Random network topologies have been proposed as a low-latency network for parallel computers. Although multicast is a common collective-communication operation, multicast algorithms each of which consists of a large number of unicasts are not well optimized for random network topologies. In this study, we firstly apply a two-opt algorithm for building efficient multicast on random network topologies. The two-opt algorithm creates a skilled ordered list of visiting nodes to minimize the total path hops or the total possible contention counts of unicasts that form the target multicast. We secondly extend to apply the two-opt algorithm for the other collective-communication operations, e.g., allreduce and allgather. The SimGrid discrete-event simulation results show that the two-opt multicast outperforms that in typical MPI implementation by up to 22% of the execution time of an MPI program that repeats the MPI_Bcast function. The two-opt allreduce and the two-opt allgather operations also improve by up to 15% and 14% the execution time when compared to those used in typical MPI implementations, respectively.