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We consider the problem of routing the lightpaths of a logical topology of a WDM network on an arbitrary physical topology, such that the logical topology remains connected even after the failure of a physical link. In a previous paper Modiano et. al. (see Proc. IEEE INFOCOM'01, 2001) introduced the notion of survivable routing and established a necessary and sufficient condition for the existence survivable routes of a logical topology in a physical topology. In this paper we show that the problem of determining whether survivable routing is possible for a logical topology in a given physical topology is an NP-complete problem. Moreover we show that the problem remains NP-complete, even when the logical topology is restricted to be a ring with a specific ordering of the nodes.