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].
There are better approximation results, all based on dynamic programming, for a number of special cases. In @cite_10 a constant approximation is given for the case that all the paths have roughly the same length. An FPTAS is described by Hartline and Koltun @cite_18 for the case that the highway has constant length (i.e., @math ). This was generalized to the case of constant-length paths in @cite_12 . In @cite_12 the authors also present an FPTAS for the case that budgets are upper bounded by a constant. An FPTAS is also known @cite_10 @cite_8 for the case that paths induce a laminar family In a laminar family of paths, two paths which intersect are contained one in the other. .
@cite_18 @cite_10 @cite_12 @cite_8
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We present efficient approximation algorithms for a number of problems that call for computing the prices that maximize the revenue of the seller on a set of items. Algorithms for such problems enable the design of auctions and related pricing mechanisms [3]. In light of the fact that the problems we address are APX-hard in general [5], we design near-linear and near-cubic time approximation schemes under the assumption that the number of distinct items for sale is constant. Novel compositions for use as cell growth-promoting materials are made by the following novel process involving the steps of: (a) slowly contacting serum or plasma with sufficient chilled perchloric acid to reach a 0.1 to 0.25 final molar concentration of said perchloric acid in said serum or plasma, (b) at a temperature of -1 DEG C. to 15 DEG C., (c) under intensive mixing which is continued until a homogeneous suspension is obtained, (d) separating the resultant precipitate, which contains the growth-promoting substances, from the supernatant, (e) eluting said growth-promoting substances from said precipitate by first resuspending said precipitate in an aqueous alkaline or salt solution, and thereafter, (f) adjusting the pH to solubilize the growth-promoting substances from the insoluble proteins, (g) separating the supernatant, which contains the growth-promoting substances, from the insoluble, undesired precipitate, (h) exchanging the solvent in the supernatant for a physiological solution, and (i) sterilizing the resultant growth-promoting material. We deal with the problem of finding profit-maximizing prices for a finite number of distinct goods, assuming that of each good an unlimited number of copies is available, or that goods can be reproduced at no cost (e.g., digital goods). Consumers specify subsets of the goods and the maximum prices they are willing to pay. In the considered single-minded case every consumer is interested in precisely one such subset. If the goods are the edges of a graph and consumers are requesting to purchase paths in this graph, then we can think of the problem as pricing computer network connections or transportation links.We start by showing weak NP-hardness of the very restricted case in which the requested subsets are nested, i.e., contained inside each other or non-intersecting, thereby resolving the previously open question whether the problem remains NP-hard when the underlying graph is simply a line. Using a reduction inspired by this result we present an approximation preserving reduction that proves APX-hardness even for very sparse instances defined on general graphs, where the number of requests per edge is bounded by a constant B and no path is longer than some constant l. On the algorithmic side we first present an O(log l + log B)-approximation algorithm that (almost) matches the previously best known approximation guarantee in the general case, but is especially well suited for sparse problem instances. Using a new upper bounding technique we then give an O(l2)-approximation, which is the first algorithm for the general problem with an approximation ratio that does not depend on B.