Pricing on Paths: A PTAS for the Highway Problem

Abstract

In the highway problem, we are given an $n$-edge path 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 to choose 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 $\mathbf{NP}$-hard only recently [K. M. Elbassioni et al., in Proceedings of the International Symposium on Algorithmic Game Theory (SAGT), 2009, pp. 275--286]. The best-known approximation is $O(\log n/\log\log n)$ [I. Gamzu and D. Segev, in Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), 2010, pp. 582--593], which improves on the previous best $O(\log n)$ approximation [M.-F. Balcan and A. Blum, in Proceedings of the ACM Conference on Electronic Commerce, 2006, pp. 29--35]. Better approximations are known for a number of special cases. Finding a constant (or better!) approximation algorithm for the general case is a challenging open problem. In this paper we present a polynomial-time approximation scheme (PTAS) for the highway problem, hence greatly improving our understanding of the complexity status of this 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 [S. Arora, J. ACM, 45 (1998), pp. 753--782]. The basic idea is to enclose 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. The same basic approach also provides PTASs for two generalizations of the 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 [I. Gamzu and D. Segev, in Proceedings of the International Colloquium on Automata, Languages and Programming (ICALP), 2010, pp. 582--593; K. M. Elbassioni et al., in Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), 2009, pp. 1210--1219].