Scheduling identical jobs on uniform machines with a conflict graph

Mallek, Amin; Bendraouche, Mohamed; Boudhar, Mourad

doi:10.1016/j.cor.2019.07.011

Cited by 14 publications

(21 citation statements)

References 14 publications

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“…In the case of uniform machines Mallek, Bendraouche, and Boudhar (2019) proved that Q|G = complete 2-partite, p j = 1|C max is NP-hard, but it may be solved in O(n) time when the number of machines is fixed. Moreover, they showed an O(mn + m 2 log m) algorithm for the particular case Q|G = star, p j = 1|C max .…”

Section: An Overview Of the Previous Workmentioning

confidence: 99%

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Pikies

Turowski

Kubale

2021

ICAPS

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In this paper we consider a problem of job scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and ∑Cj (total completion time). We consider a variety of machine types in our paper: identical, uniform, and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for the easier problems, or algorithms with a guaranteed constant worst-case approximation ratio. In particular, we fill the gap on research for the problem of finding a schedule with the smallest ∑Cj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting.

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Section: An Overview Of the Previous Workmentioning

confidence: 99%

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Pikies

Turowski

Kubale

2021

ICAPS

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“…In [23] there is 2-approximate algorithm for the problem Qm|p j = 1, G = bisubquartic|C max presented. Some further results for the problem with bipartite incompatibility graph are provided in [20]. In particular, the authors proved that the problem Q|G = complete bipartite, p j = 1|C max is NP-hard under binary encoding of graph, i.e.…”

Section: Related Workmentioning

confidence: 99%

Scheduling on uniform and unrelated machines with bipartite incompatibility graphs

Pikies¹,

Furmańczyk²

2021

Preprint

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In this paper the problem of scheduling of jobs on parallel machines under incompatibility relation is considered. In this model a binary relation between jobs is given and no two jobs that are in the relation can be scheduled on the same machine. In particular, we consider job scheduling under incompatibility relation forming bipartite graphs, under makespan optimality criterion, on uniform and unrelated machines.We show that no algorithm can achieve a good approximation ratio for uniform machines, even for a case of unit time jobs, under P ̸ = NP. We also provide an approximation algorithm that achieves the best possible approximation ratio, even for the case of jobs of arbitrary lengths pj, under the same assumption. Precisely, we present an O(n 1/2−ϵ ) inapproximability bound, for any ϵ > 0; and pj-approximation algorithm, respectively. To enrich the analysis, bipartite graphs generated randomly according to Gilbert's model G n,n,p(n) are considered. For a broad class of p(n) functions we show that there exists an algorithm producing a schedule with makespan almost surely at most twice the optimum. Due to our knowledge, this is the first study of randomly generated graphs in the context of scheduling in the considered model.For unrelated machines, an FPTAS for R2|G = bipartite|Cmax is provided. We also show that there is no algorithm of approximation ratio O(n b p 1−ϵ max ), even for Rm|G = bipartite|Cmax for m ≥ 3 and any ϵ > 0, b > 0, unless P = NP. ACM Subject ClassificationTheory of computation → Problems, reductions and completeness; Mathematics of computing → Probability and statistics; Applied computing → Operations research

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“…Moreover, R m ||C max admits a FPTAS [9]. There is (2 − 1 m )-approximation algorithm for R||C max [10]; however there is no polynomial algorithm with approximation ratio better than 3 2 , unless P = NP [11]. On the other hand, Q|p j = 1|C max and Q|| C j (with P |p j = 1|C max and P || C j as their special cases) can be solved in O(min{n + m log m, n log m}) [12] and O(n log n) [4, p. 133-134] time, respectively.…”

Section: An Overview Of Previous Workmentioning

confidence: 99%

“…In the case of uniform machines Mallek et al [3] proved that Q|G = complete 2-partite, p j = 1|C max is NP-hard, but it may be solved in O(n) time when the number of machines is fixed. Moreover, they showed an O(mn+m 2 log m) algorithm for the particular case Q|G = star, p j = 1|C max .…”

Section: An Overview Of Previous Workmentioning

confidence: 99%

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Pikies¹,

Turowski²,

Kubale³

2020

Preprint

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In this paper we consider the problem of scheduling on parallel machines with a presence of incompatibilities between jobs. The incompatibility relation can be modeled as a complete multipartite graph in which each edge denotes a pair of jobs that cannot be scheduled on the same machine. Our research stems from the work of Bodlaender et al. [1], [2]. In particular, we pursue the line investigated partially by Mallek et al. [3], where the graph is complete multipartite so each machine can do jobs only from one partition.We provide several results concerning schedules, optimal or approximate with respect to the two most popular criteria of optimality: Cmax (makespan) and Cj (total completion time). We consider a variety of machine types in our paper: identical, uniform and unrelated. Our results consist of delimitation of the easy (polynomial) and NP-hard problems within these constraints. We also provide algorithms, either polynomial exact algorithms for easy problems, or algorithms with a guaranteed constant worst-case approximation ratio or even in some cases a PTAS for the harder ones.In particular, we fill the gap on research for the problem of finding a schedule with the smallest Cj on uniform machines. We address this problem by developing a linear programming relaxation technique with an appropriate rounding, which to our knowledge is a novelty for this criterion in the considered setting.

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Scheduling identical jobs on uniform machines with a conflict graph

Cited by 14 publications

References 14 publications

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

Scheduling on uniform and unrelated machines with bipartite incompatibility graphs

Scheduling with Complete Multipartite Incompatibility Graph on Parallel Machines

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