Lawler’s minmax cost algorithm: optimality conditions and uncertainty

The well-known O(n^2) minmax cost algorithm of Lawler (MANAGE SCI 19(5):544–546, 1973 ) was developed to minimize the maximum cost of jobs processed by a single machine under precedence constraints. We propose two results related to Lawler’s algorithm. Lawler’s algorithm delivers one specific optimal schedule while there can exist other optimal schedules. We present necessary and sufficient conditions for the optimality of a schedule for the problem with strictly increasing cost functions. The second result concerns the same scheduling problem under uncertainty. The cost function for each job is of a special decomposable form and depends on the job completion time and on an additional numerical parameter, for which only an interval of possible values is known. For this problem we derive an O(n^2) algorithm for constructing a schedule that minimizes the maximum regret criterion . To obtain this schedule, we use Lawler’s algorithm as a part of our technique.