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 p_j/s_i units of time for machine i with speed s_i to process job j with processing requirement p_j. 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 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.