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Improved integer linear programming formulation for weak Roman domination problem

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Abstract

Let \(f:V \rightarrow \{0,1,2\}\) be a function, \(G=(V,E)\) be a graph with a vertex set V and a set of edges E and let the weight of the vertex \(u \in V\) be defined by f(u). A vertex u with property \(f(u)=0\) is considered to be defended with respect to the function f if it is adjacent to a vertex with positive weight. Further, the function f is called a weak Roman dominating function (WRDF) if for every vertex u with property \(f(u)=0\) there exists at least one adjacent vertex v with positive weight such that the function \(f':V \rightarrow \{0,1,2\}\) defined by \(f'(u)=1\), \(f'(v)=f(v)-1\) and \(f'(w)=f(w)\), \(w \in V \setminus \{u,v\}\) has no undefended vertices. In this paper, an optimization problem of finding the WRDF f such that \(\sum _{u \in V}{f(u)}\) is minimal, known as the weak Roman domination problem (WRDP), is considered. Therefore, a new integer linear programing (ILP) formulation is proposed and compared with the one known from the literature. Comparison between the new and the existing formulation is made through computational experiments on a grid, planar, net and randomly generated graphs known from the literature and up to 600 vertices. Tests were run using standard CPLEX and Gurobi optimization solvers. The obtained results demonstrate that the proposed new ILP formulation clearly outperforms the existing formulation in the sense of solutions’ quality and running times.

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References

  • Ahangar HA, Henning MA, Löwenstein C, Zhao Y, Samodivkin V (2014) Signed roman domination in graphs. J Comb Optim 27(2):241–255

    Article  MathSciNet  MATH  Google Scholar 

  • Alvarado JD, Dantas S, Rautenbach D (2015a) Relating \(2\)-rainbow domination to weak roman domination. arXiv preprint arXiv:1512.01067

  • Alvarado JD, Dantas S, Rautenbach D (2015b) Strong Equality of Roman and Weak Roman Domination in Trees. arXiv preprint arXiv:1507.04901

  • Alvarez-Ruiz M, Yero IG, Mediavilla-Gradolph T, Valenzuela J (2015) A stronger vision for roman domination in graphs. arXiv preprint arXiv:1502.03933

  • Burger A, De Villiers A, Van Vuuren J (2013) A binary programming approach towards achieving effective graph protection. In: Proceedings of the 2013 ORSSA annual conference, ORSSA, 2013, pp 19–30

  • Chang GJ, Chen SH, Liu CH (2014) Edge roman domination on graphs. arXiv preprint arXiv:1405.5622

  • Chapelle M, Cochefert M, Couturier JF, Kratsch D, Liedloff M, Perez A (2013) Exact algorithms for weak roman domination. In: Combinatorial Algorithms, Springer, pp 81–93

  • Chellali M, Haynes TW, Hedetniemi ST (2014) Bounds on weak roman and 2-rainbow domination numbers. Discrete Appl Math 178:27–32

    Article  MathSciNet  MATH  Google Scholar 

  • Cockayne E, Grobler P, Grundlingh W, Munganga J, Jv Vuuren (2005) Protection of a graph. Util Math 67:19–32

    MathSciNet  MATH  Google Scholar 

  • Currò V (2014) The roman domination problem on grid graphs. Ph.D. thesis, Università di Catania

  • Dreyer PA Jr (2000) Applications and variations of domination in graphs. Ph.D. thesis, Citeseer

  • Henning MA (2003) Defending the roman empire from multiple attacks. Discret Math 271(1):101–115

    Article  MathSciNet  MATH  Google Scholar 

  • Henning MA, Hedetniemi ST (2003) Defending the roman empire a new strategy. Discret Math 266(1):239–251

    Article  MathSciNet  MATH  Google Scholar 

  • Klobučar A, Puljić I (2014) Some results for roman domination number on cardinal product of paths and cycles. Kragujev J Math 38(1):83–94

    Article  MathSciNet  MATH  Google Scholar 

  • Lai YL, Lin CT, Ho HM (2011) Weak roman domination on graphs. In: Proceedings of the 28th workshop on combinatorial mathematics and computation theory, Penghu University of Science and Technology, Penghu, Taiwan, May 27–28, 2011, pp 224–214

  • Liedloff M, Kloks T, Liu J, Peng SL (2005) Roman domination over some graph classes. In: Graph-Theoretic Concepts in Computer Science, Springer, pp 103–114

  • Liu CS, Peng SL, Tang CY (2010) Weak Roman Domination on Block Graphs. In: Proceedings of The 27th workshop on combinatorial mathematics and computation theory, Providence University, Taichung, Taiwan, April 30–May 1, 2010, pp 86–89

  • Pushpam PRL, Kamalam M (2015) Efficient weak roman domination in myscielski graphs. Int J Pure Eng Math (IJPEM) 3(2):93–100

    Google Scholar 

  • Pushpam PRL, Kamalam M (2015) Efficient weak roman domination in graphs. Int J Pure Appl Math 101(5):701–710

    Google Scholar 

  • Pushpam PRL, Mai TM (2011) Weak roman domination in graphs. Discuss Math Graph Theory 31(1):161–170

    Article  MathSciNet  MATH  Google Scholar 

  • ReVelle CS, Rosing KE (2000) Defendens imperium romanum: a classical problem in military strategy. Am Math Mon 107(7):585–594

  • Shang W, Hu X (2007) The roman domination problem in unit disk graphs. In: Computational Science–ICCS 2007, Springer, pp 305–312

  • Song X, Yang J, Xie Y (2011) Weak Roman domination in 2xn grid graphs. J Henan Univ (Nat Sci) 41(1): 4–9 (in Chinese)

  • Song X, Bian J, Yin W (2013) Six safe grades on weak roman domination. J Henan Univ (Nat Sci) 5:002

    MATH  Google Scholar 

  • Stewart I (1999) Defend the roman empire!. Sci Am 281:136–138

    Article  Google Scholar 

  • Targhi M, Rad NJ, Moradi MS (2012) Properties of independent roman domination in graphs. Australas J Combin 52:11–18

    MathSciNet  MATH  Google Scholar 

Download references

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Correspondence to Marija Ivanović.

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The author declares that there is no conflict of interest regarding the publication of this paper.

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Communicated by V. Loia.

This research was partially supported by the Serbian Ministry of Education, Science and Technological Developments under the Grant No. TR36006.

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Ivanović, M. Improved integer linear programming formulation for weak Roman domination problem. Soft Comput 22, 6583–6593 (2018). https://doi.org/10.1007/s00500-017-2706-4

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