Hardness of Precedence Constrained Scheduling on Identical Machines

In 1966, Graham showed that a simple procedure called list scheduling yields a 2-approximation algorithm for the central problem of scheduling precedence constrained jobs on identical machines to minimize makespan. To date this has remained the best algorithm, and whether it can or cannot be improved has become a major open problem in scheduling theory. We address this problem by establishing a quite surprising relation between the approximability of the considered problem and that of scheduling precedence constrained jobs on a single machine to minimize weighted completion time. More specifically, we prove that if the single machine problem is hard to approximate within a factor of 2 − , then the considered parallel machine problem, even in the case of unit processing times, is hard to approximate within a factor of 2 − ζ, where ζ tends to 0 as tends to 0. Combining this with Bansal and Khot's recent hardness result for the single machine problem gives that it is NP-hard to improve upon the approximation ratio obtained by Graham, assuming a new variant of the unique games conjecture. 1. Introduction. One of the first approximation algorithms with a worst-case analysis can be traced back to 1966, when Graham [20] studied the following central scheduling problem (known as P |prec|C max in standard scheduling notation [19]): There is a set N of n jobs to be scheduled on m identical parallel machines. Each machine can process at most one job at a time, and each job j ∈ N requires p j uninterrupted units of processing on one of the machines. Jobs also have precedence constraints between them that are specified by a partial order P on N , where (i, j) ∈ P implies that job i must be completed before job j can be started. The goal is to find a schedule that minimizes the makespan C max = max j C j , where C j is the time at which job j completes in the given schedule. Arguably the simplest algorithm for many scheduling problems is the so-called list scheduling procedure, where we order the jobs in a preference list and schedule the first available job(s) from the list whenever a machine falls idle. In his seminal paper [20], Graham showed that for P |prec|C max , the list scheduling procedure has a worst-case performance guarantee of 2 − 1/m. Considering the special case with …