Abstract: In the research of service composition, it demands efficient algorithms that not only retrieve correct service compositions automatically from thousands of services but also satisfy the quality requirements of different service users. However, most approaches treat these two aspects as two separate problems, automatic service composition and service selection. Although the latest researches realize the restriction of this separate view and some specific methods are proposed, they still suffer from serious limitations in scalability and accuracy when addressing both requirements simultaneously. In order to cope with these limitations and efficiently solve the combined problem which is known as QoS-aware or QoS-driven automatic service composition problem, we propose a new graph search problem, single-source optimal directed acyclic graphs (DAGs), for the first time. This novel single-source optimal DAGs (SSOD) problem is similar to, but more general than the classical single-source shortest paths (SSSP) problem. In this paper, a new graph model of SSOD problem is proposed and a Sim-Dijkstra algorithm is presented to address the SSOD problem with the time complexity of O(n log n + m) (n and m are the number of nodes and edges in the graph respectively), and the proofs of its soundness. It is also directly applied to solve the QoS-aware automatic service composition problem, and a service composition tool named QSynth is implemented. Evaluations show that Sim-Dijkstra algorithm achieves superior scalability and efficiency with respect to a large variety of composition scenarios, even more efficient than our worklist algorithm that won the performance championship of Web Services Challenge 2009.