On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
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In the problem of minimum connected dominating set with routing cost constraint, we are given a graph G=(V,E), and the goal is to find the smallest connected dominating set D of G such that, for any two non-adjacent vertices u and v in G, the cost of routing between u and v through D (the number of internal nodes on the shortest path between u and v in G[D ∪{u,v}]) is at most α times that through V. For general graphs, the only known previous approximability result is an O( n)-approximation algorithm (n=|V|) for the case α = 1 by Ding et al. For 1 < α < 5, no non-trivial approximation algorithm was previously known even on special graphs like unit disk graphs. When α > 1, we give an O(n^1-1/α( n)^1/α)-approximation algorithm. When α≥ 5, we give an O(√(n) n)-approximation algorithm. Finally, we prove that, when α =2, unless NP ⊆ DTIME(n^poly n), the problem admits no polynomial-time 2^^1-ϵn-approximation algorithm, for any constant ϵ > 0, improving upon the Ω( n) bound by Du et al. (albeit under a stronger hardness assumption).
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