A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

Cheung, Maurice; Shmoys, David B.

doi:10.1007/978-3-642-22935-0_12

Cited by 28 publications

(61 citation statements)

References 20 publications

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“…This was also our original approach to this problem, but our rounding techniques led us to the geometric approach (which is cleaner to present as several geometric results can be used as a black-box). The same LP was used in the recent works on the special case of identical release dates [19,21,26]. We conjecture that this LP has O(1) integrality gap even for general release times.…”

Section: Resultsmentioning

confidence: 81%

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The Geometry of Scheduling

Bansal¹,

Pruhs²

2014

SIAM J. Comput.

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We consider the following general scheduling problem. The input consists of n jobs, each with an arbitrary release time, size, and monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as total weighted flow time, total weighted tardiness, and sum of flow time squared. We give an O(log log P ) approximation for this problem, where P is the ratio of the maximum to minimum job size. We also give an O(1) approximation in the special case of identical release times. These results are obtained by reducing the scheduling problem to a geometric capacitated set cover problem in two dimensions. Introduction.We consider the following general offline scheduling problem. General scheduling problem (GSP). The input consists of a collection of n jobs, and for each job j there is a positive integer release time r j , a positive integer size p j and a nondecreasing cost (or weight) function w j (t) ≥ 0 specifying a nonnegative cost for each time t > r j . (We will specify later how these weight functions are represented.) A feasible solution is a preemptive schedule, which assigns to each job j time slots [t, t + 1] (not necessarily consecutive and satisfying t ≥ r j ), during which j is run. A job is completed once it has been run for p j units of time. If job j completes at time t, then a cost of w j (t) is incurred for that job. The objective is to minimize the total cost,

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Section: Resultsmentioning

confidence: 81%

“…Cheung and Shmoys [19] proposed a primal dual algorithm for GSP with identical release times. Mestre and Verschae [26] reformulated this algorithm as a local ratio algorithm and showed that it yielded a 4-approximation.…”

Section: Techniquesmentioning

confidence: 99%

The Geometry of Scheduling

Bansal¹,

Pruhs²

2014

SIAM J. Comput.

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“…In the special case of uniform release dates, their method achieves the constant factor of 16. Cheung and Shmoys [2011] have claimed to achieve a (2 + ε)-approximation via a primal-dual approach. However, Mestre and Verschae [2014] have recently given instances where the algorithm constructs a dual solution whose value differs from the optimal integral solution cost by a factor of 4, contradicting the claimed factor of 2 + ε.…”

Section: Relatedmentioning

confidence: 99%

“…Moreover, the cost function can be interpreted as nonuniform processor speed. Due to their universality, problems addressing generalized min-sum objectives have recently attracted much attention (see, e.g., Pruhs [2010a, 2010b], Cheung and Shmoys [2011], Epstein et al [2012], Im et al [2012], Megow and Verschae [2013], and Stiller and Wiese [2010]). …”

Section: Introductionmentioning

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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“…The same analysis can be done for the Min-Exp 1|| w j U j problem with uncertain due dates and deterministic processing times. Hence and from [10], where a (4 + ǫ)-approximation algorithm, for any ǫ > 0, for this class of problems was provided, we get the following result (see also Theorem 1):…”

Section: Problems With Uncertain Due Datesmentioning

confidence: 77%

Risk-averse single machine scheduling: complexity and approximation

Kasperski

Zieliński

2019

J Sched

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In this paper a class of single machine scheduling problems is considered. It is assumed that job processing times and due dates can be uncertain and they are specified in the form of discrete scenario set. A probability distribution in the scenario set is known. In order to choose a schedule some risk criteria such as the value at risk (VaR) an conditional value at risk (CVaR) are used. Various positive and negative complexity results are provided for basic single machine scheduling problems. In this paper new complexity results are shown and some known complexity results are strengthen. 4 d i J x 1 0 0 1 0 0 2 0 0 0 2 J x 1 1 0 0 1 1 2 0 0 0 2 J x 2 1 1 0 0 0 0 2 0 0 2 J x 2 0 0 1 1 0 0 2 0 0 2 J x 3 1 1 0 0 0 0 0 2 0 2 J x 3 0 0 0 1 1 0 0 2 0 2 J x 4 0 0 1 0 1 0 0 0 2 2 J x 4 0 1 0 0 0 0 0 0 2 2

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A Primal-Dual Approximation Algorithm for Min-Sum Single-Machine Scheduling Problems

Cited by 28 publications

References 20 publications

The Geometry of Scheduling

The Geometry of Scheduling

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

Risk-averse single machine scheduling: complexity and approximation

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