Efficient Approximation Algorithms for Parallel Batch Machine Scheduling of Malleable Jobs

Cloud computing environments impose challenges in incorporating the maximum degree of parallelism constraints into Parallel Batch Machine Scheduling (PBMS). Unlike traditional PBMS that considers only fixed job widths, this paper studies generalized PBMS with malleable jobs that allow job width to be changed during the job execution, provided it does not exceed its maximum degree of parallelism. We propose a fast \(O(n \log n)\) approximation algorithm by extending the state-of-the-art PBMS algorithm by setting each job’s width to its maximum degree of parallelism, where \(n\) is the number of jobs. Due to the unique nature of malleable jobs, previous ratio proofs are inapplicable. We develop new proof techniques and establish that our algorithm achieves a ratio of \((4-\frac{2}{Bm})\) , where \(B\) and \(m\) denote machine capacities and numbers, respectively. Furthermore, by exploiting the relationship between maximum job demand and average processing capacity per processor, we refine the algorithm and achieve an improved ratio of \((4-\frac{4}{Bm})\) while preserving \(O(n \log n)\) runtime complexity. In addition, we fine-tune our algorithm to achieve a ratio of \((3-\frac{2}{Bm})\) when jobs are with identical release times.