We study an envy-free pricing problem, in which each buyer wishes to buy a shortest path connecting her individual pair of vertices in a network owned by a single vendor. The vendor sets the prices of individual edges with the aim of maximizing the total revenue generated by all buyers. Each customer buys a path as long as its cost does not exceed her individual budget. In this case, the revenue generated by her equals the sum of prices of edges along this path. We consider the unlimited supply setting, where each edge can be sold to arbitrarily many customers. The problem is to find a price assignment which maximizes vendor's revenue. A special case in which the network is a tree is known under the name of the tollbooth problem. Gamzu and Segev proposed a $\mathcal{O} \left( \frac{\log m}{\log \log m} \right)$-approximation algorithm for revenue maximization in that setting. Note that paths in a tree network are unique, and hence the tollbooth problem falls under the category of single-minded bidders, i.e., each buyer is interested in a single fixed set of goods. In this work we step out of the single-minded setting and consider more general networks that may contain cycles. We obtain an algorithm for pricing cactus shaped networks, namely networks in which each edge can belong to at most one simple cycle. Our result is a polynomial time $\mathcal{0} \left( \frac{\log m}{\log \log m}\right)$-approximation algorithm for revenue maximization in tollbooth pricing on a cactus graph. It builds upon the framework of Gamzu and Segev, but requires substantially extending its main ideas: the recursive decomposition of the graph, the dynamic programming for rooted instances and rounding the prices.

翻译：我们研究一个无嫉妒定价问题：在单一供应商拥有的网络中，每位买家希望购买连接其个人顶点对的最短路径。供应商设定各条边的价格，旨在最大化所有买家产生的总收益。只要路径成本不超过个人预算，每位顾客便会购买该路径，此时其产生的收益等于该路径上各边价格之和。我们考虑无限供应场景，即每条边可销售给任意数量的顾客。问题的核心是找到使供应商收益最大化的定价方案。当网络为树结构时，该问题被称为收费亭问题。Gamzu 与 Segev 对此场景提出了一个 $\mathcal{O} \left( \frac{\log m}{\log \log m} \right)$ 近似算法。需注意树状网络中的路径具有唯一性，因此收费亭问题属于单一意愿投标者范畴——每位买家仅对单一固定商品集合感兴趣。本文突破单一意愿设定，考虑更一般化的包含环状结构的网络。我们提出了针对仙人掌图（即每条边至多属于一个简单环的网络）的定价算法，获得了多项式时间的 $\mathcal{0} \left( \frac{\log m}{\log \log m}\right)$ 近似算法以最大化收费亭定价收益。该算法基于 Gamzu 与 Segev 的框架，但需对其核心思想（图的递归分解、根实例的动态规划以及价格舍入）进行实质性扩展。