Sublogarithmic Approximation for Tollbooth Pricing on a Cactus

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.