Approximating activation edge-cover and facility location problems

What approximation ratio can we achieve for the Facility Location problem if whenever a client u connects to facility v, opening cost of v is at most θ times service u? We show that this and many other problems are particular case Activation Edge-Cover problem. Here given multigraph G=(V,E), set R⊆V terminals, thresholds {tue,tve} each uv-edge e∈E. The goal find an assignment a={av:v∈V} nodes minimizing ∑v∈Vav, such edge Ea={e=uv:au≥tue,av≥tve} activated by covers R. obtain 1+maxx≥1⁡ln⁡x1+x/θ≈ln⁡θ−ln⁡ln⁡θ problem, where θ≤maxe∈E⁡max⁡{tue,tve}min⁡{tue,tve} parameter. This result based on simple generic algorithm sum decreasing sub-additive functions, which independent interest. As application, get same above variant Location. If all costs identical then better 1+maxk∈N⁡Hk−11+k/θ, Hk=∑i=1k1/i. For Min-Power improve 1.406 [4] (achieved iterative randomized rounding) 1.2785. unit 73/60≈1.217 15551347≈1.155.