Covering a line segment with variable radius discs

The paper addresses the problem of locating sensors with a circular field of view so that a given line segment is under full surveillance, which is termed as the disc covering problem on a line. The cost of each sensor includes a fixed component f, and a variable component that is a convex function of the diameter of the field-of-view area. When only one type of sensor or, in general, one type of disc, is available, then a simple polynomial algorithm solves the problem. When there are different types of sensors, the problem becomes hard. A branch-and-bound algorithm as well as an efficient heuristic are developed for the special case in which the variable cost component of each sensor is proportional to the square of the measure of the field-of-view area. The heuristic very often obtains the optimal solution as shown in extensive computational testing. Scope and purpose: Problems of locating facilities to cover sets of points on networks and planes have been widely studied. This paper focuses on a new covering problem that is motivated by an application where a line segment is to be kept under surveillance using different types of radars. Using reasonable assumptions, some nonlinear covering problems are formulated. Efficient exact algorithms and heuristics are developed and analyzed for "easy" and "hard" cases, respectively.