An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines

We give a new and efficient approximation algorithm for scheduling precedence-constrained jobs on machines with different speeds. The problem is as follows. We are given n jobs to be scheduled on a set of m machines. Jobs have processing times and machines have speeds. It takes pj/si units of time for machine i with speed si to process job j with processing requirement pj. Precedence constraints between jobs are given in the form of a partial order. If j?k, processing of job k cannot start until job j's execution is completed. The objective is to find a non-preemptive schedule to minimize the makespan of the schedule. Chudak and Shmoys (1997, “Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA),” pp. 581?590) gave an algorithm with an approximation ratio of O(log m), significantly improving the earlier ratio of O(m) due to Jaffe (1980, Theoret. Comput. Sci.26, 1?17). Their algorithm is based on solving a linear programming relaxation. Building on some of their ideas, we present a combinatorial algorithm that achieves a similar approximation ratio but runs in O(n3) time. Our algorithm is based on a new and simple lower bound which we believe is of independent interest.

[1]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .

[2]  Jeffrey M. Jaffe,et al.  Efficient Scheduling of Tasks without Full Use of Processor Resources , 1980, Theor. Comput. Sci..

[3]  E.L. Lawler,et al.  Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey , 1977 .

[4]  Eugene L. Lawler,et al.  Chapter 9 Sequencing and scheduling: Algorithms and complexity , 1993, Logistics of Production and Inventory.

[5]  Fabián A. Chudak,et al.  Approximation algorithms for precedence-constrained scheduling problems on parallel machines that run at different speeds , 1997, SODA '97.

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

[7]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[8]  P. H. Lindsay.  Human Information Processing , 1977 .

[9]  Efim B. Kinber.  The Inclusion Problem for some Classes of Deterministic Multitape Automata , 1983, Theor. Comput. Sci..

[10]  Jane W.-S. Liu,et al.  Bounds on Scheduling Algorithms for Heterogeneous Comnputing Systems , 1974, IFIP Congress.

[11]  Graham K. Rand,et al.  Logistics of Production and Inventory , 1995 .

[12]  Shui Lam,et al.  A Level Algorithm for Preemptive Scheduling , 1977, J. ACM.

[13]  Don Coppersmith,et al.  Matrix multiplication via arithmetic progressions , 1987, STOC.

[14]  David P. Williamson,et al.  Scheduling Parallel Machines On-Line , 1995, SIAM J. Comput..

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