Scheduling on a Batch Processing Machine with Split Compatibility Graphs

Boudhar, Mourad

doi:10.1007/s10852-005-3083-z

Cited by 12 publications

(14 citation statements)

References 13 publications

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“…Boudhar in [1,2] studied the problem of batch scheduling with complements of bipartite and split graphs, respectively. Finke et al [3] considered this problem with complements of interval graphs.…”

Section: Scheduling Of Unit-length Jobs With Bipartite Incompatibilitmentioning

confidence: 99%

Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

Furmańczyk

Kubale

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. In the paper we consider the problem of scheduling n identical jobs on 4 uniform machines with speeds s 1 ¸ s 2 ¸ s 3 ¸ s 4 , respectively. Our aim is to find a schedule with a minimum possible length. We assume that jobs are subject to some kind of mutual exclusion constraints modeled by a bipartite incompatibility graph of degree ∆, where two incompatible jobs cannot be processed on the same machine. We show that the general problem is NP-hard even if s 1 = s 2 = s 3 . If, however, ∆ • 4 and s 1 ¸ 12s 2 , s 2 = s 3 = s 4 , then the problem can be solved to optimality in time O(n 1.5 ). The same algorithm returns a solution of value at most 2 times optimal provided that s 1 ¸ 2s 2 . Finally, we study the case s 1 ¸ s 2 ¸ s 3 = s 4 and give a 32/15-approximation algorithm running also in O(n 1.5 ) time.Key words: equitable coloring, NP-hardness, polynomial algorithm, scheduling, uniform machine. edges correspond to pairs of jobs being in conflict, is a bipartite graph (without isolated vertices). For example, all graphs in our figures are bipartite. Notice that two jobs being in conflict may be executed in intersecting time intervals. A load of k jobs on M i machine requires the processing time k/s i , and all the jobs are ready for processing at the same time. Alternatively, if a load on M i is not given explicitly, we are using the notation C(M i ) to mean the schedule length on the machine M i . By definition, each load forms an independent set (color) in G. Therefore, we will be using the terms job/vertex and load/color/independent set interchangeably. Since all the tasks have to be executed, the problem is to find a 4-coloring, i.e. a decomposition of G into 4 independent sets I 1 , I 2 , I 3 , and I 4 such that the schedule length C max = maxfjI i j/s i : i = 1, …, 4g is minimized, in sym- Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machinesThere are several papers devoted to chromatic scheduling in the presence of mutual exclusion constraints. Boudhar in [1,2] studied the problem of batch scheduling with complements of bipartite and split graphs, respectively. Finke et al. [3] considered this problem with complements of interval graphs. Other models of batch scheduling with incompatibility constraints were studied in [4,5]. In all the papers the authors assumed identical parallel machines. However, to the best of our knowledge little work has been done on scheduling problems with uniform machines involved (cf. [6,7]).The rest of this paper is organized as follows. In Section 2 we show that the general problem is NP-hard even if s 1 = s 2 = s 3 . In Section 3 we show that if s 1 ¸ 12s 2 , s 2 = s 3 = s 4 , then the problem can be solved to optimality in time O(n 1.5 ) provided that the degree of G is ∆ • 4. The same algorithm returns a solution of value at most 2 times optimal provided that s 1 ¸ 2s 2 , s 2 = s 3 = s 4 . In Section 4 we study the case s 1 ¸ s 2 ¸ s 3 = s 4 and give an O(n 1.5 )-time 32/15-approximation algorithm in all such cases. Finally,...

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Section: Scheduling Of Unit-length Jobs With Bipartite Incompatibilitmentioning

confidence: 99%

Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

Furmańczyk

Kubale

2017

Bulletin of the Polish Academy of Sciences Technical Sciences

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“…This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7,4,5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.…”

Section: Introductionmentioning

confidence: 94%

Capacitated max -Batching with Interval Graph Compatibilities

Nonner

2010

Lecture Notes in Computer Science

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We consider the problem of partitioning interval graphs into cliques of bounded size. Each interval has a weight, and the cost of a clique is the maximum weight of any interval in the clique. This natural graph problem can be interpreted as a batch scheduling problem. Solving an open question from [7,4,5], we show NP-hardness, even if the bound on the clique sizes is constant. Moreover, we give a PTAS based on a novel dynamic programming technique for this case.

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“…Originally, DAC-Sum has been interpreted as a batch scheduling problem [8,7,5,3]. Specifically, we can think of each interval I ∈ J as a job and of w C as the time needed to process clique C, also called batch.…”

Section: Max-batchingmentioning

confidence: 99%

“…The objective is then to group the jobs in batches subject to the interval structure such that the total completion time is minimized. DAC-Sum is therefore also called max-batching (with interval graph compabilities), and it can be naturally extended to arbitrary graphs instead of interval graphs [5,3].…”

Section: Max-batchingmentioning

confidence: 99%

Latency Constrained Aggregation in Chain Networks Admits a PTAS

Nonner

Souza

2009

Algorithmic Aspects in Information and Management

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This paper studies the aggregation of messages in networks that consist of a chain of nodes, and each message is time-constrained such that it needs to be aggregated during a given time interval, called its due interval. The objective is to minimize the maximum sending cost of any node, which is for example a concern in wireless sensor networks, where it is crucial to distribute the energy consumption as equally as possible. First, we settle the complexity of this problem by proving its NP-hardness, even for the case of unit length due intervals. Second, we give a QPTAS, which we extend to a PTAS for the special case that the lengths of the due intervals are constants. This is in particular interesting, since we prove that this problem becomes APX-hard if we consider tree networks instead of chain networks, even for the case of unit length due intervals. Specifically, we show that it cannot be approximated within 4/3 − ǫ for any ǫ > 0, unless P=NP.

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Scheduling on a Batch Processing Machine with Split Compatibility Graphs

Cited by 12 publications

References 13 publications

Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

Scheduling of unit-length jobs with bipartite incompatibility graphs on four uniform machines

Capacitated max -Batching with Interval Graph Compatibilities

Latency Constrained Aggregation in Chain Networks Admits a PTAS

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