On Conjectures concerning the Labeled Coupon Collector Problem
Dina Barak-Pelleg, Daniel Berend
TL;DR
This work analyzes a labeled variant of the Coupon Collector Problem with group arrivals of size $k$, focusing on the case $k=2$ and two information regimes described by $Q_I(n,2)$ and $Q_{II}(n,2)$. Using a graph-based representation where drawn pairs form edges on the coupon set, the authors show that after all coupons have appeared the remaining uncertainty resides in small connected components, and they bound the residual labeling time by a polylogarithmic term via negative association techniques. They prove two conjectures of Tan et al. by establishing precise asymptotics: $E(Q_I(n,2))=\frac{1}{2} n H_n - \frac{1}{2} n + O(\log^5 n)$ and $E(Q_{II}(n,2))=\frac{1}{2} n H_n + O(\log^5 n)$, implying $E(Q_{II}(n,2)) - E(Q_I(n,2)) = \frac{1}{2} n + o(n)$ and $E(Q_{II}(n,2))=\frac{1}{2} n H_n + o(n)$. The approach leverages concentration results, a detailed analysis of size-2 components, and auxiliary propositions on conditional completion, with potential refinements to even tighter polylogarithmic bounds. The findings demonstrate the efficacy of random-graph and dependency-structure methods for labeled CCP variants and offer sharper asymptotics than previously conjectured.
Abstract
We study a labeled variant of the classical Coupon Collector Problem (CCP), recently introduced by Tan et al., where coupons arrive in groups and only the set of labels is revealed. The goal is to determine the expected number of group drawings required to uniquely identify the labeling of all coupons. We focus on the case where groups consist of pairs ($k=2$), and provide rigorous proofs for two conjectures posed by Tan et al.