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Solving the minimum M-dominating set problem by a continuous optimization approach based on DC programming and DCA

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Abstract

We propose a new optimization approach based on DC (Difference of Convex functions) programming and DCA (DC Algorithm) to the so-called Minimum M-Dominating Set problem in graphs. This problem is beforehand re-casted as a polyhedral DC program with the help of exact penalty in DC programming. The related DCA is original and computer efficient because it consists of solving a few linear programs and converges after a finite number of iterations to an integer solution while working in a continuous domain. Numerical simulations show the efficiency and robustness of DCA and its superiority with respect to standard methods.

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References

  • Basagni S, Mastrogiovanni M, Panconesi A, Petrioli C (2006) Localized protocols for ad hoc clustering and backbone formation: A performance comparison. IEEE Trans Parallel Distrib Syst 17(4):292–306

    Article  Google Scholar 

  • Brešar B, Henning MA, Rall DF (2008) Rainbow domination in graphs. Taiwan J Math 12:213–225

    MATH  Google Scholar 

  • Dai D, Yu C (2009) A 5+ϵ-approximation algorithm for minimum weighted dominating set in unit disk graph. Theor Comput Sci 410:756–765

    Article  MathSciNet  MATH  Google Scholar 

  • Dai F, Wu J (2004) An extended localized algorithm for connected dominating set formation in ad hoc wireless networks. IEEE Trans Parallel Distrib Syst 15(10):908–920

    Article  Google Scholar 

  • Dutot A, Guinand F, Olivier D, Pigné Y (2007) Graphstream: A tool for bridging the gap between complex systems and dynamic graphs. In: EPNACS: Emergent properties in natural and artificial complex systems

    Google Scholar 

  • Erdős P, Rényi A (1959) On random graphs. Publ Math 6:290–297

    Google Scholar 

  • Fomin FV, Kratsch D, Woeginger GJ (2005) Exact (exponential) algorithms for the dominating set problem. In: Graph-theoretic concepts in computer science. Lecture notes in computer science, vol 3353. Springer, Berlin, pp 245–256

    Chapter  Google Scholar 

  • Garey M, Johnson DS (1990) Computers and intractability; A guide to the theory of NP-completeness. Freeman, San Francisco

    Google Scholar 

  • Gaspers S, Kratsch D, Liedloff M (2006) Exponential time algorithms for the minimum dominating set problem. In: Proceedings of SWAT, vol 4059, pp 148–159

    Google Scholar 

  • Guha S, Khuller S (1998) Approximation algorithms for connected dominating sets. Algorithmica 20(4):374–387

    Article  MathSciNet  MATH  Google Scholar 

  • Harary F, Haynes TW (2000) Double domination in graphs. Ars Comb 50:201–213

    MathSciNet  Google Scholar 

  • Hedetniemi ST, Laskar RC (1990) Bibliography on domination in graphs and some basic definitions of domination parameters. Discrete Math 86(1–3):257–277

    Article  MathSciNet  MATH  Google Scholar 

  • Khamis SM, Daoud SS, Essa HAE (2009) A randomized algorithm for determining dominating set in graphs of maximum degree five. Theor Comput Sci 410:5122–5127

    Article  MathSciNet  MATH  Google Scholar 

  • Le Thi HA, Pham Dinh T (1997) Solving a class of linearly constrained indefinite quadratic problems by DC algorithms. J Glob Optim 11(3):253–285

    Article  MATH  Google Scholar 

  • Le Thi HA, Pham Dinh T (2003) Large scale molecular optimization from distances matrices by a dc optimization approach. SIAM J Optim 14(1):77–116

    Article  MathSciNet  MATH  Google Scholar 

  • Le Thi HA, Pham Dihn T (2005) The dc (difference of convex functions) programming and DCA revisited with DC models of real world nonconvex optimization problems. Ann Oper Res 133(1–4):23–46

    MathSciNet  MATH  Google Scholar 

  • Le Thi HA, Pham Dinh T, Le Dung M (1999) Exact penalty in DC programming. Vietnam J Math 27(2):169–178

    MathSciNet  MATH  Google Scholar 

  • Liao CS, Chang GJ (2003) k-tuple domination in graphs. Inf Process Lett 87(1):45–50

    Article  MathSciNet  MATH  Google Scholar 

  • Liedloff M (2008) Finding a dominating set on bipartite graphs. Inf Process Lett 107:154–157

    Article  MathSciNet  MATH  Google Scholar 

  • Pham Dinh T, Le Thi HA (1997) Convex analysis approach to DC programming: Theory algorithms and applications. Acta Math Vietnam 22(1):289–355

    MathSciNet  MATH  Google Scholar 

  • Pham Dinh Dinh T, Le Thi HA (1998) DC optimization algorithms for solving the trust region subproblem. SIAM J Optim 8:476–505

    Article  MathSciNet  MATH  Google Scholar 

  • Sampathkumar E (1983) The global domination number of a graph. J Math Phys Sci 23(5):377–385

    MathSciNet  Google Scholar 

  • van Rooij JMM, Bodlaender HL (2008) Design by measure and conquer: A faster exact algorithm for dominating set. In: Proceedings of STACS, pp 657–668

    Google Scholar 

  • Woeginger GJ (2003) Exact algorithms for NP-hard problems: A survey. In: Jünger M, Reinelt G, Rinaldi G (eds) Combinatorial optimization—eureka! You shrink! LNCS, vol 2570. Springer, Berlin, pp 185–207

    Chapter  Google Scholar 

  • Wu J, Li H (1999) On calculating connected dominating sets for efficient routing in ad hoc wireless networks. In: Proc third ACM int’l workshop discrete algorithms and methods for mobile computing and comm dial M, pp 7–17

    Chapter  Google Scholar 

  • Wu J, Cardei M, Dai F, Yang S (2005) Extended dominating set in ah hoc networks using cooperative communication. In: Networking. LNCS, vol 3462, pp 1393–1396

    Google Scholar 

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Authors and Affiliations

  1. SnT Interdisciplinary Center, University of Luxembourg, Kirchberg Campus, Luxembourg

    Julien Schleich

  2. LITA, University Paul Verlaine—Metz, Ile du Saulcy, 57045, Metz, France

    Hoai An Le Thi

  3. FSTC–CSC, University of Luxembourg, Kirchberg Campus, Luxembourg

    Pascal Bouvry

Authors
  1. Julien Schleich
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  2. Hoai An Le Thi
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  3. Pascal Bouvry
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Correspondence to Hoai An Le Thi.

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Schleich, J., Le Thi, H.A. & Bouvry, P. Solving the minimum M-dominating set problem by a continuous optimization approach based on DC programming and DCA. J Comb Optim 24, 397–412 (2012). https://doi.org/10.1007/s10878-011-9396-0

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  • DOI: https://doi.org/10.1007/s10878-011-9396-0

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