Scheduling Fully Parallel Jobs with Integer Parallel Units

Vincent Chau, Minming Li, Kai Wang

Abstract: We consider the following scheduling problem. We have m identical machines, where each machine can accomplish one unit of work at each time unit. We have a set of n jobs, where each job j has $$s_j$$ units of workload, and each unit workload could be executed on any machine at any time unit. A job is said completed when its whole workload has been executed. The objective is to find a schedule that minimizes the total weighted completion time $$\sum w_j C_j$$ , where $$w_j$$ is the weight of job j and $$C_j$$ is the completion time of job j. We first give a PTAS of this problem when m is constant. Then we study the approximation ratio of a greedy algorithm, Largest-Ratio-First algorithm. Any permutation is a possible outcome of this algorithm when $$w_j = s_j$$ for each job j, and for this special case we show that the approximation ratio depends on the instance size, i.e. n and m. Finally, when jobs have arbitrary weights, we prove that the upper bound of the approximation ratio is $$1 + \frac{m-1}{m+2}$$ .