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The development of efficient quality of service (QoS) routing algorithms in a high-speed network environment is a very important and at the same time very difficult task due to the need to provide divergent services with multiple QoS requirements. Heuristic algorithms based on Lagrange relaxation techniques have been proposed to resolve the contradiction between the time complexity and the quality of solution. We investigate the performance of two heuristic algorithms, LR_DCLC and NR_DCLC, for the delay-constrained least-cost (DCLC) routing problem. Algorithm LR_DCLC is based on linear relaxation, while algorithm NR_DCLC, which is proposed in this paper, is based on nonlinear relaxation. A large number of simulations demonstrate that even though both algorithms have very good performance, NR_DCLC can obtain much better solutions than LR_DCLC by running Dijkstra's (1959) algorithm on average a few more times, especially in the case when the optimal solutions are hard to find.