In chapter 3 we introduced*Rmin*(*M*), the minimum number of recombination nodes used in any ARG to derive the set of sequences*M*, and we noted that the problem of computing*Rmin*(*M*) is known to be NP-hard. Hence no provably correct, worst-case polynomial-time algorithm is known for computing*Rmin*(*M*), and we do not expect there will be one. However, several worst-case polynomial-time algorithms have been developed that compute empirically good*lower bounds*on*Rmin*(*M*), and there are several other lower-bound methods whose worst case time is not polynomially bounded, but are fast in practice. Some of the lower bounds apply