A variety of geometric network optimization problems on a set of points is presented, in which a resource bound on the total length of the network and how to maximize the number of points visited are discussed. The open problem on the approximability of the rooted `orienteering problem' is solved for the case in which the sites are given as points in the plane and the network required is a cycle. Approximation algorithms for variants of this problem in which the network required is a tree (3-approximation) or a path (2-approximation) are obtained.
