计算机科学 ›› 2021, Vol. 48 ›› Issue (4): 37-42.doi: 10.11896/jsjkx.200600064
高吉吉, 岳雪蓉, 陈智斌
GAO Ji-ji, YUE Xue-rong, CHEN Zhi-bin
摘要: 给定m台平行机(同型机),n个工件,寻找一种分配方案,使得把这n个工件分配到m台机器后,整体完工时间尽可能短,这个NP-难问题被称为经典排序问题。如果每个工件的加工时间满足一定的条件,则有望能在多项式时间内有效地得到最优的分配方案。Yue等对加工时间满足整除性质的经典排序问题考虑了一种新的算法,该算法总是能得到这种特殊情况的最优分配。该算法在多项式时间内能够得到最优分配,是对于一般的经典排序问题的近似算法。文章在此基础上,考虑该新算法在一般问题上的近似比。文中考虑了这个新算法的两种版本,分别得到了3/2和2-1/2q(q∈Z+)的近似比。紧例子表明,文中对算法的两个版本的分析都是最优的。
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[1]GRAHAM R L,LAWLER E L,LENSTRA J K,et al.Optimization and Approximation in Determini-stic Sequencing and Sche-duling:A Survey[J].Annals of Discrete Mathematics,1979,5(1):287-326. [2]GRAHAM R L.Bounds for Certain Multiprocessing Anomalies[J].Bell System Technical Journal,1966,45(9):1563-1581. [3]GRAHAM R.L.Bounds on multiprocessing time anomalies[J].SIAM Journal on Applied Mathematics,1969,17(2):416-429. [4]DEUERMEYER B L,FRIESEN D K,LANGSTON M A.Scheduling to Maximize the Minimum Processor Finish Time in a Multi-processor System[J].Siam Journal on Algebraic Discrete Methods,1982,3(2):190-196. [5]KAO T Y,ELSAYED E A.Performance of the LPT algorithm in multiprocessor scheduleing[J].Computers & Operations Research,1990,17(4):365-373. [6]CSIRIK J,KELLERER H,WOEGINGER G.The exact LPT-bound for maximizing the minimum completion time[J].Operations Research Letters,1992,11(5):281-287. [7]WOEGINGER G J.A polynomial-time approximation schemefor maximizing the minimum machine completion time[J].Operations Research Letters,1997,20(4):149-154. [8]COFFMAN E G,GAREY M R,JOHNSON D S.An Application of Bin-Packing to Multiprocessor Scheduling[J].Siam Journal on Computing,19787(1):1-17. [9]LANGSTON M A.Processor scheduling with improved heuristic algorithms[M].Texas A&M University,1981. [10]FRIESEN D K.Tighter Bounds for the Multifit ProcessorScheduling Algorithm[J].Siam Journal on Computing,1984,13(1):170-181. [11]YUE M.On the exact upper bound for the multifit processor scheduling algorithm[J].Annals of Operations Research,1990,24(1):233-259. [12]SAHN I,SARTAJ K.Algorithms for Scheduling Independent Tasks[J].Journal of the ACM,1976,23(1):116-127. [13] HOCHBAUM D S,SHMOYS D B.Using dual approximation algorithms for scheduling problems theoretical and practical results[J].Journal of the ACM(JACM),1987,34(1):144-162. [14]VEL H V D,SHIJIE S.A modification of Hochbaum andShmoys’ algorithm for scheduling problems[J].BIT Numerical Mathematics,1991,31(1):50-52. [15]JANSEN K.An EPTAS for scheduling jobs on uniform processors:using an MILP relaxation with a constant number of integral variables[J].SIAM Journal on Discrete Mathematics,2010,24:457-485. [16]KLEIN K M,JANSEN K,VERSCHAE J.Closing the gap for makespan scheduling via sparsification techniques[C]//Procee-dings of the 43rd International Colloquium on Automata,Languages,and Programming.2016,55(72):1-13. [17]CHEN L,JANSEN K,ZHANG G.On the optimality of exact and approximation algorithms for scheduling problems[J].Journal of Computer and System Sciences,2018,96(2):89-99. [18]IMPAGLIAZZO R,PATURI R,ZANE F.Which problems have strongly exponential complexity?[J].Journal of Computer and System Science,2001,63:512-530. [19]YUE X R,GAO J J,CHEN Z B.A polynomial time algorithm for scheduling on processing time constraints[C]//ACM International Conference Proceeding Series,2019 the 9th Internatio-nal Conference on Communication and Network Security.2019:109-113. [20]近似算法[M].郭效江,方志奇,农庆琴,译.北京:北京高等教育出版社,2010:74-75. |
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