The Bayesian online selection problem aims to design a pricing scheme for a sequence of arriving buyers that maximizes the expected social welfare (or revenue) subject to different structural constraints. Inspired by applications with a hierarchy of service, this paper focuses on the cases where a laminar matroid characterizes the set of served buyers. 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 showing that rounding incurs a small loss. We also study another variation, which we call the production-constrained problem. The allowable set of served buyers is characterized by a collection of production and shipping constraints that form a particular example of a laminar matroid. Using a similar LP-based approach, we design a PTAS for this problem, although in this special case the depth of the underlying laminar matroid is not necessarily a constant. The analysis exploits the negative dependency of the optimum selection rule in the lower levels of the laminar family. Finally, to demonstrate the generality of our technique, we employ the linear programming-based approach employed in the paper to re-derive some of the classic prophet inequalities known in the literature -- as a side result.

翻译：贝叶斯在线选择问题旨在设计针对到达买家的序列的定价方案，在满足不同结构约束的前提下最大化期望社会福利（或收入）。受具有服务层级结构的应用启发，本文聚焦于由层状拟阵刻画服务买家集的情形。当层状拟阵具有恒定深度时，我们首次给出该问题的多项式时间近似方案（PTAS）。我们的方法基于对线性规划松弛层级解进行舍入，该松弛能以任意精度逼近最优在线解，同时结合集中性论证表明舍入仅导致微小损失。我们还研究了另一个变体，即生产约束问题。可服务的买家集由一组生产和运输约束刻画，这些约束构成层状拟阵的一个特例。利用类似的基于线性规划的方法，我们为此问题设计了PTAS，尽管在此特例中底层层状拟阵的深度未必为常数。分析利用了层状族低层级中最优选择规则的负相关性。最后，为展示我们技术的普适性，本文采用的基于线性规划的方法还被用于重新推导文献中已知的经典先知不等式——作为辅助结果。