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The Dominated Coloring Problem and Its Application

  • Yen Hung Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8584)

Abstract

Given a graph G = (V,E), a set \(\mathcal{Z}\) of V is dominated by a vertex v ∈ V if v is adjacent to all vertices of \(\mathcal{Z}\). A proper coloring in G is an assignment of colors to each vertex of V such that any two adjacent vertices have different colors. A dominated coloring in G is a proper coloring such that each color class is dominated by a vertex in V. The dominated coloring problem is to find a dominated coloring \(\mathcal{D}\) in G such that the number of color classes in \(\mathcal{D}\) is minimized. In this paper, we provide the first possible application of the dominated coloring problem that is the development of interpersonal relationships in the social network. Since the dominated coloring problem has been listed to be NP-hard, we prove that the dominated coloring problem has a lower bound of \(\frac{|V|^{(1-\epsilon)}+1}{1.5}\) on the best possible approximation ratio in general graphs, for any ε > 0, and is NP-complete even when the graphs are bipartite graphs. Then we design the first non-trivial brute force exact algorithm to solve the dominated coloring problem in general graphs. Since the dominated coloring problem has a lower bound of \(\frac{|V|^{(1-\epsilon)}+1}{1.5}\) on the best possible approximation ratio in general graphs, we present an H(d)-approximation algorithm that is faster but still exponential-time, where d is the maximum degree of G and H(d) is the d-th harmonic number. The H(d)-approximation algorithm can run in polynomial time in bipartite graphs.

Keywords

computational complexity approximation algorithm non-trivial brute force exact algorithm dominated coloring problem development of interpersonal relationships in the social network 

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References

  1. 1.
    Berge, C.: Theory and Graphs and its Applications. Methuen, London (1962)zbMATHGoogle Scholar
  2. 2.
    Cockayne, E.J., Hedetniemi, S.T.: Towards a theory of domination in graph. Networks 7, 247–261 (1977)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Ore, O.: Theory of Graphs. American Mathematical Society Colloquium Publications (1962)Google Scholar
  4. 4.
    Biggs, N.J., Lloyd, E.K., Wilson, R.J.: Graph Theory 1736–1936. Oxford University Press (1999)Google Scholar
  5. 5.
    Chartrand, G., Zhang, P.: Introduction to Graph Theory. McGraw-Hill, Boston (2005)zbMATHGoogle Scholar
  6. 6.
    Harary, F.: Graph Theory. Addison-Wesley, MA (1969)Google Scholar
  7. 7.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, San Francisco (1979)zbMATHGoogle Scholar
  8. 8.
    Bar-Yehuda, R., Moran, S.: On approximation problems related to the independent set and vertex cover problems. Discrete Applied Mathematics 9, 1–10 (1984)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Halldorsson, M.M.: A still better performance guarantee for approximate graph coloring. Information Processing Letters 45, 19–23 (1993)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. Journal of Computer and System 9, 256–278 (1974)CrossRefzbMATHGoogle Scholar
  11. 11.
    Slavik, P.: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25, 237–254 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Wigderson, A.: Improving the performance guarantee for approximate graph coloring. Journal of the ACM 30, 729–735 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Bjorklund, A., Husfeldt, T., Koivisto, M.: Set partitioning via inclusion-exclusion. SIAM Journal on Computing 39, 546–563 (2009)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Byskov, J.M.: Enumerating maximal independent sets with applications to graph colouring. Operations Research Letters 32, 547–556 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Chang, G.J.: Algorithmic aspects of domination in graphs. In: Handbook of Combinatorial Optimization, vol. 3. Kluwer Academic Publishers, Boston (1998)Google Scholar
  16. 16.
    Eppstein, D.: Small maximal independent sets and faster exact graph coloring. Journal of Graph Algorithms and Applications 7, 131–140 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Fomin, F., Grandoni, F., Kratsch, D.: A measure & conquer approach for the analysis of exact algorithms. Journal of the ACM 56, 1–32 (2009)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)zbMATHGoogle Scholar
  19. 19.
    Lawler, E.L.: A note on the complexity of the chromatic number problem. Information Processing Letters 5, 66–67 (1976)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    van Rooij, J.M.M., Nederlof, J., van Dijk, T.C.: Inclusion/Exclusion meets measure and conquer: Exact algorithms for counting dominating sets. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 554–565. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    van Rooiij, J.M.M., Bodlaender, H.L.: Exact algorithms for dominating set. Discrete Applied Mathematics 159, 2147–2164 (2011)CrossRefMathSciNetGoogle Scholar
  22. 22.
    West, D.B.: Introduction to Graph Theory. Prentice Hall, USA (2001)Google Scholar
  23. 23.
    Eubank, S., Kumar, V.S.A., Madhav, M.V., Srinivasan, A., Wang, N.: Structural and algorithmic aspects of massive social networks. In: Proceedings of the 15th Annual ACM-SIAM symposium on Discrete Algorithms (SODA 2004), New Orleans, LA, USA, pp. 718–727 (2004)Google Scholar
  24. 24.
    Friedman, R., Kogan, A.: Efficient power utilization in multi-radio wireless ad hoc networks. In: Proceedings of the International Conference on Principles of Distributed Systems (OPODIS 2009), Nimes, France, pp. 159–173 (2009)Google Scholar
  25. 25.
    Fritsch, R., Fritsch, G.: The Four Color Theorem: History, Topological Foundations and Idea of Proof. Springer, New York (1998)CrossRefzbMATHGoogle Scholar
  26. 26.
    Hooker, J.N., Garfinkel, R.S., Chen, C.K.: Finite dominating sets for network location problems. Operations Research 39, 100–118 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Kelleher, L.L., Cozzens, M.B.: Dominating sets in social networks. Mathematical Social Sciences 16, 267–279 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    Mihelic, J., Robic, B.: Facility location and covering problems. In: Proceedings of the 7th International Multiconference Information Society, vol. D–Theoretical Computer Science, Ljubljana, Slovenia (2004)Google Scholar
  29. 29.
    Shamizi, S., Lotfi, S.: Register allocation via graph coloring using an evolutionary algorithm. In: Panigrahi, B.K., Suganthan, P.N., Das, S., Satapathy, S.C. (eds.) SEMCCO 2011, Part II. LNCS, vol. 7077, pp. 1–8. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  30. 30.
    Stecke, K.: Design planning, scheduling and control problems of flexible manufacturing. Annals of Operations Research 3, 3–12 (1985)CrossRefGoogle Scholar
  31. 31.
    Wang, F., Camacho, E., Xu, K.: Positive influence dominating set in online social networks. In: Du, D.-Z., Hu, X., Pardalos, P.M. (eds.) COCOA 2009. LNCS, vol. 5573, pp. 313–321. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  32. 32.
    Boumediene Merouane, H., Haddad, M., Chellali, M., Kheddouci, H.: Dominated coloring of graphs. In: 11th Cologne-Twente Workshop on Graphs and Combinatorial Optimization (CTW 2012), Munich, Germany, pp. 189–192 (2012)Google Scholar
  33. 33.
    Borgatti, S.P., Mehra, A., Brass, D.J., Labianca, G.: Network analysis in the social sciences. Science 323, 892–895 (2009)CrossRefGoogle Scholar
  34. 34.
    Freeman, L.C.: The Development of Social Network Analysis: A Study in the Sociology of Science. Empirical Press (2004)Google Scholar
  35. 35.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing 3, 103–128 (2007)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Woeginger, G.J.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  37. 37.
    Kann, V.: On the Approximability of NP-complete Optimization Problems. PhD thesis, Department of Numerical Analysis and Computing Science, Royal Institute of Technology, Stockholm (1992)Google Scholar
  38. 38.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43, 425–440 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  39. 39.
    Moon, J.W., Moser, L.: On cliques in graphs. Israel Journal of Mathematics 3, 23–28 (1965)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    Robson, J.M.: Algorithms for maximum independent sets. Journal of Algorithms 7, 425–440 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    Bourgeois, N., Escoffier, B., Paschos, V.T., van Rooij, J.M.M.: Fast algorithms for max independent set. Algorithmica 62, 382–415 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. MIT Press, Cambridge (2009)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Yen Hung Chen
    • 1
  1. 1.Department of Computer ScienceUniversity of TaipeiTaipeiTaiwan, R.O.C.

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