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.
The classic interval scheduling problem, where each interval requires all of the resource for its execution, is solvable in @math time @cite_25 . Storage allocation problems have been studied since the 1960's (see, e.g., @cite_10 ). The traditional goal is to contiguously store a set of objects in minimum size memory. This is the well-known dynamic storage allocation ( ) problem (see, e.g., @cite_0 and the references therein). The storage allocation problem ( ) is NP-hard, since it includes Knapsack as a special case. was first studied in @cite_25 @cite_3 . Bar-Noy al @cite_25 presented an approximation algorithm that yields a ratio of @math . Chen al @cite_11 presented a polynomial time exact algorithm for the special case where all resource requirements are multiples of @math , for some fixed integer @math . Bar-Yehuda al @cite_17 presented a randomized algorithm for with ratio @math , and a deterministic algorithm with ratio @math . The best known result is a deterministic @math -approximation algorithm due to @cite_5 .
@cite_17 @cite_3 @cite_0 @cite_5 @cite_10 @cite_25 @cite_11
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We introduce two scheduling problems, the flexible bandwidth allocation problem ( ( FBAP )) and the flexible storage allocation problem ( ( FSAP )). In both problems, we have an available resource, and a set of requests, each consists of a minimum and a maximum resource requirement, for the duration of its execution, as well as a profit accrued per allocated unit of the resource. In ( FBAP ), the goal is to assign the available resource to a feasible subset of requests, such that the total profit is maximized, while in ( FSAP ) we also require that each satisfied request is given a contiguous portion of the resource. Our problems generalize the classic bandwidth allocation problem (BAP) and storage allocation problem (SAP) and are therefore ( NP-hard ). Our main results are a 3-approximation algorithm for ( FBAP ) and a ((3+ ) )-approximation algorithm for ( FSAP ), for any fixed ( >0 ). These algorithms make nonstandard use of the local ratio technique. Furthermore, we present a ((2+ ) )-approximation algorithm for ( SAP ), for any fixed ( >0 ), thus improving the best known ratio of ( 2e-1 e-1 + ). Our study is motivated also by critical resource allocation problems arising in all-optical networks. Given @math elements with non-negative integer weights @math and an integer capacity @math , we consider the counting version of the classic knapsack problem: find the number of distinct subsets whose weights add up to at most @math . We give the first deterministic, fully polynomial-time approximation scheme (FPTAS) for estimating the number of solutions to any knapsack constraint (our estimate has relative error @math ). Our algorithm is based on dynamic programming. Previously, randomized polynomial-time approximation schemes (FPRAS) were known first by Morris and Sinclair via Markov chain Monte Carlo techniques, and subsequently by Dyer via dynamic programming and rejection sampling. In addition, we present a new method for deterministic approximate counting using read-once branching programs. Our approach yields an FPTAS for several other counting problems, including counting solutions for the multidimensional knapsack problem with a constant number of constraints, the general integer knapsack problem, and the contingency tables problem with a constant number of rows. In 1966, Graham showed that a simple procedure called list scheduling yields a 2-approximation algorithm for the central problem of scheduling precedence constrained jobs on identical machines to minimize makespan. To date this has remained the best algorithm, and whether it can or cannot be improved has become a major open problem in scheduling theory. We address this problem by establishing a quite surprising relation between the approximability of the considered problem and that of scheduling precedence constrained jobs on a single machine to minimize weighted completion time. More specifically, we prove that if the single machine problem is hard to approximate within a factor of @math , then the considered parallel machine problem, even in the case of unit processing times, is hard to approximate within a factor of @math , where @math tends to 0 as @math tends to 0. Combining this with Bansal and Khot's recent hardness result for the single machine problem gives that it is NP-hard to improve upon the approximation ratio obtained by Graham, assuming a new variant of the unique games conjecture. The multi-armed bandit (MAB) problem features the classical tradeoff between exploration and exploitation. The input specifies several stochastic arms which evolve with each pull, and the goal is to maximize the expected reward after a fixed budget of pulls. The celebrated work of , surveyed in [8], presumes a condition on the arms called the martingale assumption. In [9], A. obtained an LP-based 1 48-approximation for the problem with the martingale assumption removed. We improve the algorithm to a 4 27-approximation, with simpler analysis. Our algorithm also generalizes to the case of MAB superprocesses with (stochastic) multi-period actions. This generalization captures the framework introduced by Guha and Munagala in [11], and yields new results for their budgeted learning problems. Also, we obtain a (1 2 -- e)-approximation for the variant of MAB where preemption (playing an arm, switching to another arm, then coming back to the first arm) is not allowed. This contains the stochastic knapsack problem of Dean, Goemans, and Vondrak in [6] with correlated rewards, where we are given a knapsack of fixed size, and a set of jobs each with a joint distribution for its size and reward. The actual size and reward of a job can only be discovered in real-time as it is being scheduled, and the objective is to maximize expected reward before the knapsack size is exhausted. Our (1 2 -- e)-approximation improves the 1 16 and 1 8 approximations in [9] for correlated stochastic knapsack with cancellation and no cancellation, respectively, providing to our knowledge the first tight algorithm for these problems that matches the integrality gap of 2. We sample probabilities from an exponential-sized dynamic programming solution, whose existence is guaranteed by an LP projection argument. We hope this technique can also be applied to other dynamic programming problems which can be projected down onto a small LP.