Nearly Optimal Pricing Algorithms for Production Constrained and Laminar Bayesian Selection

Abstract

In the Bayesian online selection problem, the goal is to find a pricing algorithm for serving a sequence of arriving buyers that maximizes the expected social-welfare (or revenue) subject to different types of structural constraints. The focus of this paper is on the case where the allowable subsets of served customers are characterized by a laminar matroid with constant depth. This problem is a special case of the well-known matroid Bayesian online selection problem studied in [Kleinberg & Weinberg, 2012], when the underlying matroid is laminar. We give the first Polynomial-Time Approximation Scheme (PTAS) for the above problem. Our approach is based on rounding the solution of a hierarchy of linear programming relaxations that can approximate the optimum online solution with any degree of accuracy as well as a concentration argument that shows our rounding does not have a considerable loss in the expected social welfare. We also introduce the production constrained problem, for which the allowable subsets of served customers are characterized by joint production/shipping constraints that can be modeled by a special case of laminar matroids. We show that by leveraging the special structure of this problem, and using a similar approach as before, we can design a PTAS for this problem too even in the case where the depth of the laminar matroid is not constant. To achieve our result we exploit the negative dependence property of the selection rule in the lower-levels of the laminar family.