An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines

Abstract

We give a new efficient approximation algorithm for schedul- ing precedence constrained jobs on machines with different speeds. The setting is as follows. There are n jobs 1, . . ., n where job j requires p _j units of processing. The jobs are to be scheduled on a set of m machines. Machine i has a speed s _i; it takes p _j/s _i units of time for machine i to pro- cess job j. The precedence constraints on the jobs are given in the form of a partial order. If j ≺ k, processing of k cannot start until j’s execution if finished. Let C _j denote the completion time of job j. The objective is to find a schedule to minimize C _max = max_j C _j, conventionally called the makespan of the schedule. We consider non-preemptive schedules where each job is processed on a single machine with no preemptions. Recently Chudak and Shmoys [1] gave an algorithm with an approximation ra- tio of O(log m) significantly improving the earlier ratio of O(√m) due to Jaffe [7]. Their algorithm is based on solving a linear programming relaxation of the problem. Building on some of their ideas, we present a combinatorial algorithm that achieves a similar approximation ratio but runs in O(n ³) time. In the process we also obtain a constant factor approximation algorithm for the special case of precedence constraints induced by a collection of chains. Our algorithm is based on a new lower bound which we believe is of independent interest. By a general result of Shmoys, Wein, and Williamson [10] our algorithm can be extended to obtain an O(logm) approximation ratio even if jobs have release dates.

Supported primarily by an IBM Cooperative Fellowship. Remaining support was provided by an ARO MURI Grant DAAH04-96-1-0007 and NSF Award CCR-9357849, with matching funds from IBM, Schlumberger Foundation, Shell Foundation, and Xerox Corporation.