Coupons collecting with or without replacement, and with multipurpose coupons

The classic Coupon-Collector Problem (CCP) is generalized to the extent that each coupons serves certain “purposes”. Only basic probability theory is used. Centerpiece rather is an algorithm that efficiently counts all $k$-element transversals of a set system.

💡 Research Summary

The paper presents a substantial generalization of the classic Coupon‑Collector Problem (CCP) by allowing each coupon to satisfy multiple goals simultaneously, a situation the authors refer to as “multipurpose coupons.” Formally, a finite set of coupons (W) of size (w) and a family of non‑empty subsets (G={G_1,\dots,G_h}) are introduced, where each (G_i) contains the coupons that achieve the (i)-th goal (or property). A subset (X\subseteq W) that intersects every (G_i) is called a transversal (or hitting set) of (G); such a set represents a collection of coupons that together fulfill all goals. The central random experiment is a sequence of draws from an urn containing the coupons, either with replacement (each draw is independent and uniformly random) or without replacement (draws are a random permutation of the coupons). A trial is successful if after a certain number of draws every goal has appeared at least once; it is sharply successful if the last draw is the one that finally completes the set of goals.

The authors first derive a simple relationship between the number (\tau_k) of (k)-element transversals and the probability (q_k) that a length‑(k) trial without replacement is successful: \