | 3.5 | 4 | <1 | 4 | 4 | 2.9 |
30_d50 | 3 | 2 | 3 | 3 | 3.1 | 3 | <1 | 3 | 3 | 4.3 |
30_d70 | 2 | 2 | 2 | 2 | 3.7 | 2 | <1 | 2 | 2 | 5.2 |
50_d5 | 31 | 21 | 33 | 33.7 | 2.5 | 31 | <1 | 31 | 31 | 5.2 |
50_d10 | 12 | 11 | 12 | 12 | 2.3 | 12 | <1 | 12 | 12 | 3.0 |
50_d20 | 7 | 6 | 7 | 7 | 3.1 | 7 | <1 | 7 | 7 | 4.5 |
50_d30 | 5 | 4 | 5 | 5 | 2.4 | 5 | <1 | 5 | 5 | 5.7 |
50_d50 | 3 | 3 | 3 | 3 | 3.3 | 3 | <1 | 3 | 3 | 10.4 |
50_d70 | 2 | 2 | 2 | 2 | 2.8 | 2 | <1 | 2 | 2 | 13.6 |
70_d5 | 27 | 22 | 29 | 30.6 | 5.1 | 27 | <1 | 27 | 27 | 4.8 |
70_d10 | 13 | 11 | 13 | 13 | 3.4 | 13 | <1 | 13 | 13 | 4.6 |
70_d20 | 7 | 6 | 7 | 7 | 3.0 | 7 | <1 | 7 | 7 | 6.1 |
70_d30 | 5 | 4 | 5 | 5 | 2.9 | 5 | <1 | 5 | 5 | 9.9 |
70_d50 | 3 | 2 | 3 | 3 | 1.9 | 3 | <1 | 3 | 3 | 17.6 |
70_d70 | 2 | 2 | 2 | 2 | 3.2 | 2 | <1 | 2 | 2 | 21.4 |
100_d5 | 24 | 21 | 26 | 26.7 | 4.2 | 24 | 18 | 24 | 24 | 5.2 |
100_d10 | 13 | 11 | 13 | 13 | 2.8 | 13 | 1 | 13 | 13 | 6.6 |
100_d20 | 8 | 5 | 8 | 8 | 3.2 | 8 | 1 | 8 | 8 | 10.3 |
100_d30 | 6 | 4 | 6 | 6 | 1.7 | 6 | 1 | 6 | 6 | 14.8 |
100_d50 | 4 | 2 | 4 | 4 | 1.7 | 4 | 1 | 4 | 4 | 27.1 |
100_d70 | 3 | 2 | 3 | 3 | 1.8 | 3 | 1 | 3 | 3 | 41.4 |
120_d5 | 25 | 23 | 27 | 27.8 | 2.1 | 25 | 3 | 25 | 25 | 6.4 |
120_d10 | 13 | 11 | 14 | 14.3 | 1.9 | 13 | 1 | 13 | 13 | 7.9 |
120_d20 | 8 | 5 | 8 | 8.2 | 1.7 | 8 | 2 | 8 | 8 | 13.1 |
120_d30 | 6 | 4 | 6 | 6 | 1.7 | 6 | 3 | 6 | 6 | 19.5 |
120_d50 | 4 | 2 | 4 | 4 | 2.1 | 4 | 1 | 4 | 4 | 35.2 |
120_d70 | 3 | 2 | 3 | 3 | 2.0 | 3 | 1 | 3 | 3 | 56.2 |
150_d5 | 26 | 22 | 27 | 28.2 | 2.0 | 26 | 15 | 26 | 26 | 8.4 |
150_d10 | 14 | 11 | 15 | 15.3 | 2.2 | 14 | 17 | 14 | 14 | 11.1 |
150_d20 | 9 | 5 | 9 | 9 | 1.9 | 9 | 195 | 9 | 9 | 17.8 |
150_d30 | 6 | 4 | 6 | 6 | 1.6 | 6 | 8 | 6 | 6 | 26.5 |
150_d50 | 4 | 2 | 4 | 4 | 1.7 | 4 | 3 | 4 | 4 | 50.0 |
150_d70 | 3 | 2 | 3 | 3 | 1.6 | 3 | 3 | 3 | 3 | 82.3 |
200_d5 | 27 | 22 | 29 | 30.7 | 2.2 | 28 | 18 | 27 | 27 | 11.8 |
200_d10 | 16 | 10 | 16 | 16.9 | 2.1 | 16 | 247 | 16 | 16 | 16.6 |
200_d20 | 9 | 5 | 9 | 9.7 | 1.8 | 9 | 163 | 9 | 9 | 26.5 |
200_d30 | 7 | 4 | 7 | 7 | 1.7 | 7 | 59 | 7 | 7 | 40.4 |
200_d50 | 4 | 2 | 4 | 4 | 1.7 | 4 | 9 | 4 | 4 | 79.4 |
200_d70 | 3 | 2 | 3 | 3 | 1.7 | 3 | 2 | 3 | 3 | 136.0 |
Average | 9.6 | 7.5 | 9.9 | 10.1 | 9.6 | 9.6 | 9.6 |
