An Efficient Approximation Algorithm for Minimizing Makespan on Uniformly Related Machines

Chandra Chekuri, Michael A. Bender

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 = maxj 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.