Analyzing Collection Strategies: A Computational Perspective on the Coupon Collector Problem
Hadas Abraham, Ido Feldman, Eitan Yaakobi
TL;DR
This work tackles the Coupon Collector Problem (CCP) in its most general form, where $n$ coupon types are drawn with probabilities $\boldsymbol p$ and the goal is to obtain $t$ copies of any $k$ distinct types. It introduces a Base Algorithm (BA) that computes the exact expectation and variance via a Markov-chain DP on the full state space, and two symmetry-exploiting algorithms—UDA for uniform draws with complexity $O((t+1)^n)$ and DPSA for arbitrary distributions leveraging grouped probability symmetry—to achieve polynomial-time exact results in many practical cases. The methods provide exact moments $\mathbb{E}[\nu_t^{\boldsymbol p}(n,k)]$ and $\mathrm{Var}(\nu_t^{\boldsymbol p}(n,k))$ and demonstrate favorable performance against Monte Carlo simulations, especially when symmetry can be exploited. Collectively, the contributions enable scalable, exact CCP analysis across a range of distributions, with future work aimed at closed-form expressions and further reductions.
Abstract
The Coupon Collector Problem (CCP) is a well-known combinatorial problem that seeks to estimate the number of random draws required to complete a collection of $n$ distinct coupon types. Various generalizations of this problem have been applied in numerous engineering domains. However, practical applications are often hindered by the computational challenges associated with deriving numerical results for moments and distributions. In this work, we present three algorithms for solving the most general form of the CCP, where coupons are collected under any arbitrary drawing probability, with the objective of obtaining $t$ copies of a subset of $k$ coupons from a total of $n$. The First algorithm provides the base model to compute the expectation, variance, and the second moment of the collection process. The second algorithm utilizes the construction of the base model and computes the same values in polynomial time with respect to $n$ under the uniform drawing distribution, and the third algorithm extends to any general drawing distribution. All algorithms leverage Markov models specifically designed to address computational challenges, ensuring exact computation of the expectation and variance of the collection process. Their implementation uses a dynamic programming approach that follows from the Markov models framework, and their time complexity is analyzed accordingly.