Weighted Completion Time Minimization for Unrelated Machines via Iterative Fair Contention Resolution

Abstract

We give a 1.488-approximation for the classic scheduling problem of minimizing total weighted completion time on unrelated machines. This is a considerable improvement on the recent breakthrough of (1.5−10 −7 )-approximation (STOC 2016, Bansal-Srinivasan-Svensson) and the follow-up result of (1.5 − 1/6000)-approximation (FOCS 2017, Li). Bansal et al. introduced a novel rounding scheme yielding strong negative correlations for the first time and applied it to the scheduling problem to obtain their breakthrough, which resolved the open problem if one can beat out the long-standing 1.5-approximation barrier based on independent rounding. Our key technical contribution is in achieving significantly stronger negative correlations via iterative fair contention resolution, which is of independent interest. Previously, Bansal et al. obtained strong negative correlations via a variant of pipage type rounding and Li used it as a black box.

Nearly-Linear Time Approximate Scheduling Algorithms

“…In the correspondent integer program, x j,i ∈ {0, 1} for every (j, i) ∈ E indicates whether the job j is assigned to machine i. (20) requires that the makespan of the schedule to be at most P , (21) requires every job to be scheduled. In the linear program, we replace the requirement that x j,i ∈ {0, 1} with the non-negativity constraint (22).…”

“…(Assume the maximum over ∅ is 0.) We can then apply Theorem 4.2 with the solution x we obtained from solving LP (20)(21)(22). Clearly we have max j∈σ −1 (i) p i,j ≤ P for every i ∈ M .…”

“…Clearly we have max j∈σ −1 (i) p i,j ≤ P for every i ∈ M . So, the total load on any machine i is at most (20) is satisfied with right side replaced by (1 + O( ))P . This finishes the analysis of the algorithm for R||C max .…”

“…To overcome the barrier, they developed a novel dependent rounding scheme and a lifted SDP relaxation for the problem, leading to a (1.5 − 1/2160000)-approximation algorithm. The ratio has been improved to 1.5 − 1/6000 by Li [31] and then to the current best ratio of 1.488 by Im and Shadloo [20]. Both [31] and [20] are based on some time-indexed LP relaxation for the problem.…”

“…The ratio has been improved to 1.5 − 1/6000 by Li [31] and then to the current best ratio of 1.488 by Im and Shadloo [20]. Both [31] and [20] are based on some time-indexed LP relaxation for the problem.…”