Parameterized Approximation Scheme for the Multiple Knapsack Problem

Jansen, Klaus

doi:10.1137/1.9781611973068.73

Cited by 15 publications

(12 citation statements)

References 16 publications

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“…If the item discarded from machine i has profit p i , the remaining objects in the machine will have profit ≥ p i and by the assumption on size, there will be at least µ such items in it. We use the multiple knapsack PTAS due to Chekuri and Khanna [5] or Jansen [17] which gives a 1 − O( ) approximation to the optimum packing for a given capacity pattern, for a fixed in polynomial time.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Algorithms for the Thermal Scheduling Problem

Mukherjee

Khuller

Deshpande

2013

2013 IEEE 27th International Symposium on Parallel and Distributed Processing

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Abstract-The energy costs for cooling a data center constitute a significant portion of the overall running costs. Thermal imbalance and hot spots that arise due to imbalanced workloads lead to significant wasted cooling effort -in order to ensure that no equipment is operating above a certain temperature, the data center may be cooled more than necessary. Therefore it is desirable to schedule the workload in a data center in a thermally aware manner, assigning jobs to machines not just based on local load of the machines, but based on the overall thermal profile of the data center. This is challenging because of the spatial cross-interference between machines, where a job assigned to a machine may impact not only that machine's temperature, but also nearby machines.Here, we continue formal analysis of the thermal scheduling problem that we initiated recently [25]. In that work, the notion of effective load of a machine which is a function of the local load on the machine as well as the load on nearby machines, was introduced, and optimal scheduling policies for a simple model (where cross-effects are restricted within a rack) were presented, under the assumption that jobs can be split among different machines. Here we consider the more realistic problem of integral assignment of jobs, and allow for cross-interference among different machines in adjacent racks in the data center. The integral assignment problem with cross-interference is NP-hard, even for a simple two machine model. We consider three different heat flow models, and give constant factor approximation algorithms for maximizing the number (or total profit) of jobs assigned in each model, without violating thermal constraints. We also consider the problem of minimizing the maximum temperature on any machine when all jobs need to be assigned, and give constant factor algorithms for this problem.

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Section: Proof Of Theoremmentioning

confidence: 99%

Algorithms for the Thermal Scheduling Problem

Mukherjee

Khuller

Deshpande

2013

2013 IEEE 27th International Symposium on Parallel and Distributed Processing

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“…Another closely related problem is the Multiple Subset-Sum problems (MSS), for which PTAS has been recently developed [4,6,10]. For classical machine unavailability periods, a (2 + )-approximation can be easily derived in a straightforward way from previous PTAS's for MSS even if the instance is not periodic but verifies that each availability periods is sufficiently large.…”

Section: Periodic Onasmentioning

confidence: 99%

Single machine scheduling with small operator-non-availability periods

Rapine

Brauner

Finke

et al. 2012

J Sched

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Through an industrial application, we were confronted with the planning of experiments where human intervention of a chemists is required to handle the starting and termination. This gives rise to a new type of scheduling problems, namely problems of finding schedules with time periods when the tasks can neither start nor finish. We consider in this paper the natural case of small periods where the duration of the periods is smaller than any processing time. This assumption corresponds to experiments lasting several days whereas the operator unavailability periods are the week-ends. These problems are analyzed on a single machine with the makespan as criterion.We first prove that, contrary to the case of machine unavailability, the problem with one small operator non-availability period can be solved in polynomial time. We then derive approximation and inapproximability results for the general case of k small unavailability periods. We finally focus on the practical case of periodic and equal small unavailability periods. We prove that this problem is not in FPTAS for more than 3 periods and we derive an FPTAS for the two-period case.

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“…In total we obtain a search space of size at most np max for the target makespan; via binary search as in [4], we will find a suitable target makespan in O(log(np max )) steps which is polynomially bounded in the encoding size of the instance. For a target makespan T , we use the technique described below which involves a PTAS for MSSP [1,13] to schedule as much load as possible in the interval [0, T ). In the sequel we show that for the optimal makespan C * max , we can algorithmically find a schedule which executes almost all load in the interval [0, C * max ); the remainig load is put in the interval [C * max , ∞) via list scheduling, causing an error which will be suitably bounded however.…”

Section: An Approximation Algorithmmentioning

confidence: 99%

“…As an algorithmic building block we use a PTAS for MSSP from [1] where the knapsack capacities are permitted to be different; this approximation scheme is referred to by MSSPPTAS. Alternatively, a PTAS for the multiple knapsack problem (MKP) can be used [2,3,13]. Knapsack type problems belong to the oldest and most fundamental problems in combinatorial optimization and theoretical computer science; we refer the reader to [16,25] for in-depth surveys or the papers [1,3,12,13,19] for literature on these problems.…”

Section: Introductionmentioning

confidence: 99%

“…2.4; by guessing the assignment of large jobs to large gaps we lose again an amount of T m of processing time. In the innermost loop of our algorithm, we pack the remaining small jobs using MSSPPTAS from [1,13]. In total, for the optimal target makespan T = C * max a total amount of processing time of at least (1 − )(P (I) − 2 T m) can be scheduled in the interval [0, T ); consequently, we have only medium and small jobs with total processing time of at most 2 T m + P (I) to schedule.…”

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confidence: 99%

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Improved Approximation Algorithms for Scheduling with Fixed Jobs

Diedrich¹,

Jansen²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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We study two closely related problems in non-preemptive scheduling of sequential jobs on identical parallel machines. In these two settings there are either fixed jobs or nonavailability intervals during which the machines are not available; in either case, the objective is to minimize the makespan. Both formulations have different applications, e.g. in turnaround scheduling or overlay computing. For both problems we contribute approximation algorithms with an improved ratio of 3/2 + , respectively. For scheduling with fixed jobs, a lower bound of 3/2 on the approximation ratio has been obtained by Scharbrodt, Steger & Weisser; for scheduling with non-availability we provide the same lower bound. In total, our approximation ratio for both problems is essentially tight via suitable inapproximability results. We use dual approximation, creation of a gap structure and job configurations, and a PTAS for the multiple subset sum problem. However, the main feature of our algorithms is a new technique for the assignment of large jobs via flexible rounding. Our new technique is based on an interesting cyclic shifting argument in combination with a network flow model for the assignment of jobs to large gaps.

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Parameterized Approximation Scheme for the Multiple Knapsack Problem

Cited by 15 publications

References 16 publications

Algorithms for the Thermal Scheduling Problem

Algorithms for the Thermal Scheduling Problem

Single machine scheduling with small operator-non-availability periods

Improved Approximation Algorithms for Scheduling with Fixed Jobs

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