Before revision:  Additionally, ] \choose } \to _{\geq 0}, our approximate exchange property implies that a simple local search algorithm gives a k^{O(k)}-approximation of S \mu(S) when \mu is generated by a log-concave polynomial, and that greedy gives the same approximation ratio when \mu is strongly Rayleigh.
After revision:  In particular, given function \mu: ] \choose } \to _{\geq 0}, our approximate exchange property implies that a simple local search algorithm gives a k^{O(k)}-approximation of S \mu(S) when \mu is generated by a log-concave polynomial, and that greedy gives the same approximation ratio when \mu is strongly Rayleigh.