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( n n) [Gamzu,Segev-'10], which improves on the previous-best O( 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].
The highway and tollbooth problems belong to the family of prizing problems with single-minded customers and unlimited supply. Here we are given a set of customers: Each customer wants to buy a subset of items (), if its total prize does not exceed her budget. In the highway terminology, each driver is a subset of edges (rather than a path). For this problem a @math approximation is given in @cite_12 . This bound was refined in @cite_8 to @math , where @math denotes the maximum number of items in a bundle and @math the maximum number of bundles containing a given item. A @math approximation is given in @cite_10 . On the negative side, @cite_16 show that this problem is hard to approximate within @math , for some @math , assuming that @math for some @math .
@cite_16 @cite_10 @cite_12 @cite_8
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In the highway problem, we are given an @math -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 @math -hard only recently [K. M. , in Proceedings of the International Symposium on Algorithmic Game Theory (SAGT), 2009, pp. 275--286]. The best-known approximation is @math [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 @math approximation [M.-F. Balcan and A. Blum, in Proceedings of the ACM Conference on Electronic Commerce,... We consider the problem of maximizing the revenue raised from tolls set on the arcs of a transportation network, under the constraint that users are assigned to toll-compatible shortest paths. We first prove that this problem is strongly NP-hard. We then provide a polynomial time algorithm with a worst-case precision guarantee of @math , where mT denotes the number of toll arcs. Finally, we show that the approximation is tight with respect to a natural relaxation by constructing a family of instances for which the relaxation gap is reached. © 2005 Wiley Periodicals, Inc. NETWORKS, Vol. 46(1), 57–67 2005 The second of a two-part series, this paper derives an efficient solution to the minimal-revenue tolls problem. As introduced in Part I, this problem can be defined as follows: Assuming each trip uses only a path whose generalized cost is smallest, find a set of arc tolls that simultaneously minimizes both average travel time and out-of-pocket cost. As a point of departure, this paper first re-solves the single-origin problem of Part I, modeling it as a linear program. Then with a change of variable, it transforms the LP's dual into a simple longest-path problem on an acyclic network. The multiple-origin problem - where one toll for each arc applies to all origins - solves analogously. In this case, however, the dual becomes an elementary linear multi-commodity max-cost flow problem with an easy bundling constraint and infinite arc capacities. After a minor reformulation that simplifies the model's input to better accommodate output from common traffic assignment software, a solution algorithm is exemplified with a numerical example. We consider the problem of determining a set of optimal tolls on the arcs of a multicommodity transportation network. The problem is formulated as a bilevel mathematical program where the upper level consists in a firm that raises revenues from tolls set on arcs of the network, while the lower level is represented by a group of users travelling on shortest paths with respect to a generalized travel cost.