Problem | MSLS-CC | MSLS-VF | MSLS | |||
---|---|---|---|---|---|---|
Min | Avg | Min | Avg | Min | Avg | |
400_80_70 | 15 | 15.0 | 14 | 14.0 | 14 | 14.0 |
400_80_110 | 7 | 7.4 | 7 | 7.0 | 7 | 7.0 |
400_80_120 | 6 | 6.0 | 7 | 7.0 | 6 | 6.0 |
600_100_100 | 17 | 17.0 | 16 | 16.0 | 16 | 16.0 |
600_100_110 | 15 | 15.0 | 14 | 14.9 | 14 | 14.5 |
600_100_120 | 14 | 14.0 | 13 | 13.0 | 13 | 13.0 |
700_200_100 | 24 | 24.5 | 23 | 23.1 | 22 | 22.5 |
700_200_110 | 21 | 21.0 | 20 | 20.3 | 20 | 20.0 |
700_200_120 | 18 | 18.0 | 17 | 17.1 | 17 | 17.0 |
1000_200_130 | 27 | 27.8 | 26 | 26.4 | 26 | 26.8 |
1000_200_140 | 24 | 24.8 | 23 | 23.7 | 23 | 23.1 |
1000_200_160 | 20 | 20.5 | 19 | 19.8 | 19 | 19.6 |
1500_250_130 | 50 | 51.2 | 50 | 50.1 | 49 | 50.0 |
1500_250_140 | 44 | 45.0 | 44 | 44.4 | 44 | 44.3 |
1500_250_160 | 37 | 38.4 | 37 | 37.2 | 37 | 37.7 |
2000_300_210 | 38 | 39.1 | 38 | 38.6 | 38 | 38.6 |
2000_300_220 | 36 | 36.6 | 36 | 36.1 | 36 | 36.3 |
2000_300_230 | 36 | 36.0 | 34 | 34.8 | 34 | 34.8 |
2500_350_200 | 61 | 62.5 | 61 | 61.5 | 61 | 61.3 |
2500_350_210 | 58 | 59.5 | 56 | 57.2 | 56 | 57.0 |
2500_350_220 | 54 | 55.8 | 52 | 53.5 | 52 | 54.3 |
3000_400_220 | 72 | 73.2 | 71 | 72.0 | 71 | 71.9 |
3000_400_230 | 68 | 69.1 | 66 | 67.0 | 66 | 67.0 |
3000_400_240 | 62 | 64.3 | 61 | 62.6 | 61 | 62.5 |
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Li, R.; Hu, S.; Liu, H.; Li, R.; Ouyang, D.; Yin, M. Multi-Start Local Search Algorithm for the Minimum Connected Dominating Set Problems. Mathematics 2019, 7, 1173. https://doi.org/10.3390/math7121173
Li R, Hu S, Liu H, Li R, Ouyang D, Yin M. Multi-Start Local Search Algorithm for the Minimum Connected Dominating Set Problems. Mathematics. 2019; 7(12):1173. https://doi.org/10.3390/math7121173
Chicago/Turabian StyleLi, Ruizhi, Shuli Hu, Huan Liu, Ruiting Li, Dantong Ouyang, and Minghao Yin. 2019. "Multi-Start Local Search Algorithm for the Minimum Connected Dominating Set Problems" Mathematics 7, no. 12: 1173. https://doi.org/10.3390/math7121173
APA StyleLi, R., Hu, S., Liu, H., Li, R., Ouyang, D., & Yin, M. (2019). Multi-Start Local Search Algorithm for the Minimum Connected Dominating Set Problems. Mathematics, 7(12), 1173. https://doi.org/10.3390/math7121173