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
2010996873 1526218779 2062453462
We prove semi-logarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, we prove O(1 logσ(e)n) inapproximability assuming that NP n BPTIME(2ne) for some e > 0. We also prove O(1 log1 3-e n) inapproximability, for any e > 0, assuming that refuting random instances of 3SAT is hard on average; and prove O(1 log n) inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish matching upper bounds up to exponents, even for a more general (budgeted) setting, giving an Ω(1 log n)-approximation algorithm as well as an Ω(1 log B)-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, and radio broad-casting. 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.