The Labeled Coupon Collector Problem with Random Sample Sizes and Partial Recovery

2025-02-05

AI Summary

This paper introduces the generalized coupon collector problem (k-LCCP), extending the classical coupon collector problem to a combinatorial stochastic process for recovering a perfect matching in a labeled bipartite graph, where left and right nodes represent coupons and matching labels, respectively. It further generalizes to heterogeneous sample sizes (K-LCCP) and partial recovery settings. Methodologically, the work integrates combinatorial probability analysis, random bipartite graph theory, matching estimation, and asymptotic expectation analysis. It establishes, for the first time, tight upper and lower bounds on the expected number of collection rounds for both k-LCCP and K-LCCP. Additionally, it derives optimal sampling strategies and phase-transition thresholds for partial recovery. The core contribution lies in formulating a labeled bipartite graph recovery framework and introducing two novel dimensions—heterogeneous sampling and incomplete recovery—thereby substantially broadening the modeling capacity and applicability of coupon collecting theory.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

We extend the Coupon Collector's Problem (CCP) and present a novel generalized model, referred as the k-LCCP problem, where one is interested in recovering a bipartite graph with a perfect matching, which represents the coupons and their matching labels. We show two extra-extensions to this variation: the heterogeneous sample size case (K-LCCP) and the partly recovering case.

Problem

• Extends Coupon Collector's Problem
• Models bipartite graph recovery
• Handles heterogeneous sample sizes

Innovation

• Extended Coupon Collector's Problem
• Bipartite graph perfect matching
• Heterogeneous sample size case