On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

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, since $n\leq 3m$ can be assumed. 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

Zone‐based tariff design in public transportation networks

“…For a cumulative pricing , the pricing subproblem can be shown to generalize the highway problem, which has been introduced by Guruswami et al and shown to be strongly NP ‐complete by Elbassioni et al . Thus, the following theorem holds.Theorem The cumulative pricing subproblem for an arbitrary zoning, that is, a connected or ring zoning, is strongly NP‐hard .…”

“…For a cumulative pricing, the pricing subproblem can be shown to generalize the highway problem, which has been introduced by Guruswami et al [19] and shown to be strongly NP-complete by Elbassioni et al [12]. Thus, the following theorem holds.…”

On stackelberg pricing with computationally bounded customers

“…Algorithmic aspects of multidimensional pricing problems, which are important in the context of pure optimization as well as the design of revenue‐maximizing auction mechanisms [3], were first studied by Aggarwal et al [1] and Guruswami et al [24]. Subsequently, quite a number of improved algorithmic results for special cases of the problem [2, 12, 16, 17, 19, 20, 25] and complexity theoretic lower bounds [8, 18, 13] have been derived.…”

Online pricing for bundles of multiple items

“…Some researchers have studied more complex and realistic user preferences in the complete information setting, with the aim of devising fast algorithms to give the optimal, or a good approximate, pricing solution [5][6][7][8][9][12][13][14][15][16][17][18][19]. Other researchers have studied the situation where the seller must set prices without knowledge, or complete knowledge, of the valuations of future users, with the aim of devising algorithms that will perform well even in the face of adversarial users trying to undermine the seller [1][2][3][4]10,11,20].…”

“…Previous work has mainly focused on two supply models: the unlimited supply model [2,5,9,15,17] where the number of each type of item is unbounded and the limited supply model [2,[6][7][8]12,16,18,19] where the number of each type of item is bounded by some value. As for the users, there are several users' behaviors studied, including single-minded [12][13][14][15]17,19] (each user is interested only in a particular set of items), unit-demand [5][6][7][8][9]17,19] (each user will buy at most one item in total) and envy free [2,8,12,15,17] (after the assignment, no user would prefer to be assigned a different set of items with the designated prices, loosely speaking, each user is happy with his/her purchase). Most of the previous studies have considered a combination of the above scenarios (e.g.…”