On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

TL;DR

This paper studies profit-maximizing pricing strategies for the tollbooth and highway problems on trees, providing approximation algorithms, hardness results, and exploring variants with negative pricing.

Contribution

It introduces an $O(rac{ ext{log} n}{ ext{log} m})$-approximation for the tollbooth problem, a $(1- ext{epsilon})$-approximation for a special case, and establishes hardness results including NP-hardness and APX-hardness.

Findings

Presented an $O(rac{ ext{log} n}{ ext{log} m})$-approximation algorithm.

Developed a quasi-polynomial time $(1- ext{epsilon})$-approximation for a special case.

Proved the problem is strongly NP-hard even on a line, and APX-hard in the coupon model.

Abstract

In the \emph{tollbooth problem}, we are given a tree $\bT=(V,E)$ with $n$ edges, and a set of $m$ customers, each of whom is interested in purchasing a path on the tree. Each customer has a fixed budget, and the objective is to price the edges of $\bT$ such that the total revenue made by selling the paths to the customers that can afford them is maximized. An important special case of this problem, known as the \emph{highway problem}, is when $\bT$ is restricted to be a line. For the tollbooth problem, we present a randomized $O(\log n)$-approximation, improving on the current best $O(\log m)$-approximation. We also study a special case of the tollbooth problem, when all the paths that customers are interested in purchasing go towards a fixed root of $\bT$. In this case, we present an algorithm that returns a $(1-\epsilon)$-approximation, for any $\epsilon > 0$, and runs in quasi-polynomial time. On the other hand, we rule out the existence of an FPTAS by showing that even for the line case, the problem is strongly NP-hard. Finally, we show that in the \emph{coupon model}, when we allow some items to be priced below zero to improve the overall profit, the problem becomes even APX-hard.