The problem of finding a path that satisfies two additive constraints, such as delay and cost, has been proved to be NP-complete. Many heuristic and approximation algorithms have been developed to identify a path given a certain QoS request. Unfortunately, these algorithms cannot be applied directly in the Internet because routing in the Internet is based on table lookups and routing tables are computed before a request arrives. In this paper, we develop an approximation algorithm for computing the supported QoS going across a domain. We analyze the approximation error of our algorithm and formally prove that the approximation error of our proposed algorithm is smaller than those of the existing approaches. We further verify our performance using extensive simulations.