Polynomial algorithms for parametric minquantile and maxcovering planar location problems with locational constraints

A location is sought within some convex region of the plane for the central site of some public service to a finite number of demand points. The parametric maxcovering problem consists in finding for each R > 0 the point from which the total weight of the demand points within distance R is maximal. The parametric minimal quantile problem asks for each percentage α the point minimising the distance necessary for covering demand points of total weight at least α. We investigate the properties of these two closely related problems and derive polynomial algorithms to solve them both in case of either (possibly inflated) Euclidean or polyhedral distances.