Abstract
In this paper, we propose several integer programming approaches with a polynomial number of constraints to formulate and solve the minimum connected dominating set problem. Further, we consider both the power dominating set problem – a special dominating set problem for sensor placement in power systems – and its connected version. We propose formulations and algorithms to solve these integer programs, and report results for several power system graphs.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
References
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979)
Cheng, X., Huang, X., Li, D., Wu, W., Du, D.-Z.: A polynomial-time approximation scheme for the minimum-connected dominating set in ad hoc wireless networks. Networks 42(4), 202–208 (2003)
Li, Y., Thai, M.T., Wang, F., Yi, C.W., Wan, P.J., Du, D.-Z.: On greedy construction of connected dominating sets in wireless networks. Wirel. Commun. Mob. Comp. 5, 927–932 (2005)
Zhu, X., Yu, J., Lee, W., Kim, D., Shan, S., Du, D.-Z.: New dominating sets in social networks. J. Global Optim. 48(4), 633–642 (2010)
Wu, W., Gao, X., Pardalos, P.M., Du, D.-Z.: Wireless networking, dominating and packing. Optim. Lett. 4(3), 347–358 (2010)
Ding, L., Gao, X., Wu, W., Lee, W., Zhu, X., Du, D.-Z.: An exact algorithm for minimum CDS with shortest path constraint in wireless networks. Optim. Lett. 5(2), 297–306 (2011)
Thai, M.T., Du, D.-Z.: Connected dominating sets in disk graphs with bidirectional links. IEEE Commun. Lett. 10(3), 138–140 (2006)
Kim, D., Zhang, Z., Li, X., Wang, W., Wu, W., Du, D.-Z.: A better approximation algorithm for computing connected dominating sets in unit ball graphs. IEEE Trans. Mob. Comp. 9(8), 1108–1118 (2010)
Blum, J., Ding, M., Thaeler, A., Cheng, X.: Connected dominating set in sensor networks and MANET. In: Du, D.-Z., Pardalos, P. (eds.) Handbook of Combinatorial Optimization, pp. 329–369 (2004)
Liu, Z., Wang, B., Guo, L.: A Survey on connected dominating set construction algorithm for wireless sensor networks. Informa. Technol. J. 9, 1081–1092 (2010)
Mnif, K., Rong, B., Kadoch, M.: Virtual backbone based on mcds for topology control in wireless ad hoc networks. In: Proceedings of the 2nd ACM International Workshop on Performance Evaluation of Wireless Ad Hoc, Sensor, and Ubiquitous Networks, Quebec, Canada (2005)
Yuan, D.: Energy-efficient broadcasting in wireless ad hoc networks: performance benchmarking and distributed algorithms based on network connectivity characterization. In: Proceedings of MSWiM, Quebec, Canada (2005)
Morgan, M.J., Grout, V.: Finding optimal solutions to backbone minimisation problems using mixed integer programming. In: Proceedings of the Seventh International Network Conference (INC 2008), Boston, MA, pp. 53–63 (2008)
Wightman, P.M., Fabregasy, A., Labradorz, M.A.: An optimal solution to the MCDS problem for topology construction in wireless sensor networks. In: 2010 IEEE Latin-American Conference on Communications (LATINCOM), Belem, Brazil (2010)
Simonetti, L., da Cunha, A.S., Lucena, A.: The Minimum Connected Dominating Set Problem: Formulation, Valid Inequalities and a Branch-and-Cut Algorithm. In: Pahl, J., Reiners, T., Voß, S. (eds.) INOC 2011. LNCS, vol. 6701, pp. 162–169. Springer, Heidelberg (2011)
Pop, P.C.: A survey of different integer programming formulations of the generalized minimum spanning tree problem. Carpathian J. Mathematics 25(1), 104–118 (2009)
Haynes, T.W., Hedetniemi, S.M., Hedetniemi, S.T., Henning, M.A.: Domination in graphs applied to electric power networks. SIAM J. Disc. Math. 15, 519–529 (2002)
Aminifar, F., Khodaei, A., Fotuhi-Firuzabad, M., Shahidehpour, M.: Contingency-constrained PMU placement in power networks. IEEE Trans. Power Syst. 25, 516–523 (2010)
Aazami, A.: Domination in graphs with bounded progagation: algorithms, formulations and hardness results. J. Comb. Optim. 19, 429–456 (2010)
Miller, C.E., Tucker, A.W., Zemlin, R.A.: Integer programming formulation of traveling salesman problems. J. Assoc. Comp. Mach. 7, 326–329 (1960)
Quintao, F.R., da Cunha, A.S., Mateus, G.R., Lucena, A.: The k-cardinality tree problem: reformulations and lagrangian relaxation. Disc. Appl. Math. 158, 1305–1314 (2010)
Desrochers, M., Gilbert, L.: Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints. Oper. Res. Lett. 10, 27–36 (1991)
Martin, R.K.: Using separation algorithms to generate mixed integer model reformulations. Oper. Res. Lett. 10, 119–128 (1991)
Yannakakis, M.: Expressing combinatorial optimization problems by linear programs. J. Comp. Syst. Sci. 43(3), 441–466 (1991)
Conforti, M., Cornuéjols, G., Zambelli, G.: Extended formulations in combinatorial optimization. 4OR (8), 1–48 (2010)
Kaibel, V., Pashkovich, K., Theis, D.O.: Symmetry Matters for the Sizes of Extended Formulations. In: Eisenbrand, F., Shepherd, F.B. (eds.) IPCO 2010. LNCS, vol. 6080, pp. 135–148. Springer, Heidelberg (2010)
Dilkina, B., Gomes, C.P.: Solving Connected Subgraph Problems in Wildlife Conservation. In: Lodi, A., Milano, M., Toth, P. (eds.) CPAIOR 2010. LNCS, vol. 6140, pp. 102–116. Springer, Heidelberg (2010)
IEEE reliability test data (2012), http://www.ee.washington.edu/research/pstca/
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fan, N., Watson, JP. (2012). Solving the Connected Dominating Set Problem and Power Dominating Set Problem by Integer Programming. In: Lin, G. (eds) Combinatorial Optimization and Applications. COCOA 2012. Lecture Notes in Computer Science, vol 7402. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31770-5_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-31770-5_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31769-9
Online ISBN: 978-3-642-31770-5
eBook Packages: Computer ScienceComputer Science (R0)