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

TL;DR

The first Polynomial-Time Approximation Scheme (PTAS) for the problem when the laminar matroid has constant depth is given, based on rounding the solution of a hierarchy of linear programming relaxations that approximate the optimum online solution with any degree of accuracy.

Abstract

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.

Figures (5)

• Figure 1: The "hierarchical" map of service level network.
• Figure 2: Characterization of our hierarchy of linear programming relaxations.
• Figure 3: Bad example showing lack of negative dependency for optimal online policy of general laminar matroid.
• Figure 4: Depth-based marking for concentration.
• Figure 5: Production constrained Bayesian selection as a special case of Laminar Bayesian selection; each of the types $\circ$(red), $\times$(blue), and $\diamond$(green) has a corresponding collection of nested bins (path laminar), and these bins are inside an outer bin $[n]$ to model the shipping constraint.

Theorems & Definitions (27)

• Proposition 1
• Proposition 2
• proof
• Proposition 3
• Remark 1
• proof
• Example 1
• Theorem 1
• proof
• Remark 2: computing prices and tie-breaking probabilities in \ref{['prop:adaptive-pricing']}