Multiobjective Disk Cover Admits a PTAS

We introduce multiobjective disk cover problems and study their approximability. We construct a polynomial-time approximation scheme (PTAS) for the multiobjective problem where k types of points (customers) in the plane have to be covered by disks (base stations) such that the number of disks is minimized and for each type of points, the number of covered points is maximized. Our approximation scheme can be extended so that it works with the following additional features: interferences, different services for different types of customers, different shapes of supply areas, weighted customers, individual costs for base stations, and payoff for the quality of the obtained service. Furthermore, we show that it is crucial to solve this problem in a multiobjective way, where all objectives are optimized at the same time. The constrained approach (i.e., the restriction of a multiobjective problem to a single objective) often used for such problems can significantly degrade their approximability. We can show non-approximability results for several singleobjective restrictions of multiobjective disk cover problems. For example, if there are 2 types of customers, then maximizing the supplied customers of one type is not even approximable within a constant factor, unless P = NP.