The multi-radius cover problem

Abstract. Let G = (V, E) be a graph with a non-negative edge length lu,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 Rv of allowed transmission radii, where the cost of transmitting to radius r ∈ Rv is given by cv(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 lu,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

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.