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In this paper, we study the minimum-cost quality-of-service multicast and unicast routing problems in communication networks. For the multicast problem, we present an efficient approximation algorithm to find a balance between a minimum-cost multicast tree and a minimum-delay multicast tree, with a provably good performance under the condition that link delay and link cost are identical. For the unicast problem, we present an efficient primal-dual heuristic algorithm to find a path which balances path cost and path delay, together with an error bound. The lack of a provably good performance for the second algorithm is complemented by computational results on randomly generated networks. Our algorithm finds optimal solutions in more than 80% of the cases and finds close to optimal solutions in all other cases, while using much less time.