We discuss the multicast communication in the self-routing multistage interconnection network (MIN) for constructing the internal architecture of asynchronous transfer mode (ATM) switches. Many applications of ATM switches require multicast communications in addition to conventional point-to-point communications. This paper presents a novel approach to supporting multicast communication, on the basis of the recursive scheme that recycles a multicast packet one or more times through the network to reach at desired destinations. We also propose cost-effective multicast algorithms providing deadlock-freedom in MIN-based ATM switches. The proposed algorithms require a small and fixed number of recycling passes and a reasonable number of links used. The proposed algorithms can be easily applicable to buffered MIN-based ATM switches.