Upper and Lower Bounds on Approximating Weighted Mixed Domination
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A mixed dominating set of a graph G = (V, E) is a mixed set D of vertices and edges, such that for every edge or vertex, if it is not in D, then it is adjacent or incident to at least one vertex or edge in D. The mixed domination problem is to find a mixed dominating set with a minimum cardinality. It has applications in system control and some other scenarios and it is NP-hard to compute an optimal solution. This paper studies approximation algorithms and hardness of the weighted mixed dominating set problem. The weighted version is a generalization of the unweighted version, where all vertices are assigned the same nonnegative weight w_v and all edges are assigned the same nonnegative weight w_e, and the question is to find a mixed dominating set with a minimum total weight. Although the mixed dominating set problem has a simple 2-approximation algorithm, few approximation results for the weighted version are known. The main contributions of this paper include: [1.] for w_e≥ w_v, a 2-approximation algorithm; [2.] for w_e≥ 2w_v, inapproximability within ratio 1.3606 unless P=NP and within ratio 2 under UGC; [3.] for 2w_v > w_e≥ w_v, inapproximability within ratio 1.1803 unless P=NP and within ratio 1.5 under UGC; [4.] for w_e< w_v, inapproximability within ratio (1-ϵ) |V| unless P=NP for any ϵ >0.
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