Linear Programming Based Near-Optimal Pricing for Laminar Bayesian Online Selection

In the Bayesian online selection problem, the goal is to design a pricing scheme for a sequence of arriving buyers that maximizes the expected social-welfare (or revenue) subject to different types of structural constraints. Inspired by applications in operations management, the focus of this paper is on the cases where the set of served customers is characterized by a laminar matroid.We give the first Polynomial-Time Approximation Scheme (PTAS) for the problem when the laminar matroid has constant depth. Our approach is based on rounding the solution of a hierarchy of linear programming relaxations that approximate the optimum online solution with any degree of accuracy plus a concentration argument that shows the rounding incurs a small loss. We also study another variation, which we call the production constrained problem, for which the allowable set of served customers is characterized by a collection of production and shipping constraints forming a certain form of laminar matroid. Using a similar LP-based approach, we design a PTAS for this problem even when the depth of the laminar matroid is not constant. The analysis exploits the negative dependency of the optimum selection rule in the lower-levels of the laminar family. Finally, we conclude with a discussion of the linear programming based approach employed in the paper and re-derive some of the classic prophet inequalities known in the literature