Motivated by the cloud computing paradigm, and by key optimization problems in all-optical networks, we study two variants of the classic job interval scheduling problem, where a reusable resource is allocated to competing job intervals in a flexible manner. Each job, @math , requires the use of up to @math units of the resource, with a profit of @math accrued for each allocated unit. The goal is to feasibly schedule a subset of the jobs so as to maximize the total profit. The resource can be allocated either in contiguous or non-contiguous blocks. These problems can be viewed as flexible variants of the well known storage allocation and bandwidth allocation problems. We show that the contiguous version is strongly NP-hard, already for instances where all jobs have the same profit and the same maximum resource requirement. For such instances, we derive the best possible positive result, namely, a polynomial time approximation scheme. We further show that the contiguous variant admits a @math -approximation algorithm, for any fixed @math , on instances whose job intervals form a proper interval graph. At the heart of the algorithm lies a non-standard parameterization of the approximation ratio itself, which is of independent interest. For the non-contiguous case, we uncover an interesting relation to the paging problem that leads to a simple @math algorithm for uniform profit instances of n jobs. The algorithm is easy to implement and is thus practical.
Finally, the paper @cite_5 considers variants of and where @math , and the goal is to feasibly schedule a subset @math of the intervals of maximum total profit (namely, the amount of resource allocated to each interval @math is in @math ). The paper presents a @math -approximation algorithm for this version of , and a @math -approximation for the corresponding version of , for any fixed @math .
@cite_5
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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. We consider the following scheduling problem. The input is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single-processor nonpreemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time. We give two online algorithms with competitive ratios below @math and show several lower bounds on the competitive ratios. First, we give a barely random @math -competitive algorithm that uses only one random bit. We also show a lower bound of @math on the competitive ratio of barely random algorithms that randomly choose one of two deterministic algorithms. If the two algorithms are selected with equal probability, we can further improve the bound to @math . Second, we give a deterministic @math -competitive algorithm in the model that allows restarts, and we show that in this model the ratio @math is optimal. For randomized algorithms with restarts we show a lower bound of @math . An important question in theoretical computer science is to determine the best possible running time for solving a problem at hand. For geometric optimization problems, we often understand their complexity on a rough scale, but not very well on a finer scale. One such example is the two-dimensional knapsack problem for squares. There is a polynomial time (1 + ϵ)-approximation algorithm for it (i.e., a PTAS) but the running time of this algorithm is triple exponential in 1 ϵ, i.e., Ω(n221 ϵ). A double or triple exponential dependence on 1 ϵ is inherent in how this and several other algorithms for other geometric problems work. In this paper, we present an EPTAS for knapsack for squares, i.e., a (1+ϵ)-approximation algorithm with a running time of Oϵ(1)·nO(1). In particular, the exponent of n in the running time does not depend on ϵ at all! Since there can be no FPTAS for the problem (unless P = NP) this is the best kind of approximation scheme we can hope for. To achieve this improvement, we introduce two new key ideas: We present a fast method to guess the Ω(221 ϵ) relatively large squares of a suitable near-optimal packing instead of using brute-force enumeration. Secondly, we introduce an indirect guessing framework to define sizes of cells for the remaining squares. In the previous PTAS each of these steps needs a running time of Ω(n221 ϵ) and we improve both to Oϵ(1) · nO(1). We complete our result by giving an algorithm for two-dimensional knapsack for rectangles under (1 + ϵ)-resource augmentation. In this setting, we also improve the best known running time of Ω(n1 ϵ1 ϵ) to Oϵ(1) · nO(1) and compute even a solution with optimal profit, in contrast to the best previously known polynomial time algorithm for this setting that computes only an approximation. We believe that our new techniques have the potential to be useful for other settings as well. We study digital-goods auctions for items in unlimited supply introduced by Goldberg, Hartline and Wright. Since no deterministic algorithms are competitive for this class of auctions, one of the central research issues is how to obtain a nice probabilistic distribution over truthful algorithms. In this paper, we introduce a rather systematic approach to this goal: Consider for example the Sampling Cost Share (SCS) auction. It is well known that SCS works well if the the current bid vector produces many winners against F(2), the standard benchmark algorithm for competitive analysis. In fact, its competitive ratio is approaching to 2.0 as k (= the number of F(2) winners) grows. On the other hand, its competitive ratio becomes as bad as 4.0 for k = 2. Our new approach is to develop a sequence of similar cost-share type algorithms, DCSk, which work well for small k. Now we choose a sufficiently large constant N and run DCS1, DCS2, ..., DCSN and SCS with probabilities p1, p2, ..., pN and q, respectively. It should be noted that we can use LP to obtain optimal p1, p2,..., pN and q. By this averaging method, we can improve the competitive ratio of SCS from 4.0 to 3.531 and that of the currently best Aggregated γ3 algorithm due to Hartline and McGrew from 3.243 to 3.119.