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A key issue in any QoS routing framework is how to compute a path that satisfies given QoS constraints. We focus on the path computation problem subject to bandwidth and delay constraints. This problem can be solved easily if the exact state information is available to the node computing the path. In practice, nodes have only imprecise knowledge of the network state. Reliance on outdated information and treating it as exact can significantly degrade the effectiveness of the path selection. We adopt a probabilistic approach in which the state parameters (available bandwidth and delay) are characterized by random variables. The goal is then to find the most-probable bandwidth-delay-constrained path (MP-BDCP). We provide efficient solutions for the MP-BDCP problem by decomposing it into the most-probable delay-constrained path (MP-DCP) problem and the most-probable bandwidth-constrained path (MP-BCP) problem. MP-DCP by itself is known to be NP-hard, necessitating the use of approximate solutions. We use the central limit theorem and Lagrange relaxation techniques to provide two complementary solutions for MP-DCP. These solutions are highly efficient, requiring on average a few iterations of Dijkstra's shortest path algorithm. As for MP-BCP, it can be easily transformed into a variant of the shortest path problem. Our MP-DCP and MP-BCP solutions are then combined to obtain a set of near-nondominated paths for the MP-BDCP problem. Decision makers can then select one or more of these paths based on a specific utility function. Extensive simulations demonstrate the efficiency of the proposed algorithmic solutions and, more generally, to contrast the probabilistic path selection approach with the standard threshold-based triggered approach.