The complexity of mean flow time scheduling problems with release times

In this paper, we consider the problem of scheduling distributed biological sequence comparison applications. This problem lies in the divisible load framework with negligible communication costs. Thus far, very few results have been proposed in this model. We discuss and select relevant metrics for this framework: namely max-stretch and sum-stretch. We explain the relationship between our model and the preemptive uni-processor case, and we show how to extend algorithms that have been proposed in the literature for the uni-processor model to the divisible multi-processor problem domain. We recall known results on closely related problems, we show how to minimize the max-stretch on unrelated machines either in the divisible load model or with preemption, we derive new lower bounds on the competitive ratio of any on-line algorithm, we present new competitiveness results for existing algorithms, and we develop several new on-line heuristics. We also address the Pareto optimization of max-stretch. Then, we extensively study the performance of these algorithms and heuristics in realistic scenarios. Our study shows that all previously proposed guaranteed heuristics for max-stretch for the uni-processor model prove to be inefficient in practice. In contrast, we show our on-line algorithms based on linear programming to be near-optimal solutions for maxstretch. Our study also clearly suggests heuristics that are efficient for both metrics, although a combined optimization is in theory not possible in the general case.Keywords: Bioinformatics, heterogeneous computing, scheduling, divisible load, linear programming, stretch RésuméDans ce rapport, nous nous intéressonsà l'ordonnancement d'applications comparant de manière distribuée des séquences biologiques. Ce problème se situe dans le domaine des tâches divisibles avec coûts de communications négli-geables. Jusqu'à présent, très peu de résultats ontété publiés pour ce modèle. Nous discutons et sélectionnons des métriques appropriées pour notre cadre de travail,à savoir le max-stretch et le sum-stretch. Nous expliquons les relations entre notre modèle et le cadre mono-processeur avec préemption, et nous montrons commentétendre au cadre des tâches divisibles sur multi-processeur les algorithmes proposés dans la littérature pour le cas mono-processeur avec pré-emption. Nous rappelons les résultats connus pour des problématiques proches, nous montrons comment minimiser le max-stretch sur des machines non corrélées (que les tâches soient divisibles ou simplement préemptibles), nous obtenons de nouvelles bornes inférieures de compétitivité pour tout algorithme on-line, nous présentons de nouveaux résultats de compétitivité pour des algorithms de la littérature, et nous proposons de nouvelles heuristiques on-line. Nous nous intéressonségalement au problème de la minimisation Pareto du max-stretch. Ensuite, nousétudions, de manière extensive, les performances de tous ces algorithmes et de toutes ces heuristiques, et ce dans un cadre réa-liste. Notreétude montre que les ...

Section: Offline Max-stretch Optimizationmentioning

confidence: 99%

Minimizing the stretch when scheduling flows of divisible requests

Legrand

Vivien

2008

“…Note that problem P | r j , pmtn | C j and, therefore, problem P | r j , pmtn | T j are unary NP-hard (Baptiste et al 2007). Recall that P in the notation of the problem means that all machines have identical speeds.…”

Section: Introductionmentioning

confidence: 99%

Minimizing total tardiness on parallel machines with preemptions

Kravchenko

Werner

2010

Self Cite

The basic scheduling problem we are dealing with in this paper is the following one. A set of jobs has to be scheduled on a set of parallel uniform machines. Each machine can handle at most one job at a time. Each job becomes available for processing at its release date. All jobs have the same execution requirement and arbitrary due dates. Each machine has a known speed. The processing of any job may be interrupted arbitrarily often and resumed later on any machine. The goal is to find a schedule that minimizes the sum of tardiness, i.e., we consider problem Q | r j , p j = p, pmtn | T j whose complexity status was open. Recently, Tian et al. (J. Sched. 9:343-364, 2006) proposed a polynomial algorithm for problem 1 | r j , p j = p, pmtn | T j . We show that both the problem P | pmtn | T j of minimizing total tardiness on a set of parallel machines with allowed preemptions and the problem P | r j , p j = p, pmtn | T j of minimizing total tardiness on a set of parallel machines with release dates, equal processing times and allowed preemptions are NP-hard. Moreover, we give a polynomial algorithm for the case of uniform machines without release dates, i.e., for problem Q | p j = p, pmtn | T j .

“…3.6 It is interesting to note that problem P | pmtn | U j is binary NP-hard [27], however, its complexity status under an unary encoding is still an open question. 3.7 Problem P | r j , p j = p, pmtn | C j has been solved in [1]. It is possible to prove that for problem P | r j , p j = p, pmtn | C j , an optimal schedule can be found in the class of schedules, where each job J j is processed on machine m only within an (possibly empty) interval [S j,m , C j,m [, such that C j,m ≤ S j+1,m for each m and j < n, and C j,m ≤ S j,m−1 for each m > 1 and j.…”

Section: In [28]mentioning

confidence: 99%

“…In [25], it has been shown that an optimal solution of the above linear program can be used to obtain an optimal schedule. 3.9 Problem P | r j , pmtn | C j has been proved to be unary NP-hard by a reduction from 3-partition in [1]. 3.10 Problem P | r j , p j = p, pmtn | w j C j is unary NP-hard [32].…”

Section: Minimizementioning

confidence: 99%

Parallel machine problems with equal processing times: a survey

Kravchenko

Werner

2011

Self Cite

The basic scheduling problem we are dealing with is the following. There are n jobs, each requiring an identical execution time. All jobs have to be processed on a set of parallel machines. Preemptions can be either allowed or forbidden. The aim is to construct a feasible schedule such that a given criterion is minimized. For a couple of problems of this type, recently the complexity status has been solved and several interesting results have been presented. In this paper, we survey existing approaches for the problem class considered.