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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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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DOI: https://doi.org/10.1007/978-3-642-21204-8_18
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