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 ⊆ V of terminals, and thresholds {t^e_u, t^e_v} for each uv-edge e ∈ E. The goal is to find an assignment a = {av : v ∈ V } to the nodes minimizing ∑ _v∈_V av, such that the edge set Ea = {e = uv : au ≥ t^e_u, av ≥ t^e_v} activated by a covers R. We ln x obtain ratio (Formula presented.) − ln ln θ for the problem, where 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 the same ratio for the above variant of Facility Location. If for each facility all service costs are identical then we show a better ratio (Formula presented.) , where H_k = ∑k _i=1 1/i. For the Min-Power Edge-Cover problem we improve the ratio 1.406 of [4] (achieved by iterative randomized rounding) to 1.2785. For unit thresholds we improve the ratio 73/60 ≈ 1.217 of [4] to 1555/1347 ≈ 1.155.