Online Unrelated Machine Load Balancing with Predictions Revisited

Assumption 1 can be made with a loss of (1+ )-factor in the competitive ratio. If pi′,j pi,j ≥ m for some j ∈ J, i, i′ ∈Mj , we can change pi′,j to∞. If the job j is assigned to i′ in the optimum solution, we assign it to the machine i∗ with the minimum pi∗,j instead. Thus, the processing time of j is decreased by at least a factor of m . We apply the operation for all violations of the assumption. Then the makespan of a machine i will be increased by at most (m−1)T m/ ≤ T . This holds since the total processing time of machines other than i in the optimum solution is at most (m − 1)T . We also remark the procedure that guarantees the assumption can run online, as jobs are handled separately in the procedure.