The Power of Proportional Fairness for Non-Clairvoyant Scheduling under Polyhedral Constraints

The Polytope Scheduling Problem (PSP) was introduced by Im, Kulkarni, and Munagala (JACM 2018) as a very general abstraction of resource allocation over time and captures many well-studied problems including classical unrelated machine scheduling, multidimensional scheduling, and broadcast scheduling. In PSP, jobs with different arrival times receive processing rates that are subject to arbitrary packing constraints. An elegant and well-known algorithm for instantaneous rate allocation with good fairness and efficiency properties is the Proportional Fairness algorithm (PF), which was analyzed for PSP by Im et al. We drastically improve the analysis of the PF algorithm for both the general PSP and several of its important special cases subject to the objective of minimizing the sum of weighted completion times. We reduce the upper bound on the competitive ratio from 128 to 27 for general PSP and to 4 for the prominent class of monotone PSP. For certain heterogeneous machine environments we even close the substantial gap to the lower bound of 2 for non-clairvoyant scheduling. Our analysis also gives the first polynomial-time improvements over the nearly 30-year-old bounds on the competitive ratio of the doubling framework by Hall, Shmoys, and Wein (SODA 1996) for clairvoyant online preemptive scheduling on unrelated machines. Somewhat surprisingly, we achieve this improvement by a non-clairvoyant algorithm, thereby demonstrating that non-clairvoyance is not a (significant) hurdle. Our improvements are based on exploiting monotonicity properties of PSP, providing tight dual fitting arguments on structured instances, and showing new additivity properties on the optimal objective value for scheduling on unrelated machines. Finally, we establish new connections of PF to matching markets, and thereby provide new insights on equilibria and their computational complexity.