  In this paper, we study a very general type of online network design problem,
and generalize two different previous algorithms, one for an online network
design problem due to Berman and Coulston [4] and one for (offline) general
network design problems due to Goemans and Williamson [9]; we give an O(log
k)-competitive algorithm, where k is the number of nodes that must be
connected. We also consider a further generalization of the problem that allows
us to pay penalties in exchange for violating connectivity constraints; we give
an online O(log k)-competitive algorithm for this case as well.
