The multi-radius cover problem

Let G = (V, E) be a graph with a non-negative edge length l_u,v for every (u, v) ∈ E. The vertices of G represent locations at which transmission stations are positioned, and each edge of G represents a continuum of demand points to which we should transmit. A station located at v is associated with a set R_v of allowed transmission radii, where the cost of transmitting to radius r ∈ R_v is given by c _v(r). The multi-radius cover problem asks to determine for each station a transmission radius, such that for each edge (u, v) ∈ E the sum of the radii in u and v is at least l_u,v, and such that the total cost is minimized. In this paper we present LP-rounding and primal-dual approximation algorithms for discrete and continuous variants of multi-radius cover. Our algorithms cope with the special structure of the problems we consider by utilizing greedy rounding techniques and a novel method for constructing primal and dual solutions.