Tighter bounds on preemptive job shop scheduling with two machines

Reviewer: Zhihui Xue

The case of a two-machine preemptive job shop is studied in this paper, with the objective of minimizing the completion time of the last job (the makespan). By following the standard notation for scheduling problems, this can be denoted as 2|pmtn|C max . For this problem, Sevastianov and Woeginger [1] have provided a 1.5 approximation algorithm that runs in linear time. In this paper, the authors present a new linear-time approximation algorithm that finds a schedule with length at most P max/2 greater than the optimal one, where P max is the length of the longest job. The algorithm improves the previous performance bound in the case of many short jobs, while the worst-case overall performance bound remains the same. This improvement is achieved by matching more work for parallel processing than in Sevastianov and Woeginger [1]. Specifically, the algorithm matches work designated for both machines from each job (except two), and interleaves the work from different jobs in a deliberate way. To implement this idea initially, a directed precedence graph is used to represent the ordering requirement on the operations within a job. Then, the matching algorithm adds undirected matching edges to the precedence graph. Whenever a matching edge is identified, the operation is split into two operations with a precedence edge between them. Next, the matching edges are contracted to produce compound operations that require both machines simultaneously, resulting in an acyclic directed graph. Finally, a topological sort on the resulting directed graph will produce the desired schedule. Online Computing Reviews Service