In a network in which logical channels are used and switched for communication services, adaptive bandwidth allocation is possible, and new transfer capabilities handling elastic type of traffic such as ABR traffic have been developed in ATM for the B-ISDN. It should be, however, emphasized that appropriate network operation must be developed in order to support the new services in the most effective way from the viewpoint of both network operators and users. In particular, routing schemes suitable for the elastic traffic are needed to study and network operation methods must be investigated for the networks in which the stream traffic like CBR and the elastic traffic are supported in a mixed manner. In this paper, we treat an optimal routing problem for the elastic traffic and propose a state-dependent routing scheme called CPAR. Moreover, an optimal route selection algorithm taking into account the constraints for the routing scheme is presented and a proof of its optimality is provided. It is concluded that CPAR has desirable performance for the elastic class of traffic in cases of partial load increase, in which a route traversing lightly loaded links can be selected with more flexibility and effective load balancing is achieved.