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

Höhn, Wiebke; Jacobs, Tobias

doi:10.1145/2629652

Cited by 22 publications

(8 citation statements)

References 35 publications

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“…Our main result is an efficient PTAS (Section 3) for scheduling to minimize w j C j on a machine of varying speed (given by an oracle). This is best possible since the problem is strongly NP-hard, even when the machine speed takes only two distinct values [16]. Our results generalize recent previous results such as a PTAS on a machine with only non-decreasing speeds [28] and FPTASes for only one non-availability period [18,19].…”

Section: Our Resultssupporting

confidence: 83%

“…The problem of scheduling on a machine of varying speed is equivalent to scheduling on an ideal machine (of constant speed) but minimizing a more general global cost function w j f (C j ), where f is a nondecreasing function. In this identification, f (C) denotes the time that the varying-speed machine needs to process a work volume of C [16]. The special case of only nondecreasing (nonincreasing) speed functions corresponds to concave (convex) global cost functions.…”

Section: Previous Workmentioning

confidence: 99%

“…The special case of only nondecreasing (nonincreasing) speed functions corresponds to concave (convex) global cost functions. In a recent work, Höhn and Jacobs [16] give a formula for computing tight guarantees for Smith's rule for any convex or concave function f . They also show that the problem for increasing piecewise linear cost function is strongly NP-hard even with only two different slopes, and so is our problem when the speed function takes only two distinct values.…”

Section: Previous Workmentioning

confidence: 99%

See 2 more Smart Citations

Dual Techniques for Scheduling on a Machine with Varying Speed

Megow¹,

Verschae²

2018

SIAM J. Discrete Math.

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We study scheduling problems on a machine with varying speed. Assuming a known speed function we ask for a cost-efficient scheduling solution. Our main result is a PTAS for minimizing the total weighted completion time in this setting. This also implies a PTAS for the closely related problem of scheduling to minimize generalized global cost functions. The key to our results is a re-interpretation of the problem within the wellknown two-dimensional Gantt chart: instead of the standard approach of scheduling in the time-dimension, we construct scheduling solutions in the weight-dimension.We also consider a dynamic problem variant in which deciding upon the speed is part of the scheduling problem and we are interested in the tradeoff between scheduling cost and speed-scaling cost, which is typically the energy consumption. We observe that the optimal order is independent of the energy consumption and that the problem can be reduced to the setting where the speed of the machine is fixed, and thus admits a PTAS. Furthermore, we provide an FPTAS for the NP-hard problem variant in which the machine can run only on a fixed number of discrete speeds. Finally, we show how our results can be used to obtain a (2+ε)-approximation for scheduling preemptive jobs with release dates on multiple identical parallel machines.

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Section: Our Resultssupporting

confidence: 83%

Section: Previous Workmentioning

confidence: 99%

Section: Previous Workmentioning

confidence: 99%

See 1 more Smart Citation

Dual Techniques for Scheduling on a Machine with Varying Speed

Megow¹,

Verschae²

2018

SIAM J. Discrete Math.

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“…It is unknown whether there exists a polynomial time algorithm to compute an optimal schedule for this problem, or if it is NP hard (see [10] and the references therein). For concave or convex h, it is shown in [52] that Smith's rule (for the original problem 1|| j w j C j ) yields a ( √ 3 + 1)/2 ≈ 1.37approximation to the problem 1|| j w j h(C j ), while [28] gives explicit formulas for the exact approximation ratio of Smith's rule for this problem in terms of a maximization over two continuous variables.…”

Section: Applications To Schedulingmentioning

confidence: 99%

On Submodular Search and Machine Scheduling

2019

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Suppose some objects are hidden in a finite set S of hiding places which must be examined one-by-one. The cost of searching subsets of S is given by a submodular function and the probability that all objects are contained in a subset is given by a supermodular function. We seek an ordering of S that finds all the objects in minimal expected cost. This problem is NP-hard and we give an efficient combinatorial 2-approximation algorithm, generalizing analogous results in scheduling theory. We also give a new scheduling application 1|prec| w A h(C A ), where a set of jobs must be ordered subject to precedence constraints to minimize the weighted sum of some concave function h of the completion times of subsets of jobs. We go on to give better approximations for submodular functions with low total curvature and we give a full solution when the problem is what we call series-parallel decomposable. Next, we consider a zero-sum game between a cost-maximizing Hider and a cost-minimizing Searcher. We prove that the equilibrium mixed strategies for the Hider are in the base polyhedron of the cost function, suitably scaled, and we solve the game in the seriesparallel decomposable case, giving approximately optimal strategies in other cases.

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“…The covering version of KPSC is studied in a context of single machine scheduling with nonuniform processing speed. This problem is known to be strongly NP-hard [13], which implies that pseudo-polynomial time algorithms are unlikely to exist. This is in contrast to the fact that the classic knapsack problem and its covering version admit pseudo-polynomial time algorithms.…”

Section: Related Studiesmentioning

confidence: 99%

Submodular Maximization with Uncertain Knapsack Capacity

Kawase

Sumita

Fukunaga

2019

SIAM J. Discrete Math.

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We consider the maximization problem of monotone submodular functions under an uncertain knapsack constraint. Specifically, the problem is discussed in the situation that the knapsack capacity is not given explicitly and can be accessed only through an oracle that answers whether or not the current solution is feasible when an item is added to the solution. Assuming that cancellation of the last item is allowed when it overflows the knapsack capacity, we discuss the robustness ratios of adaptive policies for this problem, which are the worst case ratios of the objective values achieved by the output solutions to the optimal objective values. We present a randomized policy of robustness ratio (1 − 1/e)/2, and a deterministic policy of robustness ratio 2(1 − 1/e)/21. We also consider a universal policy that chooses items following a precomputed sequence. We present a randomized universal policy of robustness ratio (1 − 1/ 4 √ e)/2. When the cancellation is not allowed, no randomized adaptive policy achieves a constant robustness ratio. Because of this hardness, we assume that a probability distribution of the knapsack capacity is given, and consider computing a sequence of items that maximizes the expected objective value. We present a polynomial-time randomized algorithm of approximation ratio (1 − 1/ 4 √ e)/4 − ǫ for any small constant ǫ > 0.

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

Cited by 22 publications

References 35 publications

Dual Techniques for Scheduling on a Machine with Varying Speed

Dual Techniques for Scheduling on a Machine with Varying Speed

On Submodular Search and Machine Scheduling

Submodular Maximization with Uncertain Knapsack Capacity

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