An Improved Approximation for Scheduling Malleable Tasks with Precedence Constraints via Iterative Method

The problem of scheduling malleable tasks with precedence constraints is one of the most important strongly <inline-formula><tex-math notation="LaTeX">$\mathcal {NP}$</tex-math><alternatives> <inline-graphic xlink:href="chen-ieq1-2813387.gif"/></alternatives></inline-formula>-hard problems, given <inline-formula><tex-math notation="LaTeX">$m$</tex-math><alternatives> <inline-graphic xlink:href="chen-ieq2-2813387.gif"/></alternatives></inline-formula> identical processors and <inline-formula><tex-math notation="LaTeX">$n$</tex-math><alternatives> <inline-graphic xlink:href="chen-ieq3-2813387.gif"/></alternatives></inline-formula> tasks. A malleable task is one that runs in parallel on a varying number of processors. In addition, the processing sequences of tasks are constrained by the precedence constraints. The goal is to find a feasible schedule that minimizes the makespan (maximum completion time). This article presents an iterative method for improving the performance ratio of scheduling malleable tasks. The proposed algorithm achieves an approximation ratio of 4.4841 after 2 iterations. This improves the so far best-known factor of 4.7306 due to Jansen and Zhang. For a large number of iterations (<inline-formula> <tex-math notation="LaTeX">$>100$</tex-math><alternatives><inline-graphic xlink:href="chen-ieq4-2813387.gif"/> </alternatives></inline-formula>), the approximation ratio of the proposed algorithm is tends toward <inline-formula> <tex-math notation="LaTeX">$2+\sqrt{2}\approx 3.4143$</tex-math><alternatives> <inline-graphic xlink:href="chen-ieq5-2813387.gif"/></alternatives></inline-formula>.

[1]  Denis Trystram,et al.  Efficient approximation algorithms for scheduling malleable tasks , 1999, SPAA '99.

[2]  Klaus Jansen,et al.  Approximation Algorithms for Scheduling Parallel Jobs , 2010, SIAM J. Comput..

[3]  Jingren Zhou,et al.  SCOPE: easy and efficient parallel processing of massive data sets , 2008, Proc. VLDB Endow..

[4]  Jan Karel Lenstra,et al.  Complexity of Scheduling under Precedence Constraints , 1978, Oper. Res..

[5]  Joseph Naor,et al.  Deadline-aware scheduling of big-data processing jobs , 2014, SPAA.

[6]  Denis Trystram,et al.  Dynamic Load Balancing for Ocean Circulation Model with Adaptive Meshing , 1999, Euro-Par.

[7]  Joseph Y.-T. Leung,et al.  Complexity of Scheduling Parallel Task Systems , 1989, SIAM J. Discret. Math..

[8]  Eddy Caron,et al.  Multi-criteria malleable task management for hybrid-cloud platforms , 2016, 2016 2nd International Conference on Cloud Computing Technologies and Applications (CloudTech).

[9]  Philip S. Yu,et al.  Approximate algorithms scheduling parallelizable tasks , 1992, SPAA '92.

[10]  Mikhail Y. Kovalyov,et al.  Scheduling arbitrary number of malleable tasks on multiprocessor systems , 2014 .

[11]  Chih-Ping Chu,et al.  A 3.42-Approximation Algorithm for Scheduling Malleable Tasks under Precedence Constraints , 2013, IEEE Transactions on Parallel and Distributed Systems.

[12]  David A. Padua,et al.  Communication contention in APN list scheduling algorithm , 2009, Science in China Series F: Information Sciences.

[13]  RENAUD LEPÈRE,et al.  Approximation Algorithms for Scheduling Malleable Tasks Under Precedence Constraints , 2001, Int. J. Found. Comput. Sci..

[14]  Klaus Jansen,et al.  Scheduling malleable tasks with precedence constraints , 2012, J. Comput. Syst. Sci..

[15]  Martin Skutella,et al.  Approximation Algorithms for the Discrete Time-Cost Tradeoff Problem , 1997, Math. Oper. Res..

[16]  Denis Trystram,et al.  A 3/2-Approximation Algorithm for Scheduling Independent Monotonic Malleable Tasks , 2007, SIAM J. Comput..

[17]  Salim Hariri,et al.  Performance-Effective and Low-Complexity Task Scheduling for Heterogeneous Computing , 2002, IEEE Trans. Parallel Distributed Syst..

[18]  Klaus Jansen,et al.  A(3/2+ε) approximation algorithm for scheduling moldable and non-moldable parallel tasks , 2012, SPAA '12.

[19]  Ronald L. Graham,et al.  Bounds for certain multiprocessing anomalies , 1966 .

[20]  Klaus Jansen,et al.  Scheduling Malleable Parallel Tasks: An Asymptotic Fully Polynomial-Time Approximation Scheme , 2002, ESA.

[21]  Sanjay Ghemawat,et al.  MapReduce: Simplified Data Processing on Large Clusters , 2004, OSDI.

[22]  Klaus Jansen,et al.  Linear-Time Approximation Schemes for Scheduling Malleable Parallel Tasks , 1999, SODA '99.

[23]  Y.-K. Kwok,et al.  Static scheduling algorithms for allocating directed task graphs to multiprocessors , 1999, CSUR.

[24]  Prasoon Tiwari,et al.  Scheduling malleable and nonmalleable parallel tasks , 1994, SODA '94.

[25]  Ronald L. Graham,et al.  Bounds for Multiprocessor Scheduling with Resource Constraints , 1975, SIAM J. Comput..

[26]  Denis Trystram,et al.  An approximation algorithm for scheduling trees of malleable tasks , 2002, Eur. J. Oper. Res..

[27]  Thomas Decker,et al.  A 5/4-approximation algorithm for scheduling identical malleable tasks , 2003, Theor. Comput. Sci..

[28]  Ishfaq Ahmad,et al.  Dynamic Critical-Path Scheduling: An Effective Technique for Allocating Task Graphs to Multiprocessors , 1996, IEEE Trans. Parallel Distributed Syst..

[29]  Klaus Jansen,et al.  An Approximation Algorithm for Scheduling Malleable Tasks Under General Precedence Constraints , 2005, ISAAC.