几何定价：低成本如何帮助近似性
Geometric Pricing: How Low Dimensionality Helps in Approximability
摘要：
考虑下面的玩具问题。平面上有$m$矩形和$n$点。每个矩形$R$都是一个有预算的消费者$B_R$，如果她有足够的预算，她有兴趣购买R中最便宜的物品（点）。我们的工作是定价的项目，以最大限度地提高收入。这个问题也可以在更高的维度上定义。我们称这个问题为几何定价问题。在本文中，我们研究一类新的问题所产生的几何方面的定价问题。它直观地捕捉了在市场营销研究、医疗经济学等领域被广泛研究的典型现实假设，还有助于对其他著名的定价问题进行分类，如高速公路定价问题和平面图和二分图上的图顶点定价问题。此外，这个问题原来有密切的联系，以其他自然的几何问题，如几何版本的唯一覆盖和最大可行子系统问题。我们发现，在这个定价问题中产生的低维确实导致改进的近似比，提出次线性近似算法的两个中心版本的问题：单位需求均匀预算最小购买和一心一意的定价问题。我们的算法是通过结合算法定价和几何技术。这些结果表明，考虑几何方面可能是获得此类定价问题改进的近似算法的一个有前途的研究方向。据我们所知，这是几何学和算法定价领域交叉的少数问题之一。因此，它的研究可能会导致新的算法技术，可以使这两个领域受益。
英文摘要：
Consider the following toy problem. There are $m$ rectangles and $n$ points on the plane. Each rectangle $R$ is a consumer with budget $B_R$, who is interested in purchasing the cheapest item (point) inside R, given that she has enough budget. Our job is to price the items to maximize the revenue. This problem can also be defined on higher dimensions. We call this problem the geometric pricing problem. In this paper, we study a new class of problems arising from a geometric aspect of the pricing problem. It intuitively captures typical real-world assumptions that have been widely studied in marketing research, healthcare economics, etc. It also helps classify other well-known pricing problems, such as the highway pricing problem and the graph vertex pricing problem on planar and bipartite graphs. Moreover, this problem turns out to have close connections to other natural geometric problems such as the geometric versions of the unique coverage and maximum feasible subsystem problems. We show that the low dimensionality arising in this pricing problem does lead to improved approximation ratios, by presenting sublinear-approximation algorithms for two central versions of the problem: unit-demand uniform-budget min-buying and single-minded pricing problems. Our algorithm is obtained by combining algorithmic pricing and geometric techniques. These results suggest that considering geometric aspect might be a promising research direction in obtaining improved approximation algorithms for such pricing problems. To the best of our knowledge, this is one of very few problems in the intersection between geometry and algorithmic pricing areas. Thus its study may lead to new algorithmic techniques that could benefit both areas.
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