On‐line approximation algorithms for scheduling tasks on identical machines withextendable working time

Abstract

We study the on‐line problem of assigning tasks to identical machines whose regularworking time ‐ assumed to be unitary ‐ can be extended. If the tasks assigned to a machinedo not exceed the regular working time, then the working time of the machine is consideredto be 1; otherwise, it is the completion time of the last task assigned to the machine. Eachincoming task has to be assigned immediately to a machine and the assignment cannot bechanged later. The objective is to minimize the sum of the working times of the machines.Since the regular working time of the machines can be seen as a given capacity, the problemcan also be described through the bin packing terminology: the machines are viewed as binsand the tasks as items. A lower bound of 7/6 on the worst-case relative error of any on‐linealgorithm is shown. Then it is shown that a list scheduling heuristic which assigns theincoming task to the machine with smallest current load has worst‐case error equal to 5/4.The bound is improved to 1.228 by a new algorithm which tends to load the partially loadedmachines, as long as this does not cause an increase of the working time by more than afixed and appropriately chosen quantity x > 0.