Abstract
In this paper, we study the {k}-domination, total {k}-domin-ation, {k}-domatic number, and total {k}-domatic number problems, from complexity and algorithmic points of view. Let k ≥ 1 be a fixed integer and ε > 0 be any constant. Under the hardness assumption of \(NP\not\subseteq DTIME(n^{O(\log\log n)})\), we obtain the following results.
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1
The total {k}-domination problem is NP-complete even on bipartite graphs.
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2
The total {k}-domination problem has a polynomial time ln n approximation algorithm, but cannot be approximated within \((\frac{1}{k}-\epsilon)\ln n\) in polynomial time.
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3
The total {k}-domatic number problem has a polynomial time \((\frac{1}{k}+\epsilon)\ln n\) approximation algorithm, but does not have any polynomial time \((\frac{1}{k}-\epsilon)\ln n\) approximation algorithm.
All our results hold also for the non-total variants of the problems.
This work was supported in part by the National Basic Research Program of China Grant 2007CB807900, 2007CB807901, the National Natural Science Foundation of China Grant 61033001, 61061130540, 61073174.
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References
Aram, H., Sheikholeslami, S.M.: On the total {k}-domination and total {k}-domatic number of graphs. Bull. Malays. Math. Sci. Soc. (to appear)
Bange, D.W., Barkauskas, A.E., Host, L.H., Slater, P.J.: Generalized domination and efficient domination in graphs. Discrete Math. 159, 1–11 (1996)
Berman, P., DasGupta, B., Sontag, E.: Randomized approximation algorithms for set multicover problems with applications to reverse engineering of protein and gene networks. Discrete Appl. Math. 155, 733–749 (2007)
Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer, Berlin (1998)
Chen, J., Hou, X., Li, N.: The total {k}-domatic number of wheels and complete graphs. J. Comb. Optim. (to appear)
Cockayne, E.J., Dawes, R.M., Hedetniemi, S.T.: Total domination in graphs. Networks 10, 211–219 (1980)
Domke, G.S., Hedetniemi, S.T., Laskar, R.C., Fricke, G.: Relationships between integer and fractional parameters of graphs. In: Proceedings of the Sixth Quadrennial Conference on the Theory and Applications of Graphs, Graph Theory, Combinatorics, and Applications, Kalamazoo, MI, vol. 2, pp. 371–387 (1991)
Feige, U.: A threshold of ln n for aproximating set cover. J. ACM 45(4), 634–652 (1998)
Feige, U., Halldórsson, M.M., Kortsarz, G., Srinivasan, A.: Approximating the domatic number. SIAM J. Comput. 32(1), 172–195 (2002)
Gairing, M., Hedetniemi, S.T., Kristiansen, P., McRae, A.A.: Self-stabilizing algorithms for {k}-domination. In: Huang, S.-T., Herman, T. (eds.) SSS 2003. LNCS, vol. 2704, pp. 49–60. Springer, Heidelberg (2003)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Domination in Graphs: Advanced Topics. Marcel Dekker, New York (1998)
Haynes, T.W., Hedetniemi, S.T., Slater, P.J.: Fundamentals of Domination in Graphs. Marcel Dekker, New York (1998)
Lee, C.M.: Labelled Domination and Its Variants. PhD thesis, National Chung Cheng University (2006)
Li, N., Hou, X.: On the total {k}-domination number of Cartesian products of graphs. J. Comb. Optim. 18, 173–178 (2009)
Sheikholeslami, S.M., Volkmann, L.: The {k}-domatic number of a graph. Aequationes Math. (to appear)
Sheikholeslami, S.M., Volkmann, L.: The total {k}-domatic number of a graph. J. Comb. Optim. (to appear)
Vazirani, V.V.: Approximation Algorithms. Springer, Heidelberg (2004)
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He, J., Liang, H. (2011). Complexity of Total {k}-Domination and Related Problems. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_18
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