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The bandwidth-constrained routing problem (BCRP) asks for a route that has sufficient bandwidth for data transmission. When BCRP is defined for wired networks, it can be solved in polynomial time. On the other hand, when it is defined for wireless ad hoc networks, it is NP-complete if the underlying MAC protocol is TDMA-based or CDMA-over-TDMA-based. In this paper, we show that BCRP is still NP-complete, even if CSMA-based or contention-based CDMA MAC prot.ocols are used. Besides, we show that BCRP is polynomial-time solvable if the channel model is collision-free and the scheduling policy is FIFO. In wireless ad hoc networks, no MAC protocol was designed before, which would lead to a polynomial-time solution to BCRP. The results of this paper suggest a design for MAC protocols that can support QoS routing well.