On the Complexity of the Highway Pricing Problem

The highway pricing problem asks for prices to be determined for segments of a single highway such as to maximize the revenue obtainable from a given set of customers with known valuations. The problem is (weakly) NP-hard and a recent quasi-PTAS suggests that a PTAS might be in reach. Yet, so far it has resisted any attempt for constant-factor approximation algorithms. We relate the tractability of the problem to structural properties of customers' valuations. We show that the problem becomes NP-hard as soon as the average valuations of customers are not homogeneous, even under further restrictions such as monotonicity. Moreover, we derive an efficient approximation algorithm, parameterized along the inhomogeneity of customers' valuations. Finally, we discuss extensions of our results that go beyond the highway pricing problem.\u

On the Complexity of the Highway Pricing Problem

The highway pricing problem asks for prices to be determined for segments of a single highway such as to maximize the revenue obtainable from a given set of customers with known valuations. The problem is (weakly) NP-hard and a recent quasi-PTAS suggests that a PTAS might be in reach. Yet, so far it has resisted any attempt for constant-factor approximation algorithms. We relate the tractability of the problem to structural properties of customers'' valuations. We show that the problem becomes NP-hard as soon as the average valuations of customers are not homogeneous, even under further restrictions such as monotonicity. Moreover, we derive an efficient approximation algorithm, parameterized along the inhomogeneity of customers'' valuations. Finally, we discuss extensions of our results that go beyond the highway pricing problem.operations research and management science;

Optimal Bundle Pricing for Homogeneous Items

We consider a revenue maximization problem where we are selling a set of m items, each of which available in a certain quantity (possibly unlimited) to a set of n bidders. Bidders are single minded, that is, each bidder requests exactly one subset, or bundle of items. Each bidder has a valuation for the requested bundle that we assume to be known to the seller. The task is to find an envy-free pricing such as to maximize the revenue of the seller. We derive several complexity results and algorithms for several variants of this pricing problem. In fact, the settings that we consider address problems where the different items are `homogeneous'' in some sense. First, we introduce the notion of affne price functions that can be used to model situations much more general than the usual combinatorial pricing model that is mostly addressed in the literature. We derive fixed-parameter polynomial time algorithms as well as inapproximability results. Second, we consider the special case of combinatorial pricing, and introduce a monotonicity constraint that can also be seen as `global'' envy-freeness condition. We show that the problem remains strongly NP-hard, and we derive a PTAS - thus breaking the inapproximability barrier known for the general case. As a special case, we finally address the notorious highway pricing problem under the global envy-freeness condition.operations research and management science;

Optimal Bundle Pricing with Monotonicity Constraint

We consider the problem to price (digital) items in order to maximize the revenue obtainable from a set of bidders. We suggest a natural monotonicity constraint on bundle prices, show that the problem remains NP-hard, and we derive a PTAS. We also discuss a special case, the highway pricing problem.operations research and management science;

Prizing on Paths: A PTAS for the Highway Problem

In the highway problem, we are given an n-edge line graph (the highway), and a set of paths (the drivers), each one with its own budget. For a given assignment of edge weights (the tolls), the highway owner collects from each driver the weight of the associated path, when it does not exceed the budget of the driver, and zero otherwise. The goal is choosing weights so as to maximize the profit. A lot of research has been devoted to this apparently simple problem. The highway problem was shown to be strongly NP-hard only recently [Elbassioni,Raman,Ray-'09]. The best-known approximation is O(\log n/\log\log n) [Gamzu,Segev-'10], which improves on the previous-best O(\log n) approximation [Balcan,Blum-'06]. In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora's quadtree dissection for Euclidean network design [Arora-'98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)-ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottom-up fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. We demonstrate the versatility of our technique by presenting PTASs for two variants of the highway problem: the tollbooth problem with a constant number of leaves and the maximum-feasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09]

On Profit-Maximizing Pricing for the Highway and Tollbooth Problems

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

Pricing, Investment, and Network Equilibrium

Despite rapidly emerging innovative road pricing and investment principles, the development of a long run network dynamics model for necessary policy evaluation is still lagging. This research endeavors to fill this gap and models the impacts of road financing policies throughout the network equilibration process. The manner in which pricing and investment jointly shape network equilibrium is particularly important and explored in this study. The interactions among travel demand, road supply, revenue mechanisms and investment rules are modeled at the link level in a network growth simulator. After assessing several measures of effectiveness, the proposed network growth model is able to evaluate the short- and long-run impacts of a broad spectrum of road pricing and investment policies on large-scale road networks, which can provide valuable information to decision-makers such as the implications of various policy scenarios on social welfare, financial situation of road authorities and potential implementation problems. Some issues hard to address in theoretical analysis can be examined in the agent-based simulation model. As a demonstration, we apply the network growth model to assess marginal and average pricing scenarios on a sample network. Even this relatively simple application provides new insights into issues around road pricing that have not previously been seriously considered. For instance, the results disclose a potential problem of over-investment when the marginal cost pricing scheme is adopted in conjunction with a myopic profit-neutral investment policy.Transportation network equilibrium; Road growth; Pricing; Congestion toll; Investment; Transport policy analysis.

The Development of a Common Investment Appraisal for Urban Transport Projects.

In December 1990 we were invited by Birmingham City Council and Centro to submit a proposal for an introductory study of the development of a common investment appraisal for urban transport projects. Many of the issues had arisen during the Birmingham Integrated Transport Study (BITS) in which we were involved, and in the subsequent assessment of light rail schemes of which we have considerable experience. In subsequent discussion, the objectives were identified as being:- (i) to identify, briefly, the weaknesses with existing appraisal techniques; (ii) to develop proposals for common methods for the social cost-benefit appraisal of both urban road and rail schemes which overcome these weaknesses; (iii) to develop complementary and consistent proposals for common methods of financial appraisal of such projects; (iv) to develop proposals for variants of the methods in (ii) and (iii) which are appropriate to schemes of differing complexity and cost; (v) to consider briefly methods of treating externalities, and performance against other public sector goals, which are consistent with those developed under (ii) to (iv) above; (vi) to recommend work to be done in the second phase of the study (beyond March 1991) on the provision of input to such evaluation methods from strategic and mode-specific models, and on the testing of the proposed evaluation methods. Such issues are particularly topical at present, and we have been able to draw, in our study, on experience of:- (i) evaluation methods developed for BITS and subsequent integrated transport studies (MVA) (ii) evaluation of individual light rail and heavy rail investment projects (ITS,MVA); (iii) the recommendations of AMA in "Changing Gear" (iv) advice to IPPR on appraisal methodology (ITS); (v) submissions to the House of Commons enquiry into "Roads for the Future" (ITS); (vi) advice to the National Audit Office (ITS) (vii) involvement in the SACTRA study of urban road appraisal (MVA, ITS