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 a facility $v$,the opening cost of $v$ is at most $θ$ times the service cost of $u$? We show that this and many other problems are a particular case of the Activation Edge-Cover problem. Here we are given a multigraph $G=(V,E)$, a set $R \subseteq V$ of terminals, and thresholds $\{t^e_u,t^e_v\}$ for each $uv$-edge $e \in E$. The goal is to find an assignment ${\bf a}=\{a_v:v \in V\}$ to the nodes minimizing $\sum_{v \in V} a_v$, such that the edge set $E_{\bf a}=\{e=uv: a_u \geq t^e_u, a_v \geq t^e_v\}$ activated by ${\bf a}$ covers $R$. We obtain ratio $1+ω(θ) \approx \ln θ-\ln \ln θ$ for the problem, where $ω(θ)$ is the root of the equation $x+1=\ln(θ/x)$ and $θ$ is a problem parameter. This result is based on a simple generic algorithm for the problem of minimizing a sum of a decreasing and a sub-additive set functions, which is of independent interest. As an application, we get that the above variant of Facility Location admits ratio $1+ω(θ)$; if for each facility all service costs are identical then we show a better ratio $\displaystyle 1+\max_{k \geq 1} \frac{H_k-1}{1+k/θ}$, where $H_k=\sum_{i=1}^k 1/i$. For the Min-Power Edge-Cover problem we improve the ratio $1.406$ of Calinescu et. al. (achieved by iterative randomized rounding) to $1+ω(1)<1.2785$. For unit thresholds we improve the ratio $73/60 \approx 1.217$ to $\frac{1555}{1347} \approx 1.155$.