Complexity of total {k}-domination and related problems

Authors:
Jing He;Hongyu Liang
Affiliations:
Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China;Institute for Interdisciplinary Information Sciences, Tsinghua University, Beijing, China
Venue:
FAW-AAIM'11 Proceedings of the 5th joint international frontiers in algorithmics, and 7th international conference on Algorithmic aspects in information and management
Year:
2011

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Abstract

In this paper, we study the {k}-domination, total {k}-domination, {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 ⊄ DTIME(nO(log log n)), we obtain the following results. 1. The total {k}-domination problem is NP-complete even on bipartite graphs. 2. The total {k}-domination problem has a polynomial time ln n approximation algorithm, but cannot be approximated within ( 1/k-ε) ln n in polynomial time. 3. The total {k}-domatic number problem has a polynomial time (1/k + ε) ln n approximation algorithm, but does not have any polynomial time (1/k-ε) lnn approximation algorithm. All our results hold also for the non-total variants of the problems.