Price-setting Problems and Matroid Bayesian Online Selection

Abstract

We study a class of Bayesian online selection problems with matroid constraints. Consider a seller who has several items they wish to sell, with the set of sold items being subject to some structural constraints, e.g., the set of sold items should be independent with respect to some matroid. Each item has an offer value drawn independently from a known distribution. In a known order, items arrive and drawn values are presented to the seller. If the seller chooses to sell the item, they gain the drawn value as revenue. Given distribution information for each item, the seller wishes to maximize their expected revenue by carefully choosing which offers to accept as they arrive.

Such problems have been studied extensively when the seller's revenue is compared with the offline optimum, referred to as the "prophet". In this setting, a tight 2-competitive algorithm is known when the seller is limited to selling independent sets of a matroid [KW12]. We turn our attention to the online optimum, or "philosopher", and ask how well the seller can do with polynomial-time computation, compared to a seller with unlimited computation but with the same limited distribution information about offers.

We show that when the underlying constraints are laminar and the arrival of buyers follows a natural "left-to-right" order, there is a polynomial-time approximation scheme for maximizing the seller's revenue. We also show that such a result is impossible for the related case when the underlying constraints correspond to a graphic matroid. In particular, it is PSPACE-hard to approximate the philosopher's expected revenue to some fixed constant α < 1; moreover, this cannot be alleviated by requirements on the arrival order in the case of graphic matroids. We also show similar hardness results for both transversal and cographic matroids.

We then turn our attention to a related problem where the arrival order of items is unknown and uniformly random. In this setting, we show that there is a polynomial-time approximation scheme whenever the rank of the matroid is bounded above by a constant and all probabilities in the input are bounded below by a constant.

We additionally examine the computational complexity of computing the prophet's expected revenue. We show that this problem is #P-hard, and give a fully polynomial-time randomized approximation scheme for the problem.