Single machine scheduling with job-dependent convex cost and arbitrary precedence constraints

Carrasco, Rodrigo A.; Iyengar, Garud; Stein, Clifford

doi:10.1016/j.orl.2013.05.008

Cited by 9 publications

(7 citation statements)

References 20 publications

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“…With a modified analysis, they show that the algorithm by Cheung and Shmoys is, in fact, a pseudopolynomial time 4-approximation, yielding a (4 + ε)-approximation in polynomial time. Restricting to job-specific convex cost functions, Carrasco et al [2013] give an O(1)-speed 1-approximation algorithm for the setting with arbitrary precedence constraints. Regarding the preemptive problem variant with nonuniform release dates, Im et al [2012] show that there exists no O(1)-speed O(1)-competitive online algorithm for general cost functions.…”

Section: Relatedmentioning

confidence: 99%

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

Höhn

Jacobs²

2015

ACM Trans. Algorithms

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We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith's rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k (k−1)/(k+1) , denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time. ACM Reference Format:Wiebke Höhn and Tobias Jacobs. 2015. On the performance of Smith's rule in single-machine scheduling with nonlinear cost. ACM Trans.

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Section: Relatedmentioning

confidence: 99%

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

Höhn

Jacobs²

2015

ACM Trans. Algorithms

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“…Job scheduling has been studied for decades. In fact, the Min-WCS problem is a special case of singlemachine scheduling with a non-linear objective function under precedence constraints, which has been studied by Schulz and Verschae [15] and Carrasco et al [16]. Specifically, for any > 0, the algorithm proposed by Schulz and Verschae approximates the optimum within a factor of (2+ ) when the objective function is concave [15].…”

Section: Introductionmentioning

confidence: 99%

“…Specifically, for any > 0, the algorithm proposed by Schulz and Verschae approximates the optimum within a factor of (2+ ) when the objective function is concave [15]. When the objective function is convex, Carrasco et al proposed a (4 + )-speed 1-approximation algorithm for any > 0 [16]. 3 The solutions proposed by Schulz and Verschae [15] and Carrasco et al [16] are based on linear programming rounding.…”

Section: Introductionmentioning

confidence: 99%

“…When the objective function is convex, Carrasco et al proposed a (4 + )-speed 1-approximation algorithm for any > 0 [16]. 3 The solutions proposed by Schulz and Verschae [15] and Carrasco et al [16] are based on linear programming rounding. The objective function of the Min-WCS problem is convex, and we give a randomized 2.733-approximation algorithm for the Min-WCS problem without linear programming.…”

Section: Introductionmentioning

confidence: 99%

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Minimum Age TDMA Scheduling

Kuo

2019

IEEE INFOCOM 2019 - IEEE Conference on Computer Communications

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We consider a transmission scheduling problem in which multiple systems receive update information through a shared Time Division Multiple Access (TDMA) channel. To provide timely delivery of update information, the problem asks for a schedule that minimizes the overall age of information. We call this problem the Min-Age problem. This problem is first studied by He et al. [IEEE Trans. Inform. Theory, 2018], who identified several special cases where the problem can be solved optimally in polynomial time. Our contribution is threefold. First, we introduce a new job scheduling problem called the Min-WCS problem, and we prove that, for any constant r ≥ 1, every r-approximation algorithm for the Min-WCS problem can be transformed into an r-approximation algorithm for the Min-Age problem. Second, we give a randomized 2.733-approximation algorithm and a dynamic-programming-based exact algorithm for the Min-WCS problem. Finally, we prove that the Min-Age problem is NP-hard. I. INTRODUCTIONWe consider systems whose states change upon reception of update messages. Such systems include, for example, web caches [2], intelligent vehicles [3], and real-time databases [4]. The timely delivery of update messages is often critical to the smooth and secure functioning of the system. Moreover, since any given update is likely dependent on previous updates, the update messages should not be delivered out of order. In most cases, the system does not have exclusive access to a communication channel. Instead, it must share the channel with other systems. Hence, the transmission schedule plays a crucial role in determining the performance of the systems that share the channel.can be obtained by sorting the sender-receiver pairs according to the number of messages to be sent to the receiver [14]. However, even if the channel is TDMA-based, it remained open whether the problem can be solved optimally in polynomial time. In this paper, we therefore focus on TDMA channels. In the remainder of this paper, we refer to this scheduling problem on a TDMA channel as the Min-Age problem.In this paper, we cast the Min-Age problem as a job scheduling problem called the Min-WCS problem.The Min-WCS problem has a simple formulation inspired by a geometric interpretation of the Min-Age problem. The simplicity of the formulation also facilitates algorithm design. As we will see in Section VII, 1 The sender may serve as a relay or hub for the system and thus may not be responsible for generating update messages. 2 The buffer may be a logical one that stores the inputs to a scheduler. 3 one may solve variants of the Min-Age problem by modifying the geometric interpretation and then solving the corresponding job scheduling problem. Job scheduling has been studied for decades. In fact, the Min-WCS problem is a special case of singlemachine scheduling with a non-linear objective function under precedence constraints, which has been studied by Schulz and Verschae [15] and Carrasco et al. [16]. Specifically, for any > 0, the algorithm proposed by Schulz and...

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“…For a set of real-time tasks with precedence constraints executed on a distributed system, Mishra et al [52] proposed static and dynamic power management schemes. In reference [24], α-point scheduling techniques (discussed later) were used to compute approximate solutions for a general class of scheduling problems with each job having a convex non-decreasing cost function. Agrawal and Rao [2] showed that energy-aware scheduling is a generalization of the makespan minimization scheduling problem.…”

Section: Introductionmentioning

confidence: 99%

A modified modeling approach and a heuristic procedure for the multi-mode resource constrained project scheduling problem with activity splitting

2015

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This paper presents a new heuristic method for solving multi-mode resource constrained project scheduling problems with renewable resources. Assumptions such as resource vacation and activity splitting are also considered. The proposed heuristic determines one mode for the execution of each activity first, so that the multi-mode problem is reduced to a single-mode one. Our method is compared to two of the existing methods in terms of computational time and solution quality. The results show that while it can outperform one of them (especially as the size of the problem grows), it is outperformed by the other for tested instances with low complexity, but can yield good results for tested instances with higher values of this parameters. This quality may be useful in real-world scheduling problems. We also validate the first phase of our method, i.e., mode selection, with numerical experiments. Our results indicate that better mode vectors selected in the first phase lead to better makespans for the MRCPSP. Moreover, we correct some erroneous constraints in the mathematical model of the problem. We implement the mathematical programming model of the problem in GAMS, and show that solving it to optimality requires much more computational effort compared to our method.

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Single machine scheduling with job-dependent convex cost and arbitrary precedence constraints

Cited by 9 publications

References 20 publications

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

On the Performance of Smith’s Rule in Single-Machine Scheduling with Nonlinear Cost

Minimum Age TDMA Scheduling

A modified modeling approach and a heuristic procedure for the multi-mode resource constrained project scheduling problem with activity splitting

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