Improved Approximations for Unrelated Machine Scheduling

We revisit two well-studied scheduling problems in the unrelated machines setting where each job can have a different processing time on each machine. For minimizing total weighted completion time we give a 1.45-approximation, which improves upon the previous 1.488-approximation [Im and Shadloo SODA 2020]. The key technical ingredient in this improvement lies in a new rounding scheme that gives strong negative correlation with less restrictions. For minimizing $L_k$-norms of machine loads, inspired by [Kalaitzis et al. SODA 2017], we give better approximation algorithms. In particular we give a $\sqrt {4/3}$-approximation for the $L_2$-norm which improves upon the former $\sqrt 2$-approximations due to [Azar-Epstein STOC 2005] and [Kumar et al. JACM 2009].