We are given a set of @math jobs that have to be executed on a set of @math speed-scalable machines that can vary their speeds dynamically using the energy model introduced in [, FOCS'95]. Every job @math is characterized by its release date @math , its deadline @math , its processing volume @math if @math is executed on machine @math and its weight @math . We are also given a budget of energy @math and our objective is to maximize the weighted throughput, i.e. the total weight of jobs that are completed between their respective release dates and deadlines. We propose a polynomial-time approximation algorithm where the preemption of the jobs is allowed but not their migration. Our algorithm uses a primal-dual approach on a linearized version of a convex program with linear constraints. Furthermore, we present two optimal algorithms for the non-preemptive case where the number of machines is bounded by a fixed constant. More specifically, we consider: (a) the case of identical processing volumes, i.e. @math for every @math and @math , for which we present a polynomial-time algorithm for the unweighted version, which becomes a pseudopolynomial-time algorithm for the weighted throughput version, and (b) the case of agreeable instances, i.e. for which @math if and only if @math , for which we present a pseudopolynomial-time algorithm. Both algorithms are based on a discretization of the problem and the use of dynamic programming.
The multiple machine case where the preemption and the migration of jobs are allowed can be solved in polynomial time in @cite_24 , @cite_15 and @cite_22 . @cite_18 considered the multiple machine problem where the preemption of jobs is allowed but not their migration. They showed that the problem is polynomial-time solvable for agreeable instances when the jobs have the same processing volumes. They have also showed that it becomes strongly NP-hard for general instances even for jobs with equal processing volumes and for this case they proposed an @math -approximation algorithm. For the case where the jobs have arbitrary processing volumes, they showed that the problem is NP-hard even for instances with common release dates and common deadlines. proposed a @math -approximation algorithm for instances with common release dates, or common deadlines, and an @math -approximation algorithm for instances with agreeable deadlines. @cite_21 proposed a @math -approximation algorithm for general instances, where @math is the @math -th Bell number. Recently, the approximation ratio for agreeable instances has been improved to @math in @cite_10 . For the non-preemptive multiple machine energy minimization problem, the only known result is a non-constant approximation algorithm presented in @cite_10 .
@cite_18 @cite_22 @cite_21 @cite_24 @cite_15 @cite_10
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We consider the problem of scheduling n jobs on m parallel batching machines with inclusive processing set restrictions and non-identical capacities. The machines differ in their functionality but have the same processing speed. The inclusive processing set restriction has the following property: the machines can be linearly ordered such that a higher-indexed machine can process all those jobs that a lower-indexed machine can process. Each job is characterized by a processing time that specifies the minimum time needed to process the job, a release date before which it cannot be processed, and a machine index which is the smallest index among the machines that can process it. Each batching machine has a limited capacity and can process a batch of jobs simultaneously as long as its capacity is not violated. The capacities of the machines are non-identical. The processing time of a batch is the maximum of the processing times of the jobs belonging to it. Jobs in the same batch have a common start time and a common completion time. The goal is to find a non-preemptive schedule so as to minimize makespan (the maximum completion time). When all jobs are released at the same time, we present two fast algorithms with approximation ratios 3 and 9 4, respectively. For the general case with unequal release dates, we develop a polynomial time approximation scheme (PTAS), which is also the first PTAS even for the case with equal release dates and without processing set restrictions. We study the problem of non-preemptive scheduling to minimize energy consumption for devices that allow dynamic voltage scaling. Specifically, consider a device that can process jobs in a non-preemptive manner. The input consists of (i) the set R of available speeds of the device, (ii) a set J of jobs, and (iii) a precedence constraint Π among J. Each job j in J, defined by its arrival time aj, deadline dj, and amount of computation cj, is supposed to be processed by the device at a speed in R. Under the assumption that a higher speed means higher energy consumption, the power-saving scheduling problem is to compute a feasible schedule with speed assignment for the jobs in J such that the required energy consumption is minimized. This paper focuses on the setting of weakly dynamic voltage scaling, i.e., speed change is not allowed in the middle of processing a job. To demonstrate that this restriction on many portable power-aware devices introduces hardness to the power-saving scheduling problem, we prove that the problem is NP-hard even if aj = aj ′ and dj = dj ′ hold for all j,j ′∈ Jand |R|=2. If |R|<∞, we also give fully polynomial-time approximation schemes for two cases of the general NP-hard problem: (a) all jobs share a common arrival time, and (b) Π = ∅ and for any j,j ′ ∈ J, aj ≤ aj ′ implies dj ≤ dj ′. To the best of our knowledge, there is no previously known approximation algorithm for any special case of the NP-hard problem. We study a natural generalization of the problem of minimizing makespan on unrelated machines in which jobs may be split into parts. The different parts of a job can be (simultaneously) processed on different machines, but each part requires a setup time before it can be processed. First we show that a natural adaptation of the seminal approximation algorithm for unrelated machine scheduling [11] yields a 3-approximation algorithm, equal to the integrality gap of the corresponding LP relaxation. Through a stronger LP relaxation, obtained by applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ ≈ 1.618 is the golden ratio. This ratio decreases to 2 in the restricted assignment setting, matching the result for the classic version. Interestingly, we show that our problem cannot be approximated within a factor better than e e-1 ≈ 1.582 (unless P = NP). This provides some evidence that it is harder than the classic version, which is only known to be inapproximable within a factor 1.5 - e. Since our 1 + φ bound remains tight when considering the seemingly stronger machine configuration LP, we propose a new job based configuration LP that has an infinite number of variables, one for each possible way a job may be split and processed on the machines. Using convex duality we show that this infinite LP has a finite representation and can be solved in polynomial time to any accuracy, rendering it a promising relaxation for obtaining better algorithms. © 2014 Springer International Publishing Switzerland. In this paper we provide a tight bound for the price of preemption for scheduling jobs on a single machine (or multiple machines). The input consists of a set of jobs to be scheduled and of an integer parameter @math . Each job has a release time, deadline, length (also called processing time) and value associated with it. The goal is to feasibly schedule a subset of the jobs so that their total value is maximal; while preemption of a job is permitted, a job may be preempted no more than k times. The price of preemption is the worst possible (i.e., largest) ratio of the optimal non-bounded-preemptive scheduling to the optimal k-bounded-preemptive scheduling. Our results show that allowing at most k preemptions suffices to guarantee a Θ( +1 n, log_k+1 P ) @math . As part of the proof, we introduce the notion of the Bounded-Degree Ancestor-Free Sub-Forest (BAS). We investigate the problem of computing the maximal-value BAS of a given forest and give a tight bound for the loss factor, which is Θ(log_k+1 n)$ as well, where n is the size of the original forest and k is the bound on the degree of the sub-forest.