Optimality of the decomposition distribution for divisible package sizes

Determine whether the decomposition distribution—uniform sampling from a collection of pairwise disjoint packages of size s that partition the n coupons when s divides n—is always an optimal distribution minimizing the expected number of rounds required to collect all coupons.

Background

For the generalized Coupon Collector’s Problem, each drawing selects an s-element package from n coupons, and the distribution over the possible packages may be chosen arbitrarily. When s divides n, the decomposition distribution assigns equal probability to n/s pairwise disjoint packages whose union is the full coupon set; collecting all coupons under this distribution is equivalent to collecting all of these packages.

The paper reports that Chang and Fang established optimality for the case n even and s=n/2, including uniqueness up to relabeling, and conjectured that the analogous decomposition distribution is always optimal whenever s divides n. The results proved in the paper show that certain non-uniform distributions outperform the uniform distribution, but do not establish the claimed universal optimality of the decomposition distribution.