On the multi-radius cover problem

An instance of the multi-radius cover problem consists of a graph G = (V,E) with edge lengths l : E → R. Each vertex u ∈ V represents a transmission station for which a transmission radius ru must be picked. Edges represent a continuum of demand points to be satisfied, that is, for every edge (u, v) ∈ E we ask that ru + rv ≥ luv. The cost of transmitting at radius r from vertex u is given by an arbitrary non-decreasing cost function cu(r). Our goal is to find a cover with minimum total cost P u cu(ru). The multi-radius cover problem is NP-hard as it generalizes the well-known vertex cover problem. In this paper we present a 2-approximation algorithm for it.