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

TL;DR

This paper improves the approximation ratio for the classic unrelated machine scheduling problem with weighted completion time, achieving better than 2-approximation using a novel LP rounding with random offsets from non-uniform distributions.

Contribution

It introduces a simple LP-based rounding algorithm with non-uniform random offsets that surpasses the longstanding 2-approximation barrier for both preemptive and non-preemptive cases.

Findings

Achieved a 1.8786-approximation for the non-preemptive problem.

First better-than-2 approximation for the preemptive version.

Demonstrated the effectiveness of non-uniform random offsets in LP rounding.

Abstract

In this paper we consider the classic scheduling problem of minimizing total weighted 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 this problem, a 2-approximation is known based on a novel convex programming (J. ACM 2001 by Skutella). It has been a long standing open problem if one can improve upon this 2-approximation (Open Problem 8 in J. of Sched. 1999 by Schuurman and Woeginger). We answer this question in the affirmative by giving a 1.8786-approximation. We achieve this via a surprisingly simple linear programming, but a novel rounding algorithm and analysis. A key ingredient of our algorithm is the use of random offsets sampled from non-uniform distributions. 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-uniform distributions to give the first better than 2-approximation for this problem. This improvement also requires use of a configuration LP with variables for each job's complete schedules along with more careful analysis. For both non-preemptive and preemptive versions, we break the approximation barrier of 2 for the first time.