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We propose a distributed QoS routing scheme, called multi-constrained pre-computation based selective probing (MC-PCSP) scheme, to provide an exact solution of the complex multi-constrained shorted path (MCSP) problem. This MC-PCSP scheme is designed to support both the additive and non-additive QoS requirements in a unified mechanism, considering the practical environment where the state information available for the routing decision is not precise. It adopts and enhances the conditional selective probing approach of recently proposed PCSP scheme to reduce the message overhead without sacrificing the ideal success ratio and the cost optimally. MC-PCSP scheme pre-computes multiple best quality paths - one per each QoS metric - to support multiple QoS requirements and uses them carefully to exclude the non-optimal paths as well as the non-feasible paths from probing process, thereby guaranteeing effective and optimal search. Computer simulation reveals that the MC-PCSP scheme maintains low message overhead and guarantees ideal success ratio and optimal search for the multiple QoS requirements with imprecise state information.