Routing with end-to-end Quality-of-Service (QoS) guarantees is a key to the widespread deployment of recent emerged services. Developing QoS routing algorithm in the network is an important open topic. This paper investigates the QoS routing related problems and proposes a Fully Polynomial Time Approximation Scheme (FPTAS) for QoS routing. In the proposed FPTAS, a graph-extending based dynamic programming approach is developed, and an extended version of the proposed algorithm is studied. The theoretical analyses show that the proposed algorithms outperform the previous best-known studies.