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In this paper we consider the classic scheduling problem of minimizing totalweighted completion time on unrelated machines when jobs have release times,i.e, $R | r_{ij} | \sum_j w_j C_j$ using the three-field notation. For thisproblem, a 2-approximation is known based on a novel convex programming (J. ACM2001 by Skutella). It has been a long standing open problem if one can improveupon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman andWoeginger). We answer this question in the affirmative by giving a1.8786-approximation. We achieve this via a surprisingly simple linearprogramming, but a novel rounding algorithm and analysis. A key ingredient ofour algorithm is the use of random offsets sampled from non-uniformdistributions.  We also consider the preemptive version of the problem, i.e, $R | r_{ij},pmtn| \sum_j w_j C_j$. We again use the idea of sampling offsets from non-uniformdistributions to give the first better than 2-approximation for this problem.This improvement also requires use of a configuration LP with variables foreach job's complete schedules along with more careful analysis. For bothnon-preemptive and preemptive versions, we break the approximation barrier of 2for the first time.

Better Unrelated Machine Scheduling for Weighted Completion Time via Random Offsets from Non-Uniform Distributions