An improved lower bound for a bi-criteria scheduling problem

For the bi-criteria scheduling problem of minimizing the sum of completion times and the sum of weighted completion times, min-sum of weighted completion times, we prove that there exists no constant β>1β>1 such that (1+1/γ,β)(1+1/γ,β)-approximate schedules can be found for any γ>0γ>0. This result confirms a recently published conjecture